STAT,
"VRT
A COURSE IN
BY
EDOUARD GOURSAT PROFESSOR OF MATHEMATICS
IN
THE UNIVERSITY OF PARIS
TRANSLATED BY
EARLE RAYMOND HEDRICK PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MISSOURI
VOL.
I
DERIVATIVES AND DIFFERENTIALS EXPANSION IN SERIES
DEFINITE INTEGRALS
APPLICATIONS TO GEOMETRY
GINN AND COMPANY BOSTON
ATLANTA
NEW YORK DALLAS
CHICAGO LONDON SAN FKANCISCO
COLUMBUS
STAT.
LIBRARY
ENTERED AT STATIONERS HALL 1
COPYRIGHT,
1904,
BY
EARLE RAYMOND HEDRICK ALL RIGHTS RESERVED PRINTED IN THE UNITED STATES OF AMERICA 426.6
jgregg
GINN AND COMPANY PRIETORS BOSTON
PRO U.S.A.
AUTHOR S PREFACE This book contains, with slight variations, the material given in
my
course at the University of Paris.
I
have modified somewhat
the order followed in the lectures for the sake of uniting in a single
volume
that has to do with functions of real variables, except
all
the theory of differential equations.
being treated in the
"
The
differential notation not
Classe de Mathematiques
treated this notation from the beginning,
speciales,"
* I
have
and have presupposed only
a knowledge of the formal rules for calculating derivatives. Since mathematical analysis
tinuum,
it
logically,
is
essentially the science of the con
would seem that every course
in analysis should begin,
with the study of irrational numbers.
however, that the student
is
theory of incommensurable
well-known works f that a discussion.
As
basis of analysis,
double integral,
I
I
have supposed,
already familiar with that subject.
numbers
is
have thought
for the other
treated in so it useless to
many
The
excellent
enter upon such
fundamental notions which
lie at
the
such as the upper limit, the definite integral, the etc.,
I
have endeavored to treat them with
all
desirable rigor, seeking to retain the elementary character of the
work, and to avoid generalizations which would be superfluous in a book intended for purposes of instruction. Certain paragraphs which are printed in smaller type than the body of the book contain either problems solved in detail or else
*An
interesting account of French
methods of instruction in mathematics
will
be found in an article by Pierpont, Bulletin Amer. Math. Society, Vol. VI, 2d series
TRANS. Such books are not common in English. The reader is referred to Pierpont, Theory of Functions of Real Variables, Ginn & Company, Boston, 1905; Tannery, Lemons d arithiiietique, 1900, and other foreign works on arithmetic and on real (1900), p. 225. t
functions. iii
7814G2
AUTHOR S PREFACE
iv
supplementary matter which the reader ing without inconvenience.
may omit
Each chapter
is
at the first read
followed by a
list
of
examples which are directly illustrative of the methods treated in the chapter. Most of these examples have been set in examina tions. Certain others, which are designated by an asterisk, are
somewhat more
difficult.
The
latter are taken, for the
most
part,
from original memoirs to which references are made.
Two
of
my
and M. Jean
old students at the Ecole Normale,
I take this occasion to tender
JANUARY
M. Emile Cotton
Clairin, have kindly assisted in the correction of proofs
them
my
hearty thanks. E.
27, 1902
GOURSAT
;
TRANSLATOR The
PREFACE
S
was undertaken at the suggestion
translation of this Course
whose review of the original appeared of Professor W. in the July number of the Bulletin of the American Mathematical The lack of standard texts on mathematical sub Society in 1903. F. Osgood,
too well
known
jects in the
English language
I earnestly
hope that this book will help to
felt
is
fill
to require insistence.
the need so generally
throughout the American mathematical world.
conveniently in our in calculus,
It
may
be used
system of instruction as a text for a second course
and as a book of reference
it
will be
found valuable
an American student throughout his work. Few alterations have been made from the French
text.
to
Slight
changes of notation have been introduced occasionally for conven ience,
and several changes and .additions have been made at the sug
gestion of Professor Goursat,
work
in the
of translation.
who
has very kindly interested himself
To him
is
due
all
the additional matter
not to be found in the French text, except the footnotes which are signed,
and even
edited by him.
these,
though not of his
initiative,
I take this opportunity to express
were always
my
gratitude to
the author for the permission to translate the work and for the
sympathetic attitude which he has consistently assumed. I am also indebted to Professor Osgood for counsel as the work progressed and for aid in doubtful matters pertaining to the translation.
The
make
publishers, Messrs.
Ginn
the typography excellent.
& Company, have spared
no pains to
Their spirit has been far from com
mercial in the whole enterprise, and
it
is
their hope, as
it is
mine,
that the publication of this book will contribute to the advance of
mathematics
in
AUGUST, 1904
America.
E R HEDRICK
CONTENTS PAGE
CHAPTER I.
DERIVATIVES AND DIFFERENTIALS
1
Functions of a Single Variable Functions of Several Variables
11
The
19
I.
II.
III.
II.
Differential Notation
IMPLICIT FUNCTIONS. FUNCTIONAL DETERMINANTS.
I.
II.
III.
Implicit Functions
II.
Functional Determinants
52 61
Taylor
s
Series with a Remainder.
Singular Points.
II.
Taylor
Maxima and Minima
DEFINITE INTEGRALS I. Special Methods of Quadrature III.
.
s
Series
.
.
.
.
.
.
Definite Integrals. Allied Geometrical Concepts Change of Variable. Integration by Parts .
Integrals.
Line Integrals
V. INDEFINITE INTEGRALS
III.
VI.
Integration of Rational Functions
Double
.110
.
Methods
of
.
.
.
Evaluation.
Green
III.
Change
192
208
208 226
.236 250
s
250
Area of a Surface Double Integrals. Improper
of Variables.
Generalizations of
175
.196
Theorem II.
140
.166
........
Integrals.
134
.134
..... ....
and Hyperelliptic Integrals Integration of Transcendental Functions Elliptic
DOUBLE INTEGRALS I.
.
89
89
Improper
V. Functions defined by Definite Integrals VI. Approximate Evaluation of Definite Integrals
II.
.
.
...... ....
IV. Generalizations of the Idea of an Integral.
I.
35
35
Transformations
TAYLOR S SERIES. ELEMENTARY APPLICATIONS. MAXIMA AND MINIMA I.
IV.
CHANGE
........ .... ......... ........ ........
OF VARIABLE
III.
1
Surface Integrals
.
.
.
.......
IV. Analytical and Geometrical Applications
.
264
Integrals.
.
.
277 284
CONTENTS
viii
PAGE
CHAPTER VII.
MULTIPLE INTEGRALS.
INTEGRATION OF TOTAL DIFFER 296
ENTIALS I.
II.
VIII.
of Variables
Multiple Integrals. Change Integration of Total Differentials
.
.
INFINITE SERIES I. Series of Real Constant Terms.
296
.
.
.
.
.
.313 327
.
.
General Properties.
327
Tests for Convergence II.
Series of
III. Series of
IX.
POWER I.
II.
Complex Terms. Multiple Series Variable Terms. Uniform Convergence
SERIES.
Power Power
.
TRIGONOMETRIC SERIES
Series of a Single Variable Series in Several Variables
.
350
.
.
360
....
375
..... .
.
.
.
Analytic Curves and Surfaces IV. Trigonometric Series. Miscellaneous Series
III.
.
Implicit Functions.
.
375
.
S94 399
.
.
.411 426
X. PLANE CURVES I.
II.
III.
XI.
Envelopes Curvature
426
Contact of Plane Curves
443
SKEW CURVES I.
II.
433
........ .... ........
Osculating Plane Envelopes of Surfaces
.
.
.
.
.
.
.
Curvature and Torsion of Skew Curves IV. Contact between Skew Curves. Contact between Curves
III.
and Surfaces XII.
II.
III.
Curvature of Curves drawn on a Surface Asymptotic Lines. Conjugate Lines
459 468 486
....
Lines of Curvature
.
.
.
.
.
497
.
.
.
.
.
.514
506
526
IV. Families of Straight Lines
INDEX
453
497
SURFACES I.
453
.
541
CHAPTER
I
DERIVATIVES AND DIFFERENTIALS
FUNCTIONS OF A SINGLE VARIABLE
I.
1.
Limits.
When
the successive values of a variable x approach
nearer and nearer a constant quantity a, in such a way that the a finally becomes and remains absolute value of the difference x less
than any preassigned number, the constant a is called the This definition furnishes a criterion for
limit of the variable x.
determining whether a sary and
is
the limit of the variable
sufficient condition that it
should
x.
The neces
is
be, that, given any no matter how small, the absolute value of x a should remain less than e for all values which the variable x can
positive
number
e,
assume, after a certain instant.
Numerous examples
of limits are to be found in Geometry For example, the limit of the variable quantity x = (a 2 m 2 ) / (a m), as m approaches a, is 2 a for x 2 a will a is taken less than e. Likewise, the be less than e whenever m variable x = a where n is a positive integer, approaches the 1/n, limit a when n increases indefinitely for a x is less than e when
and Algebra.
;
;
ever n
It is apparent from these examples that greater than 1/e. the successive values of the variable x, as it approaches its limit, may is
form a continuous or a discontinuous sequence. It is in general very difficult to determine the limit of a variable quantity. The following proposition, which we will assume as selfevident, enables us, in
many
cases, to establish the existence of a limit.
variable quantity which never decreases, and which ahvays less than a constant quantity L, approaches a limit I, which
Any
remains is less
than or at most equal
to L.
Similarly, any variable quantity which never increases, and which always remains greater than a constant quantity L approaches a ,
limit
l
}
which
is
greater than or else equal 1
to
L
.
DERIVATIVES AND DIFFERENTIALS
2.
For example,
[I,
2
each of an infinite series of positive terms is than the corresponding term of another infinite series of positive terms which is known to converge, then the first series converges also for the sum 2 n of the first n terms evidently increases with n, and this sum is constantly less than the total sum if
less, respectively,
;
5 of the second
series.
2. Functions. When two variable quantities are so related that the value of one of them depends upon the value of the other, they are said to be functions of each other. If one of them be sup
posed to vary arbitrarily, it is called the independent variable. Let this variable be denoted by x, and let us suppose, for example, that it can assume all values between two given numbers a and b
Let y be another variable, such that to each value of x b). (a between a and b, and also for the values a and b themselves, there corresponds one definitely determined value of y. Then y is called a function of x, defined in the interval (a, b) and this dependence <
;
indicated by writing the equation y =/(z). For instance, it may happen that y is the result of certain arithmetical operations per formed upon x. Such is the case for the very simplest functions studied in elementary mathematics, e.g. polynomials, rational func is
tions, radicals, etc.
A function may also be defined graphically. Let two coordinate axes Ox, Oy be taken in a plane and let us join any two points A and B of this plane by a curvilinear arc .4 CB, of any shape, which ;
not cut in more than one point by any parallel to the axis Oy. Then the ordinate of a point of this curve will be a function of the abscissa. The arc A CB may be composed of several distinct por is
tions
which belong
to different curves, such as
segments of straight
lines, arcs of circles, etc.
In short, any absolutely arbitrary law may be assumed for finding the value of y from that of x. The word function, in its most gen eral sense, means nothing more nor x corresponds a value of y.
less
than this
:
to every value of
3. Continuity. The definition of functions to which the infini tesimal calculus applies does not admit of such broad generality. Let y =f(x) be a function defined in a certain interval and (a,
b),
let
x and x
ence f(x
-f
h be two values of x
in that interval.
If the differ
f(xo) approaches zero as the absolute value of h approaches zero, the function f(x} is said to be continuous for the value x From the very definition of a limit we may also say that .
-f A)
I,
FUNCTIONS OF A SINGLE VARIABLE
3]
a function f(x)
3
x if, corresponding to every continuous for x no matter how small, we can find a positive num
is
positive number ber 77, such that
e,
+
|/(*o
A)-/(*o)|<
We
shall say that for every value of h less than rj in absolute value.* a function f(x) is continuous in an interval (a, b) if it is continuous for every value of x lying in that interval, and if the differences
each approach zero when
h,
which
approaches zero. In elementary text-books
is
it
now
is
usually
rational functions, the exponential
to be taken only positive,
shown
that polynomials,
and the logarithmic function,
the trigonometric functions, and the inverse trigonometric functions are continuous functions, except for certain particular values of It follows directly from the definition of continuity the variable. or the product of any number of continuous functions and this holds for the quotient of itself a continuous function
that the is
sum
;
two continuous functions also, except for which the denominator vanishes.
for the values of the variable
seems superfluous to explain here the reasons which lead us to assume that functions which are defined by physical conditions are, It
at least in general, continuous.
the properties of continuous functions we shall now state only the two following, which one might be tempted to think were self-evident, but which really amount to actual theorems, of which
Among
rigorous demonstrations will be given later, f
If the function yf(x) is continuous in the interval (a, b), and a number between f (a) andf(b), then the equation f(x) = has at least one root between a and b. II. There exists at least one value of x belonging to the interval I.
if
N
N is
(a, b ), inclusive
of
its
end points, for which y takes on a value
M
greater than, or at least equal to, the value of the function at any other point in the interval. Likewise, there exists a value of x for which y takes on a value m, than which the function assumes no
which
is
smaller value in the interval.
The numbers
M and m are called the maximum and the minimum
values of f(x), respectively, in the interval *
The notation
t
See Chapter IV.
|
a
\
(a,
b*).
denotes the absolute value of
It is clear that
a.
DERIVATIVES AND DIFFERENTIALS
4
[I,
4
the value of x for which /(ce) assumes its maximum value M, or the value of x corresponding to the minimum m, may be at one of the
end points, a or that
N is a
if
at least one root 4.
Examples
from the two theorems above,
It follows at once
b.
number between which
lies
M and m, the equation /() = N has
between a and
of discontinuities.
b.
The functions which we
shall study
will be in general continuous, but they may cease to be so for certain exceptional values of the variable. proceed to give
We
several examples of the kinds of discontinuity
which occur most
frequently. The function y
= 1 / (x a) is continuous for every value x of x except a. The operation necessary to determine the value of y from that of x ceases to have a meaning when x is assigned the value a but we note that when x is very near to a the absolute ;
value of y is very large, and y is positive or negative with x a. As the difference x a diminishes, the absolute value of y increases indefinitely, so as eventually to become and remain greater than any
preassigned number.
y becomes
infinite
This phenomenon
when x
=
a.
is described by saying that Discontinuity of this kind is of
great importance in Analysis. Let us consider next the function y sin 1/z. As x approaches zero, I/a; increases indefinitely, and y does not approach any limit
=
it remains between + 1 and 1. The equation where A 1, has an infinite number of solutions which lie between and e, no matter how small e be taken. What ever value be assigned to y when x 0, the function under con
whatever, although
sin l/a;
= ,4,
<
\
sideration cannot be
An example
made continuous
of a still different
for x
=
0.
kind of discontinuity
is
given by
the convergent infinite series
When
x approaches
S (x~) approaches the
limit 1, although every term of the series is zero, and hence 5 (0) = 0. But if x be given a value different from zero, a geometric progression is obtained, of which the ratio is 1/(1 + a; 2 ).
5 (0)
=
0.
For,
zero,
when x
Hence
~
=
0,
FUNCTIONS OF A SINGLE VARIABLE
5]
I,
and the limit of S(x)
is
seen to be
5
Thus, in this example, the
1.
function approaches a definite limit as x approaches zero, but that 0. limit is different from the value of the function for x
=
Derivatives.
5.
Then
Let/(x) be a continuous function.
the
two
terms of the quotient
k approach zero simultaneously, as the absolute value of h approaches If this quotient approaches a limit, zero, while x remains fixed. derivative of the function /(#), and is denoted is called the this limit the notation due to Lagrange. An important geometrical concept is associated with this analytic Let us consider, in a plane XOY, the curve notion of derivative.
by y
,
or
by /
(x), in
A MB, which
represents the function y =/(#), which in the interval (a, b). Let and continuous to be
M
on this curve, in the interval
x
+
(a, b),
and
let
we
shall
assume
M
be two points their abscissas be x and
MM
The
is then slope of the straight line A, respectively. h above. Now as the zero the approaches quotient point precisely approaches the point M] and, if the function has a derivative,
M
the slope of the line
MM
MM
approaches the limit y
.
The
straight line
therefore, approaches a limiting position, which is called the tangent to the curve. It follows that the equation of the tangent is ,
Y-y = y where
To
X
and
(X-x),
Y
are the running coordinates. generalize, let us consider any curve in space,
and
let
be the coordinates of a point on the curve, expressed as functions of be two points of the curve Let and a variable parameter t.
M
M
corresponding to
two values,
equations of the chord
MM
x-f(t)
1
t
and
t
+
h, of
the parameter.
The
are then
Y
f(t + h) If
we
divide each denominator by h and then let h approach zero,
MM
evidently approaches a limiting position, which given by the equations
the chord
X -f(f) f(t)
Y4, ft)
is
DERIVATIVES AND DIFFERENTIALS
6
[i,
5
provided, of course, that each of the three functions f(t), (t), \J/ (t) The determination of the tangent to a curve possesses a derivative. thus reduces, analytically, to the calculation of derivatives.
Every function which possesses a derivative is necessarily con It is easy to give examples tinuous, but the converse is not true. of continuous functions which do not possess derivatives for par ticular values of the variable. The function y for
= xsinl/x, = =
a perfectly continuous function of x, for x 0,* and y approaches zero as x approaches zero. But the ratio y /x sinl/cc does not approach any limit whatever, as we have already seen.
example,
is
Let us next consider the function y = x*. Here y is continuous = when x = 0. But the ratio y /x = x~* a;; and y increases indefinitely as x approaches zero. For abbreviation the derivative is said to be infinite for x = the curve which repre for every value of
;
sents the function
tangent to the axis of y at the origin. Finally, the function is
y
is
continuous at x
=
limits according as it is
approaching
=
0,* but the ratio
x
zero.
is
y /x approaches two different always positive or always negative while When x is positive and small, e l/x is posi
and very large, and the ratio y /x approaches 1. But if x is negative and very small in absolute value, e l/x is very small, and the ratio y / x approaches zero. There exist then two values of the derivative according to the manner in which x approaches zero the curve which represents this function has a corner at the origin. It is clear from these examples that there exist continuous func tions which do not possess derivatives for particular values of the variable. But the discoverers of the infinitesimal calculus confi tive
:
dently believed that a continuous function had a derivative in gen eral. Attempts at proof were even made, but these were, of course, Finally, Weierstrass succeeded in settling the question conclusively by giving examples of continuous functions which do not But possess derivatives for any values of the variable whatever.! fallacious.
as these functions have not as yet been
employed
in
any applications,
=
* After the value zero has been 0. TRANSLATOR. assigned to y for x t Note read at the Academy of Sciences of Berlin, July 18, 1872. Other examples are to be found in the memoir by Darboux on discontinuous functions (Annales de
Ecole Normale Superieure, Vol. IV, 2d series). given later (Chapter IX).
I
One
of Weierstrass s examples
is
FUNCTIONS OF A SINGLE VARIABLE
6]
I,
7
them here. In the future, when we say that in the interval (a, b), we shall mean a derivative has a function f(x) for every value of x between derivative an it has that unique finite
we
shall not consider
and also f or x = a (h being positive) and f or x = b (h being is made to the contrary. negative), unless an explicit statement a and
6.
b
The
Successive derivatives.
derivative of a function f(x) is in If f (x) in turn has a deriva (x).
general another function of x,f the new function is called the second derivative of /(x), and is In the same way the third deriva or by f"(x). represented by the derivative of the second, and to be or / "(#), is defined tive y n) derivative In general, the rath so on. 7/ , or f (x), is the deriva tive,
y"
",
w
tive of the derivative of order (n
1).
If, in thus
forming the
never obtain a function which has no
we we may imagine the process carried on indefinitely. In this way we obtain an unlimited sequence of derivatives of the func Such is the case for all functions tion /(cc) with which we started. successive derivatives,
derivative,
which have found any considerable application up to the present time.
The above notation
Dn f(x),
due to Cauchy,
wth derivative. 7.
is
Leibniz
due to Lagrange. The notation Dn y, or also used occasionally to represent the
is
notation will be given presently.
The use of derivatives in the study of equa upon the following proposition, which is known as Theorem :
Rolle s theorem.
tions depends
Roue
s
=
Let a and
0. b be two roots of the equation f (x) If the function in interval (a, b~), a derivative the and continuous possesses f(x) has at least one root which lies between a and b. the equation / (#)
is
=
For the function f(x) vanishes, by hypothesis, for x
= a and x =
b.
vanishes at every point of the interval (a, b), its derivative also vanishes at every point of the interval, and the theorem is evidently If
it
fulfilled.
val, it
will
If the function f(x) does not vanish throughout the inter assume either positive or negative values at some points.
Suppose, for instance, that it has positive values. Then it will have a maximum value for some value of x, say x lf which lies between
M
a and
b (
3,
Theorem
II).
The
ratio
DERIVATIVES AND DIFFERENTIALS
8
8
[I,
where h is taken positive, is necessarily negative or else Hence the limit of this ratio, i.e. f (x^), cannot be positive f ( x i) = 0- But if we consider f (x\) as the limit of the ratio
zero. ;
i.e.
>
h where h
From
is
positive, it follows in the
these two results
same manner that f\x\) ^
evident that/
it is
^) =
0,
0.
8. Law of the mean. It is now easy to deduce from the above theorem the important law of the mean * :
Let f(x) be a continuous function which has a derivative in the interval (a,
Then
b).
m-f(a) = (b-a)f(c-),
(1)
where
c is
a number between a and
b.
In order to prove this formula, let (x) be another function which has the same properties as/(x), i.e. it is continuous and possesses a derivative in the interval (a, b). Let us determine three constants, A, B, C, such that the auxiliary function <
vanishes for x
=
=
a and for x
The necessary and
b.
sufficient
conditions for this are
A /(a)
+B
<()+
and these are
satisfied if
A =
(b),
(a)-4>
The new function in the interval (a, fore vanishes for
replacing
C
=
we
C=
B(b)+
0;
set
B =/(&) -/(a), \J/(x)
thus defined
b).
The
is
c
It is merely necessary to take which was to be proved. It is <
if/
which
their values,
(a;)
C
=/()
(*)-/(&) * (a).
continuous and has a derivative
derivative
some value
A and B by
+
Af(b)
0,
we
=x
(x)
lies
= A f (x) + B
<
between a and
(z) there b,
find a relation of the
whence form
in order to obtain the equality
to be noticed that this
demonstration does not presuppose the continuity of the derivative/ ^). des accroissements finis." as a synonym. Other English
"Formule
moyenne"
and
"
Mean value
theorem."
TRANS.
The French synonyms are
also use
Formule de la value theorem
"
"
"Average
FUNCTIONS OF A SINGLE VARIABLE
8]
I,
9
From
the theorem just proven it follows that if the derivative zero at each point of the interval (a, b), the function f(x) has the same value at every point of the interval for the applica
f (x)
is
;
x z belonging to the interval Hence, if two functions have the same (a, b), gives f(xi)=f(x.2 ). derivative, their difference is a constant and the converse is evi If a function F(x) be given whose derivative is dently true also. which have the same derivative are found by f(oc), all other functions adding to F(x) an arbitrary constant* tion of the formula to
two values
Xi,
,
;
The geometrical interpretation of the equation (1) is very simple, Let us draw the curve A MB which represents the function y = f(x) Then the ratio [/(&) /()]/ (b a) is the in the interval (a, b).
/
slope of the chord AB, while () is the slope of the tangent at a whose abscissa is c. Hence the equation (1) point C of the curve
expresses the fact that there exists a point C on the curve A MB, between A and B, where the tangent is parallel to the chord AB. If the derivative
/ (a;)
is
continuous, and
if
we
let
a and
b
approach
according to any law whatever, the number c, lies between a and b, also approaches x 0} and the equation (1) that the limit of the ratio
the same limit x
which shows
a
b
f (xo). The geometrical interpretation is as follows. Let us whose abscissa is x consider upon the curve y=f(x) a point and two points A and B whose abscissa are a and b, respectively. is
M
,
is equal to the slope of the chord The ratio [/(&) /()] / (b ) AB, while / (x ) is the slope of the tangent at M. Hence, when the two points A and B approach the point according to any law
M
whatever, the secant AB approaches, as tangent at the point M.
limiting position, the
its
* This theorem
is sometimes applied without due regard to the conditions imposed in Let/(x) and 0(^), f r example, be two continuous functions which have = If the relation / (z) (x)f(x) derivatives / (a;), (x) (x) in an interval (a, 6). is satisfied by these four functions, it is sometimes accepted as proved that the deriva - f(x) (z)] / 2 is zero, and that accordingly tive of the function// or [/ (a;) (cc)
its
statement.
4>
<
,
,
f/
is
constant in the interval
unless the function
(a, b).
But
$ (a;) does not vanish
this conclusion is not absolutely rigorous
in the interval (a, b).
Suppose, for instance,
A function/(x) equal and (x) both vanish for a value c between a and 6. to between c and b, where Cj and C2 are dif between a and c, and to ferent constants, is continuous and has a derivative in the interval (a, b), and we have
that
(a;)
<j>
C%
Ci(x)
f
(x)(x)
f(x)
interpretation
is
(x)
=
apparent.
for every value of x in the interval.
The geometrical
DERIVATIVES AND DIFFERENTIALS
10
[I,
9
This does not hold in general, however, if the derivative is not For instance, if two points be taken on the curve = on x*, y opposite sides of the y axis, it is evident from a figure that the direction of the secant joining them can be made to approach continuous.
any arbitrarily assigned limiting value by causing the two points to approach the origin according to a suitably chosen law. The equation (! ) is sometimes called the generalized law of the mean. From it de 1 Hospital s theorem on indeterminate forms fol lows at once.
x in by *
where ratio /"(#)
9.
of the
For, suppose f(a)
we
(j>
=
and
(a)
=
0.
Replacing
b
find \
between a and
x. This equation shows that if the (x) approaches a limit as x approaches a, the ratic and (a) 0. approaches the same limit, if f(a)
a^ lies
f /
(! ). \ /
(x)/(j>
(a;)
=
=
Generalizations of the law of the mean.
Various generalizations of the law
mean have been
de la Socie te
The following one is due to Stieltjes (Bulletin suggested. Mathtmatique, Vol. XVI, p. 100). For the sake of defmiteness con
sider three functions, /(x), g(x), h(x), each of which has derivatives of the first and second orders. Let a, 6, c be three particular values of the variable (a b c). <
Let
A
and
let
be an auxiliary function.
x
=
c, its
derivative
Since this function vanishes
must vanish /(a) /(&)
/ If b
of x
<
be a number defined by the equation
for
when x
some value f between
g(a)
h(a)
g(b)
h(b)
-A
6
=
b
c.
and when Hence
a2
1
a
1
b
b2
1
2f
(f)
and
be replaced by x in the left-hand side of this equation, we obtain a function which vanishes when x = a and when x = b. Its derivative therefore van
ishes for
some value
of
equation thus obtained
x between a and
6,
which we
shall call
.
The new
is
/
(a)
/
(f)
g (a)
h
(a)
= 1
0.
2f
Finally, replacing f by x in the left-hand side of this equation, we obtain a func tion of x which vanishes when x = and when x Its derivative vanishes f
=
.
FUNCTIONS OF SEVERAL VARIABLES
10]
I,
and f and therefore between a and
some value ij, which lies between Hence A must have the value
for
/
J_
(a)
11
g
c.
h (a)
(a)
1.2
where
lies
between a and
and
6,
17
lies
between a and
c.
This proof does not presuppose the continuity of the second derivatives If these derivatives are continuous, and if the values a, 6, c f"(x), g"(x), h"(x). approach the same limit XQ, we have, in the limit,
/
1
(x
)
g
(x
f (x
)
g
(xo)
f"(x
)
0"(xo)
h h
)
(x
)
(x
)
h"(x
Q)
Analogous expressions exist for n functions and the proof follows the same If only two functions /(x) and g (x) are taken, the formula? reduce to the law of the mean if we set g (x) = 1. An analogous generalization has been given by Schwarz (Annali di Mathematica, 2d series, Vol. X). lines.
II.
FUNCTIONS OF SEVERAL VARIABLES
10. Introduction. A variable quantity w whose value depends on the values of several other variables, x, y, z, -, t, which are in
dependent of each other, ent variables x, y,
w =f(x, y,z,---, is
z,
t).
a function of the
,
t;
is called a function of the independ and this relation is denoted by writing
For definiteness,
let
us suppose that w
two independent variables x and
y.
If
= f(x, y) we think
of x and y as the Cartesian coordinates of a point in the plane, each pair of values (x, y) determines a point of the plane, and con If to each point of a certain region A in the xy versely. plane,
bounded by one or more contours of any form whatever, there corresponds a value of in the region A.
w,
the function f(x, y)
is
said to be defined
M
Let (x y ) be the coordinates of a point lying in this region. The function f(x, y) is said to be continuous for the pair of values ,
(
x oi
any preassigned positive number such that
yo) if, corresponding to
positive
number
77
exists
|/C*o
+ h,
y
+ k)-f(x
,
2/ )
<
c,
another
e
|
whenever \h and \k\
may be interpreted as follows. Let us suppose constructed in the xy plane a square of side 2^ about as center, with its sides to the The axes. parallel , point
M
M
DERIVATIVES AND DIFFERENTIALS
12
[I,
11
whose coordinates are x + h, y + k, will lie inside this square, if To say that the function is continuous for the and k h rj pair of values (x T/ O ) amounts to saying that by taking this square sufficiently small we can make the difference between the value of and its value at any other point of the square less the function at <
<
|
|
|
rj.
\
,
M
than
e
in absolute value.
we may
It is evident that
as center.
replace the square by a circle about the above condition is satisfied for all
if
For, (x y ) points inside a square, it will evidently be satisfied for all points inside the inscribed circle. And, conversely, if the condition is ,
satisfied for all points inside a circle, it will also be satisfied for all
We might then points inside the square inscribed in that circle. define continuity by saying that an rj exists for every c, such that whenever V/i 2
+k I
The
<
/(<>
17
+
we
also have
h y >
+
k)
-f(x
,
definition of continuity for a function of 3, 4,
pendent variables It is clear that
variables x
and y
taken separately. 11.
2
is
,
n inde
similar to the above.
any continuous function of the two independent is
a continuous function of each of the variables
However, the converse does not always hold.*
Partial derivatives.
If
any constant value whatever be substi
tuted for y, for example, in a continuous function f(x, y), there results a continuous function of the single variable x. The deriva tive of this function of x, if it exists, is denoted by fx (x, y) or by x
.
Likewise the symbol u v or fy (x, y), is used to denote the derivative of the function f(x, y} when x is regarded as constant and y as the ,
independent variable. The functions x (x, y) and y (x, y) are called the partial derivatives of the function f(x, They are themselves, y). in general, functions of the two variables x and y. If we form their
f
partial derivatives in turn,
we
f
get the partial derivatives of the sec Thus there are four partial y).
ond order of the given function f(x,
derivatives of the second order, fa (x, y),fx (x, y),fyx (x, y),f+(x, y\ The partial derivatives of the third, fourth, and higher orders are * 2 Consider, for instance, the f unction /(x, y), which is equal to 2 xy / (x 2 y ) when the two variables x and y are not both zero, and which is zero when x 0. It is y evident that this is a continuous function of x when y is constant, and vice versa.
=
+ =
Nevertheless it is not a continuous function of the two independent variables x and y for the pair of values x = 0, y = 0. For, if the point (a-, y) approaches the origin upon the line x = y. the f unction/ (x, y) approaches the limit 1, and not zero. Such functions
have been studied by Baire
in his thesis.
FUNCTIONS OF SEVERAL VARIABLES
11]
I,
13
=
In general, given a function w defined similarly. -, f) /(x, y, z, of any number of independent variables, a partial derivative of the nth order is the result of n successive differentiations of the function /, in
in /.
a certain order, with respect to any of the variables which occur will now show that the result does not depend upon the
We
order in which the differentiations are carried out.
Let us
f =f
tjx ,
Ax, y
+ Ay) -f(x,
+
-f(x +
Ay)
Ax, y)
+ /(x,
y}
where we suppose that x, y, Ax, A?/ have Let us introduce the auxiliary function
00 =f( x
<
where v
is
+ Ax,
U = Ay replacing
<j>
u
by
(y
<
its
regarding u
-/(x,
we
<(w),
where
<
<
Ax, y
+
mean
0Ay)
U = Ax Ay/^ (x +
Ax, y
of the expression
0|
y
we
U
Ax, y
fy (u,
y
<
in x, y, Ax, Ay,
<
0Ay),
we
1.
see that
y,
+ ^ Ay),
0, and 0[ are again positive constants less than unity. these two values of U and dividing by Ax Ay, we have ing
where
Ax, y
+
find
+ 0Ay),
interchanging x and
U = Ay Aaj/q, (x +
0[
;
-f (x, y + 0Ay)].
to the function
as the independent variable,
fxy (x +
1
value,
the law of the
From the symmetry we would also have,
v),
to the function
+ 0Ay),
U = Ay [/(* + we now apply
u)
Then we may write
an auxiliary variable.
Applying the law of the mean
If
y
different forms,
definite values.
or,
Then
this let us first write the expression
U =f(x + two
:
y) be a function of the two variables x and y. provided that these two derivatives are continuous.
To prove
in
prove the following lemma
= f (x,
Let w xij
first
+ ^Ay) =f,,x (x +
Ax, y
+
Equat
0Ay).
(x, y) and fvx (x, y) are supposed continuous, of the above equation approach fxy (x, y) and
Since the derivatives /,. the two
fyx (x,
members
as Ax and Ay approach we wished to prove.
y), respectively,
the theorem which
zero,
and we obtain
DERIVATIVES AND DIFFERENTIALS
14
[I,
n
above demonstration that no hypothesis
It is to be noticed in the
made concerning the other derivatives of the second order, The proof applies also to the case where the function f^ and fy f(x, y) depends upon any number of other independent variables whatever
is
t.
besides x and y, since these other variables would merely have to be regarded as constants in the preceding developments.
now
Let us
consider a function of any
number of independent
variables, =/(>
and
let
n
y>
*)j
*>>
be a partial derivative of order n of this function.
Any
permutation in the order of the differentiations which leads to fi can be effected by a series of interchanges between two successive differentiations
tion considered. is
;
and, since these interchanges do not alter the just seen, the same will be true of the permuta
we have
result, as
have a notation which
It follows that in order to
not ambiguous for the partial derivatives of the nth order,
it is
number
of differentiations performed with respect to each of the independent variables. For instance, any nth derivative of a function of three variables, to =/(x, y, z), will be sufficient to indicate the
represented by one or the other of the notations
where p
-f-
q
+ r = n*
result of differentiating
Either of these notations represents the / successively p times with respect to x,
q times with respect to ?/, and r times with respect to 2, these oper ations being carried out in any order whatever. There are three distinct derivatives of the first order, x six of the second z
f f f ,
order, fa,
fa fa /
.
3 v
,
fa fxz
In general, a function of
;
and so
,
\
on.
p independent variables
has just as
many
n as there are distinct terms in a homo of n in p independent variables that is, order geneous polynomial distinct derivatives of order
;
as
is
shown
in the theory of combinations.
A
Practical rules. certain number of practical rules for the cal culation of derivatives are usually derived in elementary books on *
The notation /a Pyq (x, y, z) is used instead of the notation fxfyn z r (x, y, z) for Thus the notation fxy (x, y), used in place of fx y (x, y), is simpler and TRANS. equally clear.
simplicity.
..r
I,
FUNCTIONS OF SEVERAL VARIABLES
11]
A
the Calculus. its
table of such rules is appended, the function same line
derivative being placed on the
=
ax
=
ax log
1
-
;
a,
where the symbol log denotes the natural logarithm
= log x,
= y= y
=
y
->
X
cos x,
=
y
= arc tan x,
arc sin x,
JL
y = uv,
y
The
y
u f
=/(),
last
;
sin x,
y
_
and
:
y y
y
16
-\-
X
= u v 4- uv ^ u v uv
1
;
.
2/*=/>K;
two rules enable us to find the derivative of a function and that of a composite function if fu ,fv ,fw are con
of a function tinuous.
Hence we can
find the successive derivatives of the func
tions studied in elementary mathematics, and irrational functions, exponential and
polynomials, rational
logarithmic functions, trigonometric functions and their inverses, and the functions deriv able from all of these by combination.
For functions of several variables there exist certain formulae analogous to the law of the mean. Let us consider, for definiteness, a function f(x, y) of the The difference f(x h, y 4- K)
+
f(x
+ h,y +
to each part of thus find
f(x
+
h,y
where 6 and
+
k)
-f(x,
y)
two independent variables x and y. f(x, y) may be written in the form
= [/(* + h,
y
+
k)
-f(x, y
+ &)]
which we may apply the law of the mean.
k}-f(x, each
y)
= hfx (x +
6h,
y
We
+ k}+ kf (x, y + O K), v
between zero and unity. This formula holds whether the derivatives fx and / are continu
ous or not.
lie
If these derivatives are continuous, another formula,
DERIVATIVES AND DIFFERENTIALS
16
[1,512
similar to the above, but involving only one undetermined number In order to derive this second formula, con 6, may be employed.*
= f(x + ht, y + kfy, where x, y, h, sider the auxiliary function and k have determinate values and t denotes an auxiliary variable. Applying the law of the mean to this function, we find ()
Now
is <(>")
equal to
a composite function of
hfx (x
ceding formula
12.
ht,
-f-
may
Tangent plane
and
t,
derivative
its
(t)
4>
y + kf) + kfy (x + ht,
y
-f-
kt)
;
is
hence the pre
be written in the form
We
to a surface.
have seen that the derivative
of a function of a single variable gives the tangent to a plane curve. Similarly, the partial derivatives of a function of two variables occur in the determination of the tangent plane to a surface. z
(2)
. F(x,
Let
y)
be the equation of a surface S, and suppose that the function F(x, ?/), together with its first partial derivatives, is continuous at a point Let z be the corresponding value of z, (^o? yo) of the xy plane. and AT (cr 7/0 the corresponding point on the surface S. If ) ,
>
the equations
*=/(*),
(3)
z/
=
=
*
^(9
M
the represent a curve C on the surface S through the point three functions f(f), which we shall suppose continuous <j>(t),
and
differentiable,
must reduce to x y t. The tangent
,
,
of the parameter
value
t
M
given by the equations
is
x Since the curve
must hold
z
C lies on the surface t;
z , respectively, for some to this curve at the point
5)
(
Y
x
for all values of
,
"A(0>
that
S,
is,
*
the equation \j/(t)=F[f(t~),
this relation
.
must be an identity
* Another formula may be obtained which involves only one undetermined number 0, and which holds even when the derivatives/^, and/, are discontinuous. For the applica tion of the law of the mean to the auxiliary function =f(x + ht,y + k) +f(x, y + kt) <j>(t)
gives <(!)
-0(0)
or
f(x
+ h,y +
k)
-f(x,
y)
= (0), = hfx (x +
0<0<1.
6h,
y
+
k)
+ kfy (x,
y
+
6k),
0<0<1.
The operations performed, and hence the final formula, all hold provided the deriva tives fx and fy merely exist at the points (x + ht, y + k), TRANS. (x,y + kt),0^t^\.
FUNCTIONS OF SEVERAL VARIABLES
13]
I,
in
17
Taking the derivative of the second member by the rule for t = t we have
t.
the derivative of a composite function, and setting (5)
(t
<j,
We
can
and
(5),
now
eliminate
and the
f (t
(
<
~),
)
i//(
)>
)FVa
(t
.
between the equations
(4)
result of this elimination is
Z-z = (X-
(6)
)FXo +
)=fi(t
,
ar
)
F + Xg
(Y
-y)
F^.
is the equation of a plane which is the locus of the tangents to It is called the tan curves on the surface through the point
This
M
all
gent plane
.
to the surface.
We
13. Passage from increments to derivatives.
have defined the successive from
derivatives in terms of each other, the derivatives of order n being derived It is natural to inquire whether we those of order (n 1), and so forth.
may
not define a derivative of any order as the limit of a certain ratio directly, with We have already done some out the intervention of derivatives of lower order. thing of this kind for fxy is the limit of the ratio
f(x
as
+ Ax,
y
11); for the
(
+ Ay) -/(x +
Ax and Ay both approach
second derivative
/"
zero.
demonstration given above shows that/rj,
Ax, y)-f(x, y Ax Ay
It
+
Ay)
+ /(x,
y)
can be shown in like manner that the
of a function f(x) of a single variable is the limit of the
ratio
/(x
+
+
hi
h*)
-f(x +
hi)
-f(x
^1^2 as hi
and h 2 both approach
zero.
For, let us set
/i(x)=/(z and then write the above
ratio in the
+
Ai)
form
h\
f
>
(x
+
+ hi
The
limit of
derivative
is
this ratio
is
therefore the second derivative
/",
provided that
continuous.
Passing now to the general case, let us consider, for definiteness, a function of Let us set three independent variables, w =f(x, y, 2).
AW =/(x + h, y, AW =/(x, y + k, A^w =/(x,
z)
-/(x,
2)
-/(x,
y, 2),
y, 2 -f 1)
-/(x,
y, z),
y, 2),
w. If we consider ^, k, I as given functions of x, y, 2, themselves are increments first then these three constants, and we may form the relative increments of these functions corresponding to
where A* w, A* w, A z u are the^irsi increments of l
DERIVATIVES AND DIFFERENTIALS
18
13
[I,
increments
This gives us the second increments, hi, ki, ^ of the variables. A* 1 A* w A* A w Tnis process can be continued indefinitely an increment v of order n would be defined as a first increment of an increment of order (n 1). Since we may invert the order of any two of these it will be suffi 1
>
;
operations, cient to indicate the successive increments given to each of the variables. An increment of order n would be indicated by some such notation as the following :
+
where p
+
q
r
AX
=
A<->
=
AX
A*p A*
1
A^/(z,
and where the increments
y, z),
be either equal or This increment may be expressed in terms of a unequal. partial derivative of order n, being equal to the product hihy
hpki
n,
may
I
lr
kgl\
+
h, k,
+ d,,hp y + eiki + + Oq kq z + ffi li + + Kir), where every 6 lies between and 1. This formula has already been proved for first and for second increments. In order to prove it in general, let us assume that it holds for an increment of order (n 1), and let x
fx p*z
(x
+
*i Ai
,
=
(X, y, 2)
,
AX
Ah/ Ajt
A**
1
f.
Then, by hypothesis, $(x,y,z)
=
h z ---hp ki-- -k q li--
fxp-i
Ir
if, i
r(x + 0sh2 +
----\-6 ---P hp, y-\
,H----
).
But the nth increment considered is equal to 0(x + hi, y, z) y, z); and if we apply the law of the mean to this increment, we finally obtain the formula sought. (x,
Conversely, the partial derivative fxT^ zr
AX
hp ki k 2
hi h?
as all the increments h, k, It is interesting to
the usual definition.
function of x and y,
no
first
I
approach
kg
---
A/-/ lr
li
zero.
notice that this definition
sometimes more general than + ^(y) is a Then u also has
is
Suppose, for example, that w =/(x, y) where neither nor ^ has a derivative.
(x)
and consequently second derivatives are out
derivative,
the ordinary sense. Nevertheless, tive fxy is the limit of the fraction in
/(x
the limit of the ratio
is
.-AX-.. -AX
-
+
h,
y
+
-/(x +
k)
if
we adopt
h, y)
the
-/(x, y
+
of the question,
new definition, k)
+/(x,
the deriva
y)
hk which
is
equal to h)
+
t( V
+
k)
(x
+
h)
hk
But the numerator of this ratio is identically zero as a limit, and we find/xy = 0.* *
A
similar
zero.
Hence the
remark may be made regarding functions of a = xs cosl/x has the derivative
ratio
approaches
single variable.
example, the f unction /(K)
f and f
(x)
(x)
has no derivative for x
=
3 x 2 cos -
0.
+
But the
xsin-i ratio
/(2ar)-2/(tt)+/(0) o"
or 8
a cos (I/ 2 a)
2 a cos (I/
or),
has the limit zero
when a approaches
zero.
For
THE DIFFERENTIAL NOTATION
)14 ]
l
19
THE DIFFERENTIAL NOTATION
III.
which has been in use longer than any it is by no means indispensable, Although other,* it possesses certain advantages of symmetry and of generality which
The
differential notation, is
due to Leibniz.
are convenient, especially in the study of functions of several varia This notation is founded upon the use of infinitesimals. bles.
which approaches zero as small a limit is called an infinitely quantity, or simply an infinitesi the that The condition mal. quantity be variable is essential, for not an infinitesimal unless it is zero. is a constant, however small,
Any
14. Differentials.
variable quantity
which approach zero Ordinarily several quantities are considered standard of compari as the is chosen them of One simultaneously. Let called the principal infinitesimal. Then infinitesimal. another infinitesimal, and ft son,
and
be the principal said to be an
is
is
infinitesimal of higher order with respect to a, if the ratio ft/a On the other hand, ft is called an infini zero with a.
approaches of
tesimal
the
order with respect to a,
first
approaches a limit this case
K
different
^ where
=K+
ft=a(K + c)= Ka
if
a.
Hence
+ at,
Ka
The complementary term is called the principal part of ft. an infinitesimal of higher order with respect to a. In general, such that ft we can find a positive power of a, say
and at
the ratio ft/a In zero.
e,
another infinitesimal with respect to
c is
if
from zero as a approaches
is
a",
zero,
ft
is
K
/a"
from zero as a approaches Then called an infinitesimal of order n with respect to a.
approaches a
finite
limit
different
we have
=K+ 4 a ;
e,
or ft
The term
=
an (K
-f e)
= Ka* +
".
again called the principal part of ft. these definitions, let us consider a continuous func Having given Let Aa; be an tion y=f(x), which possesses a derivative (x). Ka"
is
f
*
With the
possible exception of
Newton
s
notation.
TRANS.
DERIVATIVES AND DIFFERENTIALS
20 increment of
From
x,
and
let A?/
14
[I,
denote the corresponding increment of
the very definition of a derivative,
y.
we have
approaches zero with Ace. If Ax be taken as the principal infinitesimal, AT/ is itself an infinitesimal whose principal part is f (x) Ax.* This principal part is called the differential of y and is
where
c
denoted by dy.
dy=f(x)&x.
When /(x)
reduces to x
itself, the above formula becomes dx and hence we shall write, for symmetry,
= Ax
;
where the increment dx of the independent variable x is to be given the same fixed value, which is otherwise arbitrary and of course variable, for all of the several
functions of x which
may
dependent
be under consid
eration at the same time.
Let us take a curve C whose equation is y = f(x), and consider two points on it, and whose abscissae are x and x -f dx, In the triangle MTN we have respectively.
M
M
,
NT = MN tan Z TMN = dxf (x). Hence NT represents the differential
while
Ay
is
equal to
NM
.
It
is
evident from the figure that
dy,
MT
an infinitesimal of higher order, in general, with respect to NT, is parallel to the x axis. approaches M, unless Successive differentials may be defined, as were successive deriv atives, each in terms of the preceding. Thus we call the differ is
as
M
MT
ential of the differential of the first order the differential of the second order, where dx is given the same value in both cases, as 2 above. It is denoted by d
y:
d*y
= d (dy) =
[/"(x)
Similarly, the third differential
d*y
dx] dx
=
f"(x)
= d(d*y) = [_f(x)dx*]dx
=f"(x)(dx)*,
* Strictly speaking, we should here exclude the case ever, convenient to retain the same definition of dy
even though
it is
(dx}*.
is
not the principal part of Ay.
where f
=f
TRANS.
(x)
= 0.
It is,
(x)&x in this case
how also.
THE DIFFERENTIAL NOTATION
14]
I,
and so (n
1)
The
21
In general, the differential of the differential of order
on. is
/ (or),
derivatives
/"(a),
-,
f
(n
...
\x),
the other hand, in terms of differentials, and tion for the derivatives
can be expressed, on a new nota
we have
:
~ dy
t
y
_ ~
,,
dx
dx
To each of the
M
2
t>
rules for the calculation of a derivative corresponds
a rule for the calculation of a differential.
= mx m dx d log x = d xm
,
dn y ~dtf
For example, we have
da x =
l
dx,
,
d sin x
j
=
a x log a dx cos
x dx
SC ,
.
aarcsmcc
dx
=
Vl -
darctanx
>
a;
=-
2
1
;
;
;
dx
+x
2
Let us consider for a moment the case of a function of a function.
y
=/(), where u
is
a function of the independent variable
x.
whence, multiplying both sides by dx, we get
yx dx =/(M) X ux dx; that
is,
dy =f(u)du.
The formula
for dy is therefore the same as if u were the inde variable. This is one of the advantages of the differential pendent notation. In the derivative notation there are two distinct formulae,
yx=f(u)uxy
&=/(*)>
y with respect to cc, according as y is given directly as a function of x or is given as a function of x by means of an auxiliary function u. In the differential notation the to represent the derivative of
same formula applies in each case.* If y = f(u, v, w) is a composite function, we have Vx at least
if
fu ,f ,fw v
= U xfu + Vx f + Wx fn v
,
are continuous, or, multiplying
yx dx =
u x dxfu
+
v x dxfv
+ wx dxfw
by dx,
;
* This particular advantage is slight, however for the last formula ahove well a general one and covers both the cases mentioned. TRANS. ;
is
equally
DERIVATIVES AND DIFFERENTIALS
22 that
[I,
15
is,
dll
Thus we have,
= f du + f dv +fw dw. u
v
for example,
V du,
V
The same
rules enable us to calculate the successive differentials. Let us seek to calculate the successive differentials of a function
y
= /(u),
We
for instance.
have already
dy=f (u}du. In order to calculate d?y, it must be noted that du cannot be regarded as fixed, since u is not the independent variable. We must then calculate the differential of the composite function f du, where u (u)
We
and du are the auxiliary functions.
To
thus find
calculate d*y, we must consider d*y as a composite function, with d2 u as auxiliary functions, which leads to the
u, du,
expression
d*y
and so
=f
8
+
"(u)du
+f (u)d*u
3f"(u)dud*u
;
It should be noticed that these formulae for d*y, d*y, etc., are not the same as if u were the independent variable, on account of the terms d*u, d z u, etc.*
A
on.
similar notation
is
of several variables.
f(x, is
y, s),
which
is
used for the partial derivatives of a function
Thus the
f
xf>flzr
n of
in our previous notation,
represented by
____
in the differential notation.f
in
partial derivative of order
represented by
This notation is purely symbolic, and no sense represents a quotient, as it does in the case of functions
of a single variable.
.
Let w =f(x,
15. Total differentials.
three independent variables x, o
du * This disadvantage
=
o
/
dx ^ex
+
y,
z)
be a function of the
The expression
y, z.
Q
/
/
^ dy + dz -^ dz dy -
would seem completely
to offset the
advantage mentioned
TRANS. Strictly speaking, we should distinguish between d^y and d?uy, etc. t This use of the letter d to denote the partial derivatives of a function of several variables is due to Jacob! Before his time the same letter d was used as is used for
above.
.
the derivatives of a function of a single variable.
I,
THE DIFFERENTIAL NOTATION
15]
23
where dx, dy, dz are three fixed called the total differential of otherwise which are arbitrary, assigned to the three increments, is
o>,
independent variables
The three products
x, y, z.
8f TT-
df j ~ dz
df j dy,
7
dx.
ex
cz
dy
are called partial differentials. The total differential of the second order ential of
the"
dx, dy, dz remaining the
=
d2 u
same
d*
the total differ
is
order, the increments
first
as
we pass from one
dx
-f -^
differential to
Hence
the next higher.
or,
the
total differential of
_
,
7
.
d(dta)
ddia
=
cdw
ddia
Ox
oy
dy
+ -=cz
dz
;
expanding, ex cz
dx oy
ex*
!
Oy Oz
+
-
2
Ox oy
If cPf be replaced by becomes the square of
We may
dxdy df
2 ,
+
dxdz
2
ox cz
+
2 = dy dz. Oy Oz
the right-hand side of this equation
then write, symbolically,
0x
cy
oz
2 being agreed that df* is to be replaced by 8 f after expansion. of the total differential In general, if we call the total differential order the total differential of of order n, and denote it by
it
(n
1)
dn (a, we may write,
in the
same symbolism,
*.-(*** \0x
where df n
is to
Oy
+*)", Oz /
be replaced by d nf after expansion
ordinary notation,
;
that
is,
in our
DERIVATIVES AND DIFFERENTIALS
[I,
15
where n
A pqr is
p\q\r\
the coefficient of the term ap &
r
c
in the
development of
(a.
+ b + c)
n .
We
n will show that it then For, suppose this formula holds for d w. and this will prove it in general, since we have holds for dn+l
n
for
it
already proved
=
From
2.
the definition,
we
find
dn+l w=d(dn (o)
r
-
zn+if
whence, replacing
en + 1
d n+l f
+
n
/by cf
+ l the right-hand side becomes ,
f
( 1
dx
7T-
dy
-f 7f-
^V
C7
+
rf
I
,
<7
or
cz
cy
\ox
I
Hence, using the same symbolism,
cy
we may -
-
cz
cy Note.
Let us suppose that the expression for dw, obtained in any
way whatever,
is
= P dx
dw
(7)
where P,
write
Q,
R
-f-
Q dy
are any functions x, rfw
=
d
^dx
+
+ R dz, Since by definition
y, z.
8
ay cy
+
d
^cz
dz,
we must have
where dx, dy, dz are any constants.
/\ (o)
S
-
"5~
^X
The
P
.r,
go)
^~
^
-
Hence n y,
8
-
P
-ft.
"5~~
KB
single equation (7) is therefore equivalent to the three separate and it determines all three partial derivatives at once. equations (8)
;
I,
THE DIFFERENTIAL NOTATION
16]
In general,
if
25
the nth total differential be obtained in any
whatever,
= 2 Cpqr
w
d"
dx" dy"
dz
way
r ;
Cyqr
then the coefficients
are respectively equal to the corresponding nth derivatives multiplied by certain numerical factors. Thus all
these derivatives are determined at once.
We
shall
have occasion
to use these facts presently.
written
down
as follows
w
:
dia_dFdii
dFdv
dFdw
du dx
dv dx
dw dx
_d_F_d_u du dy
d_F_d_v_
___
dy
dv dy
dw dy
dz
du dz
do dz
dw
dw
dF du
dF dv
dF dw
du
dv dt
dw
dx d>
=
Let w F(u, v, w~) being themselves functions of the The partial derivatives may then be
16. Successive differentials of composite functions.
be a composite function, u, v, independent variables x, y, z, t.
d_F_d_v
_
dt
dt
dz
dt
If these four equations be multiplied by dx, dy, dz, dt, respectively, and added, the left-hand side becomes d(
+
3- dx dx
that
is, do*
;
and the
,
W
-r-
dy
7
dy
+
^<
,
-^-
dz
7
dz
W
J. +,^-^ dt, d
coefficients of
0F
d]F do
d_F
du
dw
on the right-hand side are du, dv, dw, respectively. do)
(9)
Hence
= ^dF du + dF dv + ^dF dw, -r
cu
dv
cw
and we
see that the expression of the total differential of the first order of a composite function is the same as if the auxiliary functions were the independent variables. This is one of the main advantages
The equation (9) does not depend, in form, either upon the number or upon the choice of the independent variables and it is equivalent to as many separate equations as there are independent variables. of the differential notation.
;
To
calculate
that the second u, v,
d2 w,
let
us apply the rule just found for dta, noting of (9) involves the six auxiliary functions
member
w, du, dv, dw.
We
thus find
DERIVATIVES AND DIFFERENTIALS
26
=
-i^-
4-
dudv
4-
-z
cucu
Ctr 4-
-
du dv d
2
F
d2
du 2
^ cv 2
F
+ ^gw du dw
d
2
4- o
F
du dv
du dw + + cucw
dv 2
+
dv
Q
dw
[I,
dF
lu
dz u
-^-
en
dvdw + ^ ff cw cv
cu
F
d2 TT-^1
-f
+
t?w
Cw
^y^M>
dF ^ cw
simplifying and using the same symbolism as above,
or,
d2 w
= [7^- du+ 7
Vc/w
,
^ dv ^^y
+
^ +
dw\
,
CM;
/
+^
*
TT-
cu
c?
Co
2
,
w 4- ^
cw
.
This formula is somewhat complicated on account of the terms in d 2 u, d z v, dz w, which drop out when u, v, w are the independent This limitation of the differential notation should be variables. 2 borne in mind, and the distinction between d w in the two cases s To determine d w, we would apply the same rule carefully noted. to
2 <2
o>,
2 noting that d w depends upon the nine auxiliary functions
2
2
2
w, du, dv, dw, d u, d v,d w; and so forth. The general expres sions for these differentials become more and more complicated u, v,
dn w
an integral function of du, dv, dw, d 2 u, the terms containing dn u, dn v, dn w are is
dF cu If, in
dF
7
dn u
the expression for
4-
d"
dF
,
dn v
cv
w, u, v,
4-
cw
,
;
dn u, d n v, dn w, and
7
d n w.
w, du, dv, dw,
be replaced by
n their values in terms of the independent variables, d t becomes an coefficients are equal in whose dx, dy, dz, integral polynomial
(cf.
by
Note, 15) to the partial derivatives of w of order n, multiplied thus obtain all these derivatives certain numerical factors.
We
at once.
Suppose, for example, that we wished to calculate the first and second derivatives of a composite function
=
If we calculate function of two independent variables u (x, y). these derivatives separately, we find for the two partial derivatives of the first order
1ft
8w
dx
_
8w du
du dx
8u)
dy
_
du>
du
du dy
Again, taking the derivatives of these two equations with respect to x, and then with respect to y, we find only the three following distinct equations, which give the second derivatives :
THE DIFFERENTIAL NOTATION
27
du\*
(11)
du
ex
dx*
dx dy
U CU
du* Cx dy
C
d*
du dx dy
&u
d-
22
da
dy
t
,2*
The second
of these equations is obtained by differentiating the of equations (10) with respect to y, or the second of them with In the differential notation these five relations (10) respect to x.
first
and (11) may be written
in the
form
en (12)
cu If
du and d*u
in these formulae be replaced
du
and
dy
TT-
-^ a
by
dx dy
in the first give the first respectively, the coefficients of dx and dy z of dx , 2 dx dy, and coefficients while the of o, partial derivatives
dy
2
in the second give the second partial derivatives of w.
The formula for the total differential n of a composite function becomes considerably simpler in certain special cases which often arise in practical applications. Thus, let us seek the differential of order n of the product of two = uv. For the first values of n we have functions 17. Differentials of a product.
of order
o>
dw
=
v dti
and, in general, d"
w
=
v
+
u
it is
d"
u
dv,
d* a)
= v d* u +
2 du dv -f ud* v,
;
evident from the law of formation that
4- r,
dr d n ~^u
+ Cd*v d
n
~2
n
-f
+
It might be shown by alge where C lt C 2 are positive integers. braic induction that these coefficients are equal to those of the ,
but the same end may be reached by the expansion of (a + following method, which is much more elegant, and which applies do not depend to many similar problems. Observing that C l C 2 &)"
;
,
,
upon the particular functions n and v employed,
let
us take the
DERIVATIVES AND DIFFERENTIALS
28
[I,
17
u = e*, v = &, where x and y are the two inde pendent variables, and determine the coefficients for this case. We special functions
thus find
w
=
e x+y ,
dw
=
e x+y
(dx
+ dy),
du =
e*dx,
dv
e v dy,
dn
-,
<*
=
d u = ex dx*, d*v = e dy
x e +y
(dx
+ dy)n
,
z
,
2
y
,
;
and the general formula, after division by ex+ J becomes ,
(dx
+ di/} n =
dx*
+
C^dydx*-
Since dx and dy are arbitrary,
r Cl
_n ~l
n(n-l)
1
C t dy 2 dx n - 2
+
[-dp.
-\
follows that
it
~ n(n -1)
"
p
1.2
-
(n
- p + 1)
1.2-..p
and consequently the general formula may be written n (13) d (uv)
= vd u+^dud 1 n
n-
l
u
+
^
7
1
.
^ 4
d 2 vdn
~2
u
-\
\-ud*v.
This formula applies for any number of independent variables.
u and v are functions of a single variable x, we n by dx the expression for the nth derivative of the product of two functions of a single variable. It is easy to prove in a similar manner formulae analogous to (13) for a product of any number of functions. Another special case in which the general formula reduces to a simpler form is that in which u, v, w are integral linear functions In particular,
if
have, after division
,
of the independent variables x, y,
where the have
u=
ax
v = w=
a"x
coefficients a,
a
a x
z.
+ cz+f, + b y + c z +/ + + by
-f
,
c"z
+/",
b"y
,
= dv = dw = du
a",
b,
b
a dx
+
a dx
-f-
a"dx
+
are constants.
,
b b
+ dy + + dy
b"dy
c dz, c
dz,
c"dz,
the differentials of higher order d u, dn v, dn iv, where n>l, vanish. Hence the formula for dn is the same as if u, v, w were
and
n
For then we
all
<j>
the independent variables
;
that
is,
I,
THE DIFFERENTIAL NOTATION
18]
dn w
We proceed
8F = (dF -5- du + -T- dv
homogeneous
of degree m, <(w,
is
4- 5
,
dw
V">
I
.
to apply this remark.
A
Homogeneous functions.
18.
8F
.
.
29
u
=
(x,
w)=
v,
when we
identically satisfied
function
v
tx,
is
y, z)
said to be
the equation
if
t
m
$(x,
y, z)
set
w=
= ty,
tz.
equate the differentials of order n of the two sides of this with respect to t, noting that u, v, w are linear in t, and that equation
Let
xis
du
The remark
ihi
If
+
x
+ set
"
>==
*^) #
=
z dt.
shovvs that (
d
y fo
dw =
= y dt,
dv
dt,
made
just
we now
=
m(m ~ 1}
1, w, v,
the development of the
first
w
"
(m
reduce to
"
#,
n
?/,
+1 ) 2,
a;
<m
""*(
y*)-
and any term of
member,
becomes d"<>
whence we may
which reduces,
write, symbolically,
for
n
= 1,
to the well-known formula
We
have then, altogether, three systems of nota Various notations. tion for the partial derivatives of a function of several variables, Each of that of Leibniz, that of Lagrange, and that of Cauchy. these
is
somewhat inconveniently long, especially in a complicated For this reason various shorter notations have been Among these one first used by Monge for the first and
calculation.
devised.
DERIVATIVES AND DIFFERENTIALS
30
[I,
19
second derivatives of a function of two variables is now in common use. If z be the function of the two variables x and y, we set
P and the
t
ex 2
dy total differentials dz
and d
=
ex 8y 2
z are given
= p dx + q dy, d z = r dx 2 s dx dy +
o if
by the formulae
dz 2
2
t
-f-
dy~.
is now coming into general use is the Let z be a function of any number of independent vari following. x n then the notation ables x 1} x z x 3)
Another notation which
,
,
;
ex is
l
ex.
where some of the indices a lt
used,
ox
2
a.2)
a n may be
,
zeros.
19. Applications. Let y f(x) be the equation of a plane curve C with The equation of the tangent at a point respect to a set of rectangular axes.
M(x,
is
y)
Y-y = y The
slope of the normal,
tangency,
l/y
is
;
which
(X-x).
perpendicular to the tangent at the point of and the equation of the normal is, therefore, is
N
P
Let be the foot of the ordinate of the point Jlf, and let T and be the points of intersection of the x axis with the tangent and the normal, respectively. The distance is called the subnormal
PN
;
the subtangent; MN, the normal; and T, the tangent. From the equation of the normal the ab
FT,
M
scissa of the point
N
subnormal
.
+
is
x
If
we
yy
,
whence the
agree to call the the subnormal, and to attach the length sign + or the sign according as the direc tion is positive or negative, the subnormal is
yy
PN
PN
will
C.
The
lengths
Various problems a.
.
MN and M T are given by the triangles MPN and MPT:
may
instance, all the curves for
number
always be yy for any position of the curve Likewise the subtangent is y /y
be given regarding these lines. Let us find, for is constant and equal to a given
which the subnormal
This amounts to finding
the equation yy
=
a.
The left-hand
all
the functions
side
is
y=f(x) which
the derivative of
2 2/
satisfy
/2, while the
I,
EXERCISES
EXS.]
These functions can therefore
right-hand side is the derivative of ax. only by a constant whence ;
y
= 2ax +
2
31 differ
C,
the equation of a parabola along the x axis. Again, if we seek the curves for which the subtangent is constant, we are led to write down the equa
which
is
y /y
tion
= l/;
whence log2/
which
is
tote.
To
=
-
a
+ logC,
or
y
=
Ce?,
the equation of a transcendental curve to which the x axis is an asymp find the curves for which the normal is constant, we have the equation
or
/a2
The
first
member
is
the derivative of
(x
which
C)
y*
- Vo^-
y
=
a2
2
+
y
2
2 ;
hence
,
the equation of a circle of radius a, whose center lies on the x axis. curves for which the tangent is constant are transcendental curves, which
is
The
we
+
-
shall
later.
study
= f(x)
Y
and let and F(x) be the equations of two curves C and In order that be the two points which correspond to the same value of x. the two subnormals should have equal lengths it is necessary and sufficient that Let y
M,
M
C",
YY that
is,
that
Y
2
directed in like
=yy
;
y + C, where the double sign admits of the normals being This relation is satisfied by the cirfves or in opposite senses. 2
and also by the curves
which gives an easy construction for the normal
to the ellipse
and
to the hyperbola.
EXERCISES 1.
= f(6) be the equation of a plane curve in polar coordinates. Through
Let p
the pole O draw a line perpendicular to the radius be the points where this vector OM, and let T and
N
line cuts the tangent
and the normal.
sions for the distances
OT, ON,
MN,
Find expres
and
MT
in
terms of /(0) and / Find the curves for which each of these distances, (
in turn, 2.
is
constant.
Let y
skew curve
= f(x), T,
i.e.
z(x)
be the equations of a Let
of a general space curve.
N
FIG. 3
DERIVATIVES AND DIFFERENTIALS
32
be the point where the normal plane at a point meets the z axis and tangent at
lar to the
M to
PN and JOT, 3.
P
;
the z axis.
in turn, is constant.
These curves
[Note.
that
let
Exs.
is, the plane perpendicu be the foot of the perpen Find the curves for which each of the distances
.M",
dicular from
Af,
[I,
lie
on paraboloids of revolution or on spheres.]
Determine an integral polynomial /(z) of the seventh degree in x, given + 1 is divisible by (x - I) 4 and f(x) - 1 by (x+1)*. Generalize the
that f(x)
problem. 4.
Show
that
if
the two integral polynomials
Vl -p-t
= Q Vl -
P and x
Q
satisfy the relation
2 ,
then
dP
ndx Vl - x2
Vl - p* where n
a positive integer.
is
From
[Note.
the relation
l-P2 = Q2(l-x)
(a) it
follows that
- 2 PP = Q [2 Q (l - x*) -
(b)
The equation by Q-] and
where R\
Show
where
let
(t) is
x
shows that
A;
is
prime to
P
and
;
(b)
shows that
P
divisible
all dif
a polynomial of the fourth degree and P / Q is a rational function. U/ satisfies a relation of the form
V
dx
kdt
VR(X)
Vfl!()
_
[JACOBI.]
Each root of the equation R(U/ V) = 0, since must cause VU and hence also dx/dt,
UV
Show
is
V
a constant.
is
to vanish,
Q
be a polynomial of the fourth degree whose roots are = U / be a rational function of t, such that
that the function
[Note.
6*.
E (x)
Let
5*.
ferent,
(a)
2 Qx].
,
it
cannot cause
R
(x)
to vanish.]
that the nth derivative of a function y = $ (u), where may be written in the form
u
is
a func
tion of the independent variable x,
where
~ ctx
1.2
(*=1,
2,
[First notice that the nth derivative coefficients
AI, A*,
-,
An
may be written in the form (a), where the are independent of the form of the function <j>(u).
I,
EXERCISES
EXS.]
To
find their values, set
Show
7*.
A^,
it
An
,
+
2 )
(>(x
,
n(n
the form
and
solve the
(b).]
X )n- 2P 0("-P)(x2 )
+
,
p
ju
.
is
is
)
(2 1
successively,
result
-V--( n -*P+V
n (n
+
(x
2
dx n
,
The
.
2
that the nth derivative of
"
un
2
equal to w,
(M)
A
resulting equations for
33
where p varies from zero to the last positive integer not greater than n/2, and where 0(0 (x2 ) denotes the ith derivative with respect to x.
Apply
this result to the functions er 3?, arc sin x, arc tan x.
8*.
x
If
=
show that
cos u,
d-i(l
-x 2
=
)">-*
S
dx m ~ l
m ~i 1.3.5-
-(2m-
,
l)
(
m
1)
.
sin
mu.
[OLINDE RODRIGUES.] 9.
Show that Legendre
s
polynomial,
2 satisfies
4
.
.
6
n
dx"
the differential equation
ax
ax-
1
Hence deduce the 10.
2
Show
coefficients of the polynomial.
that the four functions
yt
y2
= =
sin (n arc sin x),
2/3
cos (n arc sin x),
2/4
= sin (n arc cos x), = cos (n arc cos x),
satisfy the differential equation (1
-
-
x2 )
Hence deduce the developments
y"
xy
+
ri*y
=
Q.
when they reduce
of these functions
to poly
nomials. 11*. Prove the formula i
dn
= _(x-iei) dx V
G* -
(-!)"
x+! [HALPHEN.]
12.
Every function
of the
form rx
The function
z
+ ^.
= x0(x +
y)
=
z
and
whatever be the functions 13.
2
and
\f/.
+
ty*
=
+ y^(x + y) -2s+t=
r
whatever be the functions
x$(y/x) + $ (y/x)
2 sxy
satisfies the
0,
satisfies 0,
the equation
equation
DERIVATIVES AND DIFFERENTIALS
34 14.
The function
=f[x +
z
satisfies the
(y)]
equation ps
[I,
=
qr,
Exs
whatever
be the functions / and 0. 15.
The function
z
=
+
rx 2
and
<j>
Show y
where fa
+
2 sxy
whatever be the functions 16.
+
x<j>(y/x)
y~ n ^(y/x)
ty
+ px +
2
satisfies the
qy
=
equation
n 2 z,
\f/.
that the function
-
x
-
ai 0! (x) |
+
x
-
a z fa (x) \
+
+
x
-
|
an
\
n (x),
(x), (x), together with their derivatives, 0i (x), (x), 0n(x), are continuous functions of x, has a derivative which is discontinuous for x = a\ Oz an
fa
,
,
,
n (x),
,
,
17. Find a relation between the first and second derivatives of the function =/(&! M), where M = 0(x2 x 3 ); x t x2 x 3 being three independent variables, and /and two arbitrary functions. ,
18.
Let/"(x)
,
be the derivative of an arbitrary 1 d*u,
u dx 2 where u 19*.
may
=
[/ (x)]-i and
The nth
,
=/(x) [/
_
1
#2
v
dx 2
f unction
/(x).
Show
that
(x)]-*.
derivative of a function of a function
u-
where y
= ^ (x),
be written in the form
^1.2,
where the sign of summation extends over all the positive integral solutions of the equation i + 2 j + 3 h + Ik = n, and where p = i + j + + k. -\
[FA A DE BRUNO, Quarterly Journal of Mathematics, Vol.
.
I, p.
359.]
CHAPTER
II
FUNCTIONAL DETERMINANTS CHANGE OF VARIABLE
IMPLICIT FUNCTIONS
IMPLICIT FUNCTIONS
I.
20.
A
We
particular case.
frequently have to study functions for
which no explicit expressions are known, but which are given by means of unsolved equations. Let us consider, for instance, an equation between the three variables x, y, z, F(x,
(1)
y, z)
= 0.
This equation defines, under certain conditions which we are about to investigate, a function of the two independent variables x and y.
We
shall prove the following
Let x
=x
y
,
=.
y
,
z
=
theorem
z b&
a
set
:
of values which satisfy the equa
us suppose that the function F, together with its first in the neighborhood of this set of values* is continuous derivatives, z vanish for x = x y there F does not derivative the y z z If
and
tion (1),
let
,
=
=
,
,
and only one continuous function of the independent variables x and y which satisfies the equation (1), and which assumes the value z when x and y assume the values x and y respectively. exists one
,
=
=
=z
x y The derivative Fz not being zero for x y z Since is that it F, Fx positive. suppose, for defmiteness, I
sets of values x, y, z (2)
and
,
,
,
let
Fv F ,
z
us are
us choose a positive are continuous for all functions four so small that these
supposed continuous in the neighborhood,
number
,
\x-x
which
satisfy the relations
\y-y
\
\*-z
\
that, for these sets of values of x, y,
F
z
*Iu a recent
let
(x,y,z}
>
z,
P,
Mathematique de France, Vol. XXXI, Goursat has shown, by a method of successive approximations, that it is not necessary to make any assumption whatever regarding x and F even as to His general their existence. His proof makes no use of the existence of Fx and Fy TRANS. theorem and a sketch of his proof are given in a footnote to 25. 35 article (Bulletin de la Societe
190. ?, pp. 184-192)
F
t/
.
,
FUNCTIONAL RELATIONS
36
[II,
20
P is some positive number. Let Q be another positive num ber greater than the absolute values of the other two derivatives Fx Fy in the same region. where
,
x, y, z
Giving
down
write
F
- F(*o,
V, *)
(*>
values which satisfy the relations
the following identity 7/0,
= F(x, y, z} - F(x
)
(2),
we may then
:
-F(x
,
T/o,
z)
,
+
y, z)
+F(x
,
F(x y, F(x
z)
,
z)
7/0j
,
7/0,
)
5
applying the law of the mean to each of these differences, and = 0, observing that F(x y ) or,
,
F(x>y>*)
,
=
T
*o)-F a r[o
(*
+ - yo) Fv [*., + (z - ) F 2
Hence
-F(cc,
S
(3)
T/,
F^
2) is of
the form
^=
A (x y +B(x, y,
y
I
>
>
[>o,
+
8(x
2/o ?/o,
ff(y
C (x,
where the absolute values of the functions A(x, C(x,
y, z)
y,
*] ]
),
0"0
^ (x ~ xd - )+ z) (y T/O
o),
-y *o +
+
)].
y, z) (z y, z),
-*
B(x,
),
y, z),
satisfy the inequalities M|
\B\
\C\>P
which satisfy (2). Now let c be a number less than and the smaller of the two numbers positive Z, rj I and x that and Pe/2Q. Suppose y in the equation (1) are given definite values which satisfy the conditions for all sets of values of x, y, z
and that we seek the number of roots of that equation, z being regarded as the unknown, which lie between z e and z + c. In the expression (3), for F(x, y, z} the sum of the first two terms is always less than 2Qrj in absolute value, while the absolute value of the third term is greater than Pe when z is e. From replaced by z the manner in which 77 was chosen it is evident that this last term determines the sign of F.
and F(x, y, z + e ) which lies between z >
since the derivative
and
z
+
only one
X Q and
?/
;
It
follows, therefore, that F(x, y, z e) hence the equation (1) has at least one root <
e
F
z
is
and
z
+
e.
Moreover this root
is
unique,
positive for all values of z between z
e
It is therefore clear that the equation (1) has one and root, and that this root approaches ZQ as x and y approach e.
,
respectively.
IMPLICIT FUNCTIONS
20]
II,
37
Let us investigate for just what values of the variables x and y Let h be the root whose existence we have just proved is denned. the foregoing reason the smaller of the two numbers I and PI/2Q; x and of the variables y satisfy the ing shows that if the values one h, the equation (1) will have inequalities \x
<
x^\
and only one root which
between
lies
and z y ), with
z
I
about the point M (x
-f
Let
I-
R
be a
sides parallel , square of side 2 h, this square, inside lies As long as the point (x, y) to the axes. x and of function a determines y, which the
equation (1) uniquely I and z remains between z the above, at the point
M
M
,
+
This function
I.
its
is
continuous, by any other
this is likewise true for
and
by the hypotheses made regarding the func tion F and its derivatives, the derivative Ft (x lf y l} i) will be posi z i~ z x since \x l tive at the point lt ya\
of
l
R;
for,
M
\
M
o\
M
,
consideration will be continuous for
and hence the root under
defined only in the interior of the function. region R, we have thus far only an element of an implicit out In order to define this function
Since the root considered
is
we proceed by successive Let L be a con follows.
side of R, steps, as
tinuous path starting at the point (x y and ending at a point (X, F) Let us suppose that outside of R. ,
~)
the variables x and y vary simul taneously in such a way that the point If
we
z
of
(x, y)
describes the path L.
start at (x z,
,
we have
y
Fm
4
with the value
)
a definite value of this root as long as we remain Let 1 (x l y^ be a point of the path inside R,
M
inside the region R.
and
~
,
z t the corresponding value of
x
=x
y=y
The
z.
=z
conditions of the theorem
there exists another region lt v l} being R l} about the point MI, inside which the root which reduces to z l for satisfied for
x
= Xi,
y
= yi
is
z
,
This
uniquely determined.
new
region #! will
M
Taking then such a point t but outside R, we may repeat the same con
have, in general, points outside of R.
on the path L, inside R struction and determine a new region
R2
,
inside of
which the solu
this process could be defined; (1) find as as we did not a set of values of repeated indefinitely, long z F which for the present for 0. shall content ourselves x, y, z
tion of the equation
=
is
We
and
FUNCTIONAL RELATIONS
38
[II,
21
with these statements; we shall find occasion in later chapters to treat certain analogous
problems in
detail.
Let us return
Derivatives of implicit functions.
21.
R, and to the solution z
= (x,
to the region
y) of the equation (1),
which
is
a
continuous function of the two variables x and y in this region. This function possesses derivatives of the first order. For, keeping y fixed, let us give x an increment Ax. Then z will have an incre
ment
Az, and
we
find,
by the formula derived
As, y,z
+ A) -
F(x,
= Az Fx (x +
0Az, y,e
+ Az)
F(x
+
in
20,
z)
ij,
-f
Aa
F
t
z (x, y,
+
Az)
= 0.
Hence
and when A# approaches zero, As does also, since z is a continuous The right-hand side therefore approaches a limit, a;. and z has a derivative with respect to x
function of
:
In a similar manner
we
find
If the equation F = functions of the variables x and
is
y,
of degree
m
in
z,
it
defines
m
and the partial derivatives cz/cx,
values of the variables 3z/dy also have m values for each set of x and y. The preceding formulas give these derivatives without ambiguity, if the variable z in the second member be replaced by the value of that function whose derivative is sought.
For example, the equation
defines the
two continuous functions
+ Vl for values of
The
x2
y*
Vl
and
x2
y
2
x and y which satisfy the inequality x-
first partial
derivatives of the
first
are
-y
+
y
2 <
1.
II,
IMPLICIT FUNCTIONS
2]
and the
39
the second are found by merely chang partial derivatives of The same results would be obtained by using the
ing the signs. formulae
dz
_
x
Cz
z
cy
dx replacing z by
its
y z
two values, successively. If
Applications to surfaces.
22.
_
we
interpret x, y, z as the Cartesian
coordinates of a point in space, any equation of the form
F(x,y,
(4)
z)=0
represents a surface S. Let (cc y z^) be the coordinates of a point A of this surface. If the function F, together with its first deriva tives, is continuous in the neighborhood of the set of values x yw z ,
,
,
,
three of these derivatives do not vanish simultaneously at the point A, the surface S has a tangent plane at A. Suppose,
and
if all
for instance, that
F
is
z
not zero for x
=x
,
y
=y
,
z
=
Accord
.
ing to the general theorem we may think of the equation solved for z near the point A, and we may write the equation of the surface in the form z 4(x, y},
=
(x, y)
is
a continuous function
tangent plane
at
A
where
;
and the equation of the
is
Replacing dz /dx and dz /dy by the values found above, the equation of the tangent plane becomes
If
F = z
0,
but
Fx
=
pendent variables and
0, at a;
A
,
we would
consider y and z as inde
as a function of them.
We
would then
same equation (5) for the tangent plane, which dent a priori from the symmetry of the left-hand side. find the
the tangent to a plane curve F(x, y)
=
0,
at a point (x
,
is
also evi
Likewise
y
~),
is
If the three first derivatives vanish simultaneously at the point A.
dF
FUNCTIONAL RELATIONS
40
[II,
23
We
shall see later the preceding reasoning is no longer applicable. to the various curves which lie on that the tangents (Chapter III)
the surface and which pass through not a plane.
A
form, in general, a cone and
In the demonstration of the general theorem on implicit functions Our geometrical that the derivative F^ did not vanish.
we assumed
intuition explains the necessity of this condition in general. For, is parallel to the % axis, the if but 3= tangent plane 0, F^ F^
=
line parallel to the z axis and near the line x = x w y = y meets the surface, in general, in two points near the point of Hence, in general, the equation (4) would have two tangency. roots which both approach z when x and y approach x and y
and a
,
respectively. If the sphere
y = 0, with
e
23.
x ;
=1+
a;
2
+y
-+-
we
find
they are real
if c
c,
1 = 0, for instance, be cut by the line two values of z, which both approach zero is negative, and imaginary if c is positive. 2
2
Successive derivatives.
In the formulae for the
3z = _Fx dx~ F,
= cy~ dz_
first
derivatives,
_FJL F,
consider the second members as composite functions, z being an auxiliary function. We might then calculate the successive deriv The atives, one after another, by the rules for composite functions. existence of these partial derivatives depends, of course, upon the
we may
existence of the successive partial derivatives of F(x, y, K). The following proposition leads to a simpler method of determin ing these derivatives.
If several functions of an independent variable satisfy a relation
F=
obtained by equating to 0, their derivatives satisfy the equation zero the derivative of the left-hand side formed by the rule for differ vanishes For it is clear that if entiating composite functions.
F
when the
variables which occur are replaced by func identically tions of the independent variable, then the derivative will also van ish identically.
which
The same theorem holds even when
satisfy the relation
F=
the functions
depend upon several independent
variables.
Now suppose that we wished to calculate the successive derivatives of an implicit function y of a single independent variable x defined by the relation
II,
IMPLICIT FUNCTIONS
23]
We
41
find successively
dF~ T~ ox
+2 ^
*
dxdy
v y
d2
+
F
v
2
+
SF~
,
cy
+^
dF y
v"
dy
-dy
=0
2
2
ox dy
dx*
3
y
ox dy*
*
dxdy
dy 32
from which we could calculate successively y
s
7,1
,
y
y",
Example. Given a function y =/(x), we may, inversely, consider y as the independent variable and x as an implicit function of y defined by the equation y=f(x). If the derivative / (x) does not vanish for the value XQ, where 2/o =/(zo)i there exists, by the general theorem proved above, one and only one function of y which satisfies the relation y
= f(x)
and which takes on the value
XQ for y = 2/0- This function is called the inverse of the f unction /(x). To cal of this function, we need merely culate the successive derivatives x y x y t, ay, differentiate, regarding y as the independent variable, and we get ,
1
= / =
(x)
=/"
(x) (x y )*
xy
,
2
/"(x)
(X,)
+ / (x) ay, + 3f"(x)x y x? +/
(x)x
3,
2/
whence 1
_8[/"(x)]-
f"(z)
~[7w
~7^)
[/
It
should be noticed that these formulae are not altered
/
(x), Xy2
the
and /"(x),
Xj,s
and
/"
(x),
,
for
it is
(
if
we exchange
x v and
evident that the relation between
= /(x) and x = (y) is a reciprocal one. application of these formulae, let us determine all those functions which satisfy the equation
two functions y
As an
y=f(x)
-
"
yy
3y"*
=
0.
Taking y as the independent variable and x as the function, becomes
=
this equation
0.
Xj/>
But the only functions whose third derivatives are zero are polynomials of most the second deree. Hence x must be of the form
at
C2 , C3 are three arbitrary constants. Solving this equation for y, see that the only functions y = /(x) which satisfy the given equation are of the form where Ci,
we
y
=
a
_
V bx + c,
FUNCTIONAL RELATIONS
42 where a, 6, whose axis
c is
two
variables,
Let us now consider an implicit function
denned by the equation
= 0.
F(x,y,z)
(6)
The
24
This equation represents a parabola
are three arbitrary constants. parallel to the x axis.
24. Partial derivatives.
of
[II,
first order are given, as partial derivatives of the
we have
seen,
by the equations
9F.9F9* p 7T-
(7)
o
?l
u,
9* 0z
0x
-^
h
dFdz_ ;p
fl
u.
0#
partial derivatives of the second order we need two equations (7) again with respect to x and the differentiate only with respect to y. This gives, however, only three new equations,
To determine the
for the derivative of the first of the equations (7) with respect to y is identical with the derivative of the second with respect to x.
The new equations
are the following:
dx 2 d
2
F
.OJL.? + dxdzdx d2
d*F dz ^
,
F
d_z_
""
~F +
d2
d2
F
dz
"
n
fe dy
ftr
+
d2
F
dz
2
dz
\dx] dz dz
r\
dx
/!?)%-
dz 2
r\
o
I
dx dy 2
(dz\ (dy)
dx 2
d*z d_F O O O c~
dF
+^
=o _
^J
d2 z :
dy
2
third and higher derivatives may be found in a similar manner. By the use of total differentials we can find all the partial deriva
The
tives of a given order at the
following theorem
z,
the relation
This depends upon the
w, of any number of independent vari 0, the total differentials satisfy satisfy a relation F total differential 0, which is obtained by forming the
If several functions ables x, y,
same time.
:
u, v,
=
dF=
of F as if all the variables which occur in
F ivere independent variables.
=
be the given relation between In order to prove this let F(u, v, w) the three functions u, v, iv of the independent variables x, y, z, t. The first partial derivatives of M, v, w satisfy the four equations
dFdu __
__
du dx d_Fd_u
du dy
dv dx
dw dx
dv dy
dw dy
,
IMPLICIT FUNCTIONS
II,24J
dFdv ~
dFdu ~ o
d_F_d-w_
~o
CM
tf
dFdu
a?
CZ =
a^
a<
st
Multiplying these equations by dx, dy, dz,
we
adding,
^)
~a
^^! =
d_Fd_
&
aw
o OW
I
tfS
43
dt,
respectively, and
find d = dF=0. + -/-dw + ^-dv -^du dv OW du
This shows again the advantage of the differential notation, for the preceding equation
is
independent of the choice and of the number To find a relation between the second
of independent variables.
we need merely apply the general theorem to the considered as an equation between u, v, w, du, 0, equation of higher order than the The differentials forth. and so dv, dw, first of those variables which are chosen for independent variables total differentials,
dF =
must, of course, be replaced by zeros. Let us apply this theorem to calculate the successive total differ entials of the implicit function defined by the equation (6), where x and y are regarded as the independent variables. We find *
Fi
dx
ox
dF
T- dx dx
dF
+ -z-
dy
F ^ + 7
cy
8F
dy
3F j ^ = + 7cz dz
j a;/
+ -rdz
dz
V
2 >
+ ,
)
/
dF n
cz
0,
= 0,
two of these equations may be used instead of the five equations (7) and (8) from the expression for dz we may find the two first derivatives, from that for d^z the three of the second order, and the
first
;
etc.
Consider for example, the equation
Ax 2 which
gives, after
two
+
A
y*
+
A"z*
=
l,
differentiations,
Ax dx + A ydy + A dz = 2 A dx + A dy 2 + A + A zd*z = "z
2
0,
"
"dz
whence dg
~
-- Axdx + A
0,
ydy,
TTi
A"z
and, introducing this value of dz in the second equation,
A
(A x*
+A
2 "z
}
dx*
+ 2AA xy dx dy + A
(A y*
we
+A
find 2
"z
}
dy*
FUNCTIONAL RELATIONS
44 Using Monge
s notation,
A(Ax*
+
24
we have then
Ax
p=
[II,
~IV AA
_~
A"z*)
A y
~
q
~IV
xy
"**
evidently general, whatever be the number of the derivatives which independent variables or the order of the partial it is desired to calculate.
This method
is
Example. Let z = /(x, y) be a function of x and y. Let us try to calculate 2 the differentials of the first and second orders dx and d x, regarding y and z as the independent variables, and x as an implicit function of them.
First of
all,
we have dz
=
+
dx
dy.
dx
Since y and z are
now
the independent variables,
d*y
and consequently a second
=
dy
=
dxdy
dy*
Monge s notation, using p, q, r, s, equations may be written in the form
= we
find ,
dx=
r
q dy,
+
dx2
-
dz
for the derivatives of /(x, y), these
t
p dx +
dz
first
0,
^dx + 2 ^- dxdy + ?^-dy* + ^d?x. dx x
In
the
set
differentiation gives
2
From
=
d2 z
we must
q
2 s dx
dy
+
tdy*
+ p d2 x.
dy
-,
P
and, substituting this value of dx in the second equation,
The z,
first
-
rdz*
and second
+ 2(ps-qr)dydz + (q*r -2pqs partial derivatives of x, regarded as a function of
therefore, have the following values
dx dz
d*x dz 2
As an which
_
r
p
8
dz x
dy dz
_
1
8x
p
dy
_qr
ps
p
3
_
q
p d2 x
dy
_ 2pqs
2
application of these formulae, let us find
satisfy the equation
y and
:
=
pH
q*r
ps all
those functions /(x, y)
2pqs.
=/(x, y), x be considered as a function of the two inde pendent variables y and z, the given equation reduces to Xyt = 0. This means If, in the
equation z
IMPLICIT FUNCTIONS
25]
II,
that x v
is
function of
independent of y z.
which shows that x
-
y
and hence xv
;
This, in turn,
may
is (z)
=
45
where
0(z),
is (z)
an arbitrary
be written in the form
independent of
Hence we may write
y.
It is clear, therefore, that all the is another arbitrary function of z. functions z =/(x, y) which satisfy the given equation, except those for which fx This equation represents vanishes, are found by solving this last equation for z.
where ^ (z)
a surface generated by a straight line which
The general theorem.
25.
is
always parallel to the xy plane.
Let us consider a system of n equations }
xpi u
" l>
**ll
i
(E) *
x
(x
u
x
u
w)
un x l} x y between the n-\-p variables u i} u 3 , that these equations are satisfied for the values x v ,
u
=
wj,
j
un
= un
that the functions
;
first partial derivatives
this system
;
F
i
= ,
,
xp
Suppose
.
xp = xp are continuous and possess x\,
,
,
which are continuous, in the neighborhood of
of values; and, finally, that the determinant
du
does not vanish for
uk
x,-
=
Under these conditions there
=
i(xi,
for
x,
= x\,
,
= p
and only one system of con x2 xp un n (x x^), un and which reduce to u\, u\,
exists one
x2 tinuous functions u^ which satisfy the equations (E) x* * x
,
,
,
<}>
1,
,
,
,
~)
,
p
his paper quoted above (ftn., p. 35) Goursat proves that the same conclusion be reached without making any hypotheses whatever regarding the derivatives remain cFi/dXj of the functions { with regard to the x s. Otherwise the hypotheses exactly as stated above. It is to be noticed that the later theorems regarding the would not follow, however, without existence of the derivatives of the functions some assumptions regarding dFf /dXj. The proof given is based on the following
*In
may
F
4>
FUNCTIONAL RELATIONS
46
The determinant A ables u l}
is
called the Jacobian,* or the Functional Deter
n functions
minant, of the it?,
un
,
.
Fu F
2,
Fn
-,
with respect to the n vari by the notation
It is represented
D(F F2 lf
We
26
[II,
,
...,F,,)
by proving the theorem in the special case of a two equations in three independent variables x, y, z and two unknowns u and v. will begin
system of
= 0, Fi(x,y,z,u,v) = Q. y, z, u, v)
Fi(x,
(9)
(10)
=x
These equations are satisfied, by hypothesis, for x u = u v = v and the determinant ,
,y
= y ,z = z
,
;
_
dF\ dFj du cv
dF\ dFt dv cu
does not vanish for this set of values.
It follows that at least
one
of the derivatives dF^/dv, dF2 /dv does not vanish for these same values. Suppose, for definiteness, that oFl /8u does not vanish. According to the theorem proved above for a single equation, the relation (9) defines a function v of the variables x, y,
v
which reduces
to v
in the equation (10) x, y, z,
and
=f( x
=x
y,
,
x
by
this function,
,
*>
y=y
for
z, u,
)>
z
,
we
= zw
u
=
u
Replacing v
.
obtain an equation between
u,
$(,
y, z, u}
=F
t
[x, y, z, u,
f(x,
y, z,
)]
=
0,
lemma: Let f\(x\,3kt,---,v p MI, u 2 u n ) be n ,u n ), ,/(!, x?, ,x p MI, u 2 functions of the n + p variables X{ and u^, which, together with the n 2 partial deriva tives cfi/GUfr, are continuous near Xi x p = 0, HI = 0, u n = 0. If 0, x z = 0, the n functions f{ and the n 2 derivatives dfi/^Uf. all vanish for this system of values, then the n equations ,
;
;
,
i=/i.
2
=/2,
,
,
,
=/
"
admit one and only one system of solutions of the form
where all
continuous functions of the p variables Xi, x 2 xp which 1( 2 n a approach zero as the variables all approach zero. The lemma is proved by means of m ~l -1) >
,
>
a suite of functions
where
M^
O)
= 0.
limiting function solution.
u^ =f (x i
It is
U
{
,
1
,x z
,
,x p
\
u[
\
w^"
,
,
u^
^)
(i
,
= l,
2,
,
n),
shown that the suite of functions u\ m) thus denned approaches a which 1) satisfies the given equations, and 2) constitutes the only
The passage from
the
lemma to
the theorem consists in an easy transforma
tion of the equations (E) into a form similar to that of the *JACOBI, Crelle s Journal, Vol. XXII.
lemma.
TRANS.
IMPLICIT FUNCTIONS
25]
II,
which
for
is satisfied
x
=x du
and from equation
y
,
^
y
,
t
= zw
z
47
=u
u
Now
.
.
,
dv du
8u
(9),
du
ou
ov
whence, replacing df/du by this value in the expression for
WP we
nhfaun obtain d
D(u,
~
~du
v)
dF^ dv
It is evident that this derivative does not vanish for the values z o>
y<
Hence the equation
uo-
=
is satisfied
when u
by a certain continuous function u = (x, y, ), which and, replacing u by y = y z = z MO when x = x <
<
,
,
f(x, y,
we obtain
z, ?/),
;
x
,
is
replaced
is
equal to
(x, y, z)
in
continuous function.
for v also a certain
The proposition
is then proved for a system of two equations. can show, as in 21, that these functions possess partial derivatives of the first order. Keeping y and z constant, let us
We
give x an increment Ax, and let AM and Ay be the corresponding increments of the functions u and v. The equations (9) and (10)
then give us the equations
+
-
+ where
e,
e
,
e",
rj,
+ A.
+
.
+
+ A.
+
,
+ A,
.
,
rj
approach zero with
rj"
+
A,,
=
."
+ ,- = Aa-,
A,
0,
It follows
Av.
that ,
^+
\ da;
A;/.
8Fi
,
c
^+
/ \ go
A/^
,
7
77"
/
,\
^+
/^i
77
aF2 4-V II p t] / \ ou
"V
,
\-
y
When Ax
2
V- +
/ \ Ox
V g
f.
approaches zero, AM and Av also approach zero and hence do so at the same time. The ratio Aw /Ax therefore e, e 77, 77 approaches a limit that is, u possesses a derivative with respect to x ,
e",
;
,
77"
:
;
FUNCTIONAL RELATIONS
48
dF cF2
dF dF2
du
dx
dv
dx
dx
dF dF2
dFi
dFz
du
dv
du
l
It follows in like
l
dv l
dv
manner that the
26
[II,
ratio
Av/Aa; approaches a
finite
limit dv /dx, which is given by an analogous formula. Practically, these derivatives may be calculated by means of the two equations
dF dv
dFj du du dx
8 Ft
dx
dF2
dF^du
o
"T"
CU <~\
OX
~r\~
OX
T
dx
dF^dv ~
==
~^~~
"
OX
CV
and the partial derivatives with respect a similar manner. In order to prove the general theorem that
_
l
dv
!
to
y and z may be found
it
will be sufficient to
in
show
the proposition holds for a system of (n 1) equations, it will hold also for a system of n equations. Since, by hypothesis, the functional determinant A does not vanish for the initial values if
of the variables, at least one of the first minors corresponding to the is different from zero for these same values.
elements of the last row
Suppose, for definiteness, that it is the minor which corresponds to is not zero. This minor is precisely
dFn /dun which
D(F F and, since the theorem
l}
2,
-,F _
D(UI,
2,
...,
we may obtain
equations, of the equations (E) in the form
M1
=
^
1
,*; Mn ),
(a? 1 , a;,,
where the functions <,.
by the functions ^ 1? a
new equation ai
>*,;
-,
un _
=
l
are continuous. ,<_!
=F
,,-..,
is
<^i5 >
n
continuous.
xp
un
-i>
we would
1)
;
,
1)
un ),
;
,
we
w n _,
obtain
,
^
t
,
<^
2,
.,
show that the derivative
^0*i, *a,
xp
-,
,
Then, replacing u^
not vanish for the given set of values x\, x 2 can solve this last equation in the form
=
fa^fa, x 2
in the last of equations (E),
(x lt
n
It only remains for us to
where $
).
solutions of the first (n
for the determination of
un )
1
_,)
assumed to hold for a system of (n
is
clear that
it is
n
,
-,
xp
,
<;
B
_,,
MII )
d
=
0.
does
for, if so,
we
*p),
Then, substituting this value of un ir obtain certain continuous functions foi
II,
25]
HI,
u2
,
IMPLICIT FUNCTIONS -,
un _ l
49
In order to show that the derivative
also.
in
ques
tion does not vanish, let us consider the equation
-
The
derivatives
(n
1) equations
.
.
.
d un-i ^ un
dui dun
dun 8<j>
}
/8un
,
r
~7,
7J
d<jj
*
/du n
2
"
T
d* n _i/d
,
,
o
dun
~o
n
are given by the
V}
o
i
(12) n-1
tj
*Pl
i
i
g -^n-l
^yn-1
^-^n-1
r\
.
and we may consider the equations (11) and (12) as n linear equa tions for n -i/du n d/du n) from which we find n d!/du
,
,
D (MU
cu n
,
d
,-
t<2,
It follows that the derivative
D (M!,
i)
d/du
?<
2,
,
wn )
does not vanish for the initial
n
and hence the general theorem
is proved. successive derivatives of implicit functions defined by several equations may be calculated in a manner analogous to that used in
values,
The
When there are several independent the case of a single equation. variables it is advantageous to form the total differentials, from which the
same order may be found.
partial derivatives of the
Consider the case of two functions u and v of the three variables x, y, z defined
by the two equations F(x,
The total differentials of the two equations
3F -%-
.
dx
dx
d -zSec
+ ,
0F
_
^- dy
.
+
d$ -^-
cy
..
du
first
SF
.
dz ^dz
+
-^- dz
d& ^s
,
v)=0,
order du and dv are given by the
+
dy
dx
y, z, u,
.
+ + .
8F -5-
du
d -r
du
,
du
,
du
.dF
+
+
-r-
dv
cv d -,
,
-5
c?v
= 0, = A0.
cv
z Likewise, the second total differentials d u and d*v are given by the
equations
FUNCTIONAL RELATIONS
50
-
dx
+
-..
V
dF
+
2)
dF G
dF
+ /-d*u+ du
cv dv) 2 >
+
[II,
=
d*v
co
26
0,
d&
d&
= + ~d*v CV
?-*d*u CU
0,
and so forth. In the equations which give dn u and dn v the deter minant of the coefficients of those differentials is equal for all vahies of n to the Jacobian D(F, v), which, by hypothesis, does not <)/Z>(w,
vanish. Let MI, 2 u n be n functions of the n independent vari xn) n, such that the Jacobian D(UI, 2, u,,)/D(xi, x 2
26. Inversion.
ables xi, x 2 ,
,
,
,
/i
,
,
,
The n equations
does not vanish identically. g\
=
Un
(
n (X 1(
X2
Xn )
-,
,
,.
x n as functions of u\, M 2 define, inversely, Xi, For, taking any x, for which the Jacobian does not vanish, and system of values x?, x, denoting the corresponding values of MI, w2 Un by uj, w!j, M, there x2
,
,
,
,
,
,
which
when
,
according to the general theorem, a system of functions
exists,
satisfy (13),
MI
tions 0i,
=
wj, 2
,
,
-,
and which take on the values -, x, respectively, u n = unn These functions are called the inverses of the func and the process of actually determining them is called n x",
x",
.
,
an inversion. In order to compute the derivatives of these inverse functions Thus, in the case of two functions
we need merely
apply the general rule.
u=f(x,
v
y),
= (x,y)
t
we
consider u and v as the independent variables and x and y as inverse functions, we have the two equations if
,
du
= 8f,dx + ,
dx
Bf
,
,
dv
dy,
=
, - dx
30-
+
d
dx
cy
,
dy.
cy
whence ^0 ,
dx
=
%f
j
ait
dy
have then,
, ,
dy
=
.
du
H
dx
a/^0_?/a0
a/ a0
dy dx
dx dy
dx dy
We
c0
-i
av
dy
df -
dv
dx
_
^/ ^0 dy dx
finally, the formulae
50
dx du
_
dy
dx
8/a0_a/c0
dv
dx dy
dy dx
d_f
dy d_fdj>_d_fd_$
dx dy
dy dx
II,
IMPLICIT FUNCTIONS
27]
51 8f
dx
eu
_ cf
df
eB^
d
27. Tangents to skew curves. sented by the two equations
let
Let us consider a curve C repre
l*i(*,y,)-0,
(14)
and
ey ex
ex ey
ey ex
ex e^
ex
dy
~"
JF,(a5,y,) = <
x
,
T/O ,
5
be the coordinates of a point
2
M
of this curve, such
that at least one of the three Jacobians
dF1 dFt_dF\
dF\dF^ _d_F\d_F\ dy
dz
dz
dx
vz
dy
dx
8F*
8F gF2
dF_i
dz
ex
dij
l
dy
dF dx
replaced by x %, z ot respectively. that D(Fl} Fj/D(y, z) is one which does Suppose, for defmiteness, the equations (14) may be solved Then the not vanish at n point
does not vanish
when
x, y, z are
M
in the form
y
where $ and
\j/
=
,
.
^(x)
z
)
= t( x
)>
are continuous functions of x which reduce to y
=
The tangent to the curve C z respectively, the two equations point 3/o is therefore represented by ,
when x
x
X-x = 1
where the derivatives
<#>
(cc)
.
F-7/Q ^ (x ) and
and
at the
= Z -g f (x ) "
i//(ce)
may
be found from the two
equations
In these two equations let us set x = x 0) y = y ,z = 2^, and replace - )/(X - x ), * (*) and ^o) by ( F - T/O ) / (X *) and (Z The equations of the tangent then become respectively.
FUNCTIONAL RELATIONS
62
28
[II,
or
X
Y
xa
^(y,
*)
The geometrical
o
Z
y
o
)
^>(,
z
fl(,y
interpretation of this result
is
The
very easy.
two equations (14) represent, respectively, two surfaces Sj and S2 of which C is the line of intersection. The equations (15) represent the two tangent planes to these two surfaces at the point 1/ and ,
;
the tangent to C is the intersection of these two planes. The formulae become illusory when the three Jacobians above all
vanish at the point
M
In this case the two equations (15) reduce and the surfaces Si and S2 are tangent at the The intersection of the two surfaces will then consist, in .
to a single equation,
point A/
.
general, as
M
point
we
shall see, of several distinct branches through the
.
II.
FUNCTIONAL DETERMINANTS
We
28. Fundamental property. have just seen what an important role functional determinants play in the theory of implicit functions.
All the above demonstrations expressly presuppose that a certain Jacobian does not vanish for the assumed set of initial values.
Omitting the case in which the Jacobian vanishes only for certain particular values of the variables, we shall proceed to examine the very important case in which the Jacobian vanishes identically.
The following theorem Let
x \i x ii
HI, ")
a relation
u2 x
,
,
n"
w2
fundamental.
un
In
II (M I}
is
>
be n functions of the n independent variables order that there exist between these n functions u n) == 0, which does not involve explicitly any >
of the variables x ly x z functional determinant
,
,
xn
,
it is
D(UI, M 2
necessary
,
,
and
sufficient that the
?y)
should vanish identically.
In
th.3 first
place this condition
is necessary. For, if such a rela between the n functions HI, u%, un the following n equations, deduced by differentiating with respect to each of the z s in order, must hold
tion TL(UI, w 2 ,
,
wn )
=
exists
,
:
,
II,
FUNCTIONAL DETERMINANTS
28.1
8udu
end Ul ^
7;
jfi
~T~
"^
we cannot
__
"
,
/-,
=
,
Q.
du n dx n
have, at the same time,
=
= <7U
!
^5 = CU H
2
would
since the relation considered it is
Q
du2 dx n
^5 =
identity,
an 8u n
T
~r
jf2
,
dui 8xn
and, since
2
"o
53
in that case reduce to a trivial
clear that the determinant of the coefficients,
which
is
the theorem, must vanish.*
of precisely the Jacobian The condition is also sufficient.
To prove
this,
we
shall
make
use of certain facts which follow immediately from the general theorems.
x, is
w be three functions of
the three independent variables functional determinant D(u, v, w)/D(x, y, z) y, z, such that the Then no relation of the form not zero. 1)
Let
u, v,
A du
+
/u,
dv
+ v dw =
can exist between the total differentials du, dv, dw, except for = p, = v = 0. For, equating the coefficients of dx, dy, dz in the
X
three equations for foregoing equation to zero, there result
which have no other solutions than X 2)
Let
w, u, v,
variables x, y, is
u, v,
t
/u,
X,
p.,
v
= v = 0.
w be four functions of the three independent such that the determinant D (u, v, w} / D (x, y, s)
We
can then express x, y, z inversely as functions of we obtain and substituting these values for x, y, z in
not zero.
w
z,
=
o>,
a function a,
of the three variables u,
= $ (u,
v, ^v.
w)
v,
If by any process ivhatever we can
obtain a relation of the form (16)
du
= P du +
+ R dw
the Professor Osgood has pointed out, the reasoning here supposes that ^H / dUn do not all vanish simultaneously to vanish. This supposition U (u lf u 2 is solved for one of the variables u t II relation the when certainly justified
*As
/,
partial derivatives an / Si/i dU for any system of values which cause ,
is
Q dv
,
,
=
-,)
.
FUNCTIONAL RELATIONS
54
[II,
28
between the total differentials dw, du, dv, dw, taken with respect to the independent variables x, y, z, then the coefficients P, Q, R are equal, respectively, to the three first partial derivatives of
d& P= o
= d$
Q
Cu
=
**
~o
cv
<
w)
(u, v,
:
8
o ow
For, by the rule for the total differential of a composite function (
we have
16),
= d& du +
du
d&
d
dv
-^
-|-
cw
cv
dw
:
and there cannot exist any other relation of the form (16) between du, dv, dw, for that would lead to a relation of the form
d
A.
where
X,
/t,
v
do not
+
du
p.
do
+ v dw = We
all vanish.
0,
have just seen that this
is
impossible. It is clear that these
remarks apply to the general case of any
number
of independent variables. Let us then consider, for definiteness, a system of four functions of four independent variables
X=F
l
(x,y,z,
Y=Fi (x, Z = F (x,
(17)
3
T=F
t
*),
y, z, t), y, z, t),
(x,y,z,
t),
where the Jacobian
D(F F F3 Fi )/D(x,
zero
and
by hypothesis
;
l}
2
let
us suppose,
,
,
is identically that one of the first
y, z, t)
first,
We may then minors, say D(F^ F2 Fs )/D(x, y, z), is not zero. think of the first three of equations (17) as solved for x, y, z as functions of X, Y, Z, t and, substituting these values for x, y, z in ,
;
the last of equations (17), (18)
We
obtain
T
as a function of
A
,
Y, Z, t:
T=*(X,Y,Z,t).
proceed to show that this function $ does not contain the vari For this purpose t, that is, that 8$ /dt vanishes identically. us consider the determinant
able let
we
FUNCTIONAL DETERMINANTS
II,
55
A=
If, in
this determinant, dX, dY, dZ,
ox
dT
be replaced by their values
Ct
tiy
the determinant be developed in terms of dx, dy, dz, dt, it turns out that the coefficients of these four differentials are each zero the first three being determinants with two identical columns, while the
and
if
;
Hence A
last is precisely the functional determinant.
=
0.
But
if
we develop this determinant with respect to the elements of the last column, the coefficient of dTis not zero, and we obtain a relation of the form
dT = P dX + Q dY + R dZ. By side dt,
the remark
is
equal to
hence d&/dt
made
d
= 0.
above, the coefficient of dt in the right-hand But this right-hand side does not contain
It follows that the relation (18) is of the
which proves the theorem
form
stated.
It can be shown that there exists no other relation, distinct from that just found, between the four functions X, Y, Z, T, independent
of x, y, z, t. For, if one existed, and if we replaced T by $>(X, Y, Z) in it, we would obtain a relation between X, Y, Z of the form which is a contradiction of the hypothesis that
U(X, D(X,
Y,
Z)=0,
Y,
Z)/D(x,
y, z)
does not vanish.
first minors of the pass to the case in which all the one of the second least at where but Jacobian vanish identically, the first two of Then zero. not is minors, say D(Flt F^)/D(x, y}, of X. Y, z, t, functions x as and for solved be y equations may
Let us
now
(17)
and the
last
two become
Z = *! (X,
Y, z,
t),
T=
.
2
(A
,
Y, z, t).
FUNCTIONAL RELATIONS
56
On
we can show,
the other hand
ex
as before, that the determinant
dX fy
dY
ex ex
[n,
dZ dy
vanishes identically and, developing it with respect to the elements of the last column, we find a relation of the form ;
dZ = FdX + QdY, whence
it
follows that
In like manner
it
can be shown that
!r= and there exist in
this case
two
dt
= 0;
distinct relations between the four
functions X, Y, Z, T, of the form
There
exists,
however, no third relation distinct from these two;
X
there were, we could find a relation between and Y, which would be in contradiction with the hypothesis that D(X, Y} D(x, y)
for, if
/
is
not zero. Finally, if all the second minors of the Jacobian are zeros, but all four functions X, Z, Y, T are constants, three of them are
not
The above reasoning is evidently general. n functions F1} F2 FH of the n independ
functions of the fourth. If the Jacobian of the
ent variables x ly x 2
,
x n together with
,
+
r (n 1) -rowed minors, vanishes identically, but at least one of the (n r)- rowed minors is not zero, there exist precisely r distinct relations between
the n functions
,
,
,
all its
and certain r of them can be expressed in terms of the remaining (n r), between which there exists no relation. The proof of the following proposition, which is similar to the above demonstration, will be left to the reader. The necessary and condition that n n sufficient functions of p independent variables be connected by a relation which does not involve these variables is that ;
+
every one of the Jacobians of these
n functions, with
respect to
any n
II,
FUNCTIONAL DETERMINANTS
28]
57
In par of the independent variables, should vanish identically. and sufficient condition that two functions
ticular, the necessary
Fi(#i
,
other
xz
,
,
CC
and
B)
that the
is
dF2 /dXf should
F
2
x z ,---, #) should be functions of each dF1 /dx i and
(x l ,
corresponding partial derivatives
be proportional.
Fn in the foregoing theorems may Note. The functions F19 F2 involve certain other variables y 1} y 2) besides x l , x 2 xn -, ym Fn )/D(x l x 2 -, oj n ) is zero, the If the Jacobian D(Fl} Fz ,
,
,
,
,
,
,
,
.
,
Fn are connected by one or more relations functions JF\, F2 which do not involve explicitly the variables x 1} x 2 x n but which may involve the other variables y 1} y 2 ym ,
,
,
,
,
,
.
,
The funda Applications. The preceding theorem is of great importance. mental property of the logarithm, for instance, can be demonstrated by means of it, without using the arithmetic definition of the logarithm. For it is proved at the beginning of the Integral Calculus that there exists a function which is defined for
positive values of the variable, which is zero is l/x. Let/(x) be this function, and let
all
when x
1,
and
whose derivative
v
u=f(x)+f(y),
=
xy.
Then
D (u, D (x, Hence there
v)
_
x
y)
exists a relation of the
y x
y
=0.
form
f(x)+f( V ) = we need only
and to determine since x
is
set
y
=
which gives f(x)
1,
= <j>
(x).
Hence,
arbitrary,
f(z)+f(y)=f(xy). It is clear
that the preceding definition might have led to the discovery of had they not been known before the
the fundamental properties of the logarithm Integral Calculus.
As another application
let
us consider a system of n equations in n
2
(MI,
unknowns
,
(19) .
where *i
J/i,
JT2
)
)
H
n are
Xmi which
*2
F
may
Fn(Ul, W 2
,
constants or functions of certain other variables also occur in the functions
F
If the
.F,-.
Jacobian
un ) vanishes identically, there exist between 2 n )/D(u\, 2i , Z)(Fi, the n functions F, a certain number, say n fc, of distinct relations of the form ,
>
,
Ft
)
t
,
F = U n - k (F!, lt
,
Fk
).
FUNCTIONAL RELATIONS
58
In order that the equations (19) be compatible,
H
t+
l
=
Hi (Hn
= U H - t (Hi,
n
k
29
evidently necessary that
it is
.,H ),..-,H
[II,
,
Hk
),
reduce to k distinct equations. We and, if this be true, the n equations (19) have then the same cases as in the discussion of a system of linear equations. 29. Another property of the Jacobian. The Jacobian of a system of n functions of n variables possesses properties analogous to those of the derivative of a function of a single variable. Thus the preceding
theorem may be regarded as a generalization of the theorem of 8. The formula for the derivative of a function of a function may be Fn be a system of n func extended to Jacobians. Let Flf F2 u n and let us suppose that u^ w 2 tions of the variables M I} u 2 ,
,
,
-,
x
,
,
>
u a themselves are functions of the n independent variables x lf x. Then the formula
D(F
l
D(x
-,
,
Fn ) D( UI
,
D(x 1}
lt
follows at once from the rule for the multiplication of determinants and the formula for the derivative of a composite function. For, let
us write
down
the
two functional determinants cj\ ou
dF in the second
where the rows and the columns changed.
The
first
dx u
dx n
du
element of the product
dF
l
,
cx
have been inter
equal to
is
i
du,,
that
is,
to
30. Hessians.
?!,
and similarly for the other elements.
Let/(x,
?/,
z)
be a function of the three variables x, y,
the functional determinant of the three
first partial
df/dz, a2
/
ax 2 a 2/
ax cy a2
/
ex cz
a2
/
ex 5y a2
a2
/
dx az
/
a2/
2
dydz
a?/
a 2/
cy oz
a2
/
cz-
z.
Then
derivatives cf/dx, Sf/cy,
II,
FUNCTIONAL DETERMINA NTS
30]
59
is called the Hessian of f(x, y, z). The Hessian of a function of n variables is defined in like manner, and plays a role analogous to that of the second deriva tive of a function of a single variable. proceed to prove a remarkable
We
invariant property of this determinant. Let us suppose the independent vari ables transformed by the linear substitution
(19
aX+ X+
(X= y=
)
a
where X, F, Z are the transformed
[3Y+ yZ,
Y+
p
y Z,
and
variables,
or,
0, 7,
,
7"
are constants
such that the determinant of the substitution,
A=
a
J8
a
/3
a"
7 7 7"
/3"
This substitution carries the function /(x, y, z) over into a new function F(X, Y, Z) of the three variables X, Y, Z. Let II (X, F, Z) be the Hessian of this new function. We shall show that we have identically is
not zero.
II (X, F, Z)
where x, ?/, z are supposed replaced For we have
fZF
H=
dF BY
D(X, and
if
dF^ cZ )
= A 2 /t(x,
7
their expressions
by
^
?I\
~\dX aT
aZ/
Y, Z)
D(x,
we consider cf/cx, cf/dy, df/dz,
we may
z),
?/,
in /i(x, y, z)
D(x,
D(X,
y, z)
for a
moment,
cz
By
^, ^, Kl cx
Y, Z)
as auxiliary variables,
But from the
relation
D(x,
U(X,
y,
Y, Z)
dz /
dy
F(X,
Y, Z) =f(x, y,
dX
ex
dY 3F = a/ + 7 -^ c
:
Z
cx
we
z),
dF = cf a + a ,cf +
find
,,Bf a"
,
dz
cy
dy
7
,
a/ cy
+ ,
7
c/ cz
whence
dx
and hence,
d_F
dF
er
ez
dy
dz
a"
= 7
7
finally,
H=
D(x,
D(X, It is clear
a a
that this theorem
is
general.
- = A 2 /i-
y, z)
Y, Z)
).
y,
write
D(x,y,z)
from (19
A;
FUNCTIONAL RELATIONS
60 Let us
now
=
+
ox3
3 bx*y
+
be a given binary cubic form whose coefficients Then, neglecting a numerical factor,
=
ax bx
+ +
and the Hessian
by
bx
4
cy
cy
ex
+
dy
is
=
- &2 )x2 +
(ac
+
3 cxy 2
dy*
a, b, c,
(ad
-
30
Let
consider an application of this property of the Hessian. /(x, y)
h
[II,
d are any constants.
+
bc)xy
-
(bd
c 2 )y 2 ,
seen to be a binary quadratic form. First, discarding the is a perfect square, we may write it as the product of
case in which the Hessian
two linear factors
:
h
=
+
(mx
ny) (px
qy).
now, we perform the linear substitution
If,
mx + ny = X,
F(X, Y) = AX* + whose Hessian
this
the
form
3
=
Y,
form,
BX Y+3 CXY + DY 2
2
8 ,
is
2 H(X, Y) = (AC - B
and
px + qy
new
the form/(x, y) goes over into a
If
+
X
2
)
+ (AD - EC] XY + (BD - C2 F 2 )
,
must reduce, by the invariant property proved above, to a product of KXY. Hence the coefficients A, B, C, D must satisfy the relations
one of the two coefficients B, shall have
C
be different from zero, the other must be so,
and we
-? F(X, Y) =
(B*X*
+ 3 B2 CX* Y + 3 BC* XY 2 +
whence F(X, Y), and hence /(x, particular case,
F(X, Y)
it is
-f-
evident that
will be of the canonical
y), will
we
shall
C*
B Y =
Y
(
3
)
^+^
)\
be a perfect cube. Discarding this and the polynomial have B = C = ;
form
AX* + DY 3
.
Hence the reduction of the form /(x, y) to its canonical form only involves the solution of an equation of the second degree, obtained by equating the Hessian are precisely the two The canonical variables X, of the given form to zero.
Y
factors of the Hessian. It is
AX
3
easy to see, in like manner, that the form/(x, y) is reducible to the form When the Hessian van the Hessian is a perfect square.
+ BX 2 Y when
ishes identically /(x, y)
is
a perfect cube /(x, y)
=
:
(ax
II,
TRANSFORMATIONS
31]
TRANSFORMATIONS
III.
It often happens, in
61
many problems which
arise in
Mathematical
we
are led to change the independent variables. It Analysis, that therefore becomes necessary to be able to express the derivatives
with respect to the old variables in terms of the derivatives with We have already considered a problem respect to the new variables.
Let us now consider the
of this kind in the case of inversion.
question from a general point of view, and treat those problems which occur most frequently. 31.
and x=
be
<().
respect to respect to
Let
I. Let y be a function of the independent variable x, a new independent variable connected luith x by the relation It is required to express the successive derivatives of y with
Problem
let t
x in terms of
t
and the
successive derivatives of
y with
t.
be the given function, and F(t) =/[<()] the func in the given function. By the by replacing x by
y=f(x)
tion obtained
<j>(t)
rule for the derivative of a function of a function,
dy
37
=
at
we
find
dy X ,. 9 m, ax .
~r~
whence
dy dt
y
t
To find the derivative of y
This result
may
be stated as follows
with respect
to x,
take the derivative of that function with respect
:
and divide it by the derivative of x with respect to t. The second derivative d 2 y/dx* may be found by applying rule to the expression just found for the first derivative.
d
Ll =
dx*
_ -y^
We
to t
this
find
-I
w)
(0-y^"(0. [>
(0]
and another application of the same rule gives the third derivative
:
FUNCTIONAL RELATIONS
62
[H,
32
performing the operations indicated,
or,
_
6
<*
[>
The remaining
derivatives
may
(OJ be calculated in succession by
In general, the nth deriva repeated applications of the same rule. x be to tive of y with respect expressed in terms of may (), (n) the first n successive derivatives of y with respect to and (), t. These formulae may be arranged in more symmetrical form. <}>
<j>"(t),
<
,
Denoting the successive differentials of x and y with respect to t by z dn y, and the successive derivatives of y -, dx, dy, d x, d*y, (n with respect to x by y y \ we may write the preceding d"x,
,
y",
,
formulae in the form
y 9
7
.
f
dx
_ dx d
2
dy 6?
y dx 3
(20)
~~
y
x2
- 3 d?y dx d
z
+ 3dy (d*x)* -
x
dy d*x dx
5
The independent variable t, with respect to which the differentials on the right-hand sides of these formulae are formed, is entirely arbitrary and we pass from one derivative to the next by the ;
recurrent formula ,
,
<>=
the second
member being regarded
as the quotient of
two differen
tials.
32.
Applications.
These formulas are used
in the
study of plane
when
the coordinates of a point of the curve are expressed in curves, terms of an auxiliary variable t.
=/(*)
y
=
* co
in order to study this curve in the neighborhood of one of its points of y it is necessary to calculate the successive derivatives y But the preceding formulas with respect to x at the given point. ,
?/",
give us precisely these derivatives, expressed in terms of the succes without the necessity sive derivatives of the functions f(t) and (#), <j>
II,
TRANSFORMATIONS
32]
63
of having recourse to the explicit expression of y as a function of x, Thus the which it might be very difficult, practically, to obtain. first
formula
=
= dx
y>
f (t)
The value of occurs in an impor gives the slope of the tangent. tant geometrical concept, the radius of curvature, which is given by y"
the formula
which we shall derive
In order to find the value of R, when
later.
the coordinates x and y are given as functions of a parameter t, need only replace y and by the preceding expressions, and y"
we we
find
R=
2
(dx .
4-
^r~
,
dy^Y
"
where the second member contains only the x and y with respect to t.
first
and second deriva
tives of
The following interesting remark is taken from M. Bertrand s Traitt de Calcul differentiel et integral (Vol. I, p. 170). Suppose that, in calculating some geometrical concept allied to a given plane curve whose coordinates x and y are supposed given
in
terms of a parameter F(x,
y, dx, dy,
d2 x,
,
we had obtained
d 2 y,
-,
the expression
d n x, dy),
all the differentials are taken with respect to t. Since, by hypothesis, concept has a geometrical significance, its value cannot depend upon the choice of the independent variable t. But, if we take x = t, we shall have
where
this
dx
dt,
dz x
=
d3 x
=
=
da x
=
0,
f(x, y, y
which
is
and the preceding expression becomes ( ,
y",
>
2/
)
;
same as the expression we would have obtained by supposing at the was solved with respect to y in the case to the case where the inde this To return from particular *().
the
start that the equation of the given curve
form y = pendent variable from the formulae
is
arbitrary,
(20).
we need only
Performing
replace y
,
y",
by their values
this substitution in
with which we should get back to the expression F(x, y, dx, dy, d 2 x, d 2 y, we started. If we do not, we can assert that the result obtained is incorrect. )
For example, the expression
dxd 2 y + dyd2 x
FUNCTIONAL RELATIONS
64
[II,
33
cannot have any geometrical significance for a plane curve which is independent of the choice of the independent variable. For, if we set x = t, this expression 2 reduces to /(I + y )$ and, replacing y and by their values from (20), we do not get back to the preceding expression. y"
;
y"
The formulae (20)
33.
differential
determine
are also used frequently in the study of Suppose, for example, that we wished to
equations. functions y of the independent variable x,
all the
which
satisfy the equation
(21)
(1
_^*_ e eg +
=
.
,
where n is a constant. Let us introduce a new independent variable = cos t. Then we have t, where x
dy
d?y dx 2
dy
dt
dx
sin
d*y smt-jfi. at*
and the equation (21) becomes,
t
cost
dy dt <
sin
8
1
after the substitution,
(22)
It is easy to find all the functions of
for it
may
t
which
satisfy this equation,
be written, after multiplication by 2 dy /dt,
whence
where a
is
an arbitrary constant.
Consequently
or
71
= 0.
II,
TRANSFORMATIONS
34]
The left-hand
side
is
that this difference
65
the derivative of arc sin (y/a)
y
=
a sm(nt
It follows
nt.
must be another arbitrary constant
b,
whence
+ &),
which may also be written in the form y
=A
sin nt
+ B cos nt.
Returning to the original variable x, we see that all the functions of x which satisfy the given equation (21) are given by the formula
y
where
=A
A and B
are
sin (n arc cos a)
+ B cos (n arc cos a),
two arbitrary constants.
To every relation between x and y there corresponds, a relation by means of the transformation x = f(t, u), y = u*), 34.
Problem
II.
and
u.
(t,
between
t
respect to
It is required to express the derivatives
x in terms of t,
This problem
is
and
u.
the derivatives of
u with
seen to depend upon the preceding
of y with
respect to
when
t.
it is
noticed that the formulae of transformation,
give us the expressions for the original variables x and y as func t if we imagine that u has been replaced in
tions of the* variable
,
We
these formulas by its value as a function of t. need merely apply the general method, therefore, always regarding x and y as composite functions of t, and u as an auxiliary function of t.
We
find then,
first, 8<jt
dy
_dy
dx
dx
dt
dt
8
du
du dt
dt
df
df du
dt
du dt
and then d?y 2 dx"
or,
__
dx
d (dy\ \dxj
dt
dt
performing the operations indicated, ,
_
_
SuBtdt
du*\dt) t
gu
du dt
dfr*
+
,
,
\ dt
du dt/\dt*
FUNCTIONAL RELATIONS
66
[II,
33
is expressible in terms of t, u, and 2 dn u/dtn the derivatives du/dt, d?vi/dt , , Suppose, for instance, that the equation of a curve be given in The formulae for the rectangular coor polar coordinates p
In general, the nth derivative y (n)
.
=
/(o>).
dinates of a point are then the following
x
Let p
,
= p cos
p sin w.
y
:
be the successive derivatives of p with respect to w, From the preceding formulae
p",
considered as the independent variable.
we
find
= dy = d x = dx
p sin w
cos (a dp sin
2
o>
cosu)
+
dp
dz p
sinw d 2 p
d2 y
+
p cos w
e?w, d
2 sin w
dai
dp
p cos
w
2 cosw
rfw ffy
p sin
w
da?, 7(o
2 ,
whence 2 <&e
+ dij
1
dif
d^x
= =
2
dp
+p
2 du dp
The expression found above
p
2
rfw
,
p
d
+p
d2 p
2
c?w
3 .
for the radius of curvature
becomes
+ *p m -pp
35. Transformations of plane curves.
m
2
2
Let us suppose that to every
M
of the same plane cor we make another point point If we denote the coordinates known construction. some respond by of a plane
m
of the point by (x, y) in general, two relations
M
and those of by (X, F), there will exist, between these coordinates of the form
X=f(x,y),
(23)
Y=4>(x,
y}.
These formulae define a point transformation of which numerous examples arise
in
Geometry, such as projective transformations, the
transformation of reciprocal radii, a curve c, the corresponding point
When
etc.
the point
m describes
M describes another curve C, whose
properties
may
be deduced from those of the curve
nature of the transformation employed. cessive derivatives of y with respect to x, sive derivatives of F with respect to X. is y",
necessary to be able to express -.
and we
This find
is
F
precisely the problem
Let y and F
c
,
?/",
,
F",
and from the be the suc the succes
To study the curve C in
terms of
it
y
,
which we have just discussed
;
,
F",
x, y,
II,
TRANSFORMATIONS
36]
67
dY
It is seen that Y depends only on x, y, y so forth. Hence, the transformation (23) be applied to two curves c, c which are tangent at the point (x, ?/), the transformed curves C, C will also
and
.
if
,
be tangent at the corresponding point (A F). This remark enables us to replace the curve c by any other curve which is tangent to it in questions which involve only the tangent to the transformed ,
curve C.
Let us consider, for example, the transformation defined by the formulae
Y= which
is
x2
+
2
y
the transformation of reciprocal radii, or inversion, with the cor Let m be a point of a curve c and
M
the origin as pole.
responding point of the curve C. In order to find the tangent to this curve
C we need
only apply the result of ordinary Geometry, that an inversion carries a straight line into a circle
through the pole. Let us replace the curve tangent mt.
The
c
inverse of
by
mt
its is
a
through the two points Mand O, whose center lies on the perpendicular circle
<-
A FIG. 5
upon mt. The tangent MT to this circle the angles Mmt and mMT are equal, and AM, the complement of the angle mOt. The tangents mt
Ot let fall from the origin is
perpendicular to
since each
and
is
MT are therefore antiparallel with respect to the radius
36. Contact
transformations.
vector.
The preceding transformations
are
most general transformations which carry two tangent Let us suppose that a point curves into two other tangent curves. is determined from each point m of a curve c by a construction not the
M
FUNCTIONAL RELATIONS
68
[II,
36
which depends not only upon the point m, but also upon the tangent to the curve c at this point. The formulae which define the trans formation are then of the form (24)
X = /(*,
1
y, y-),
and the slope Y of the tangent by the formula
y,
Y=<j>(x,
to the
dx
y)
;
transformed curve
is
given
y
dy
dy>
In general, F depends on the four variables apply the transformation (24) to two carves
and if we y which are tangent C will have a point
x, y, c,
,
y"
;
c
at a point (x, y~), the transformed curves C, (X, Y) in common, but they will not be tangent, in general, unless happens to have the same value for each of the curves c and c .
y"
In order that the two curves
C and C should always
be tangent, it should not depend on that is, and (x, y, y ) should satisfy the
is necessary and sufficient that Y that the two functions f(x, y, y )
y";
<
condition
In case this condition
is
satisfied,
the transformation
is
called a
contact transformation. It is clear that a point transformation particular case of a contact transformation.*
is
a
Let us consider, for example, Legendre s transformation, in which the point M, which corresponds to a point (x, y) of a curve c, is given by the equations
X=y
from which we
,
Y=xy -y;
find
Y
,
_dY _xjf _ ~ -
dX
y"
which shows that the transformation In like manner we find
dY dX V
11
a contact transformation.
is
dx
1
y"dx
y"
y
=r
dX *Legendre and Ampere gave many examples of contact transformations. Sophus Lie developed the general theory in various works see in particular his Geometric der Beruhrungstransformationen. See also JACOBI, Vorlesungen iiber Dynamik. ;
TRANSFORMATIONS
37]
II,
and so
From
forth.
x
the preceding formulae
=Y
y
,
= XY -Y,
69
it
y
follows that
=
X,
which shows that the transformation is involutory.* All these prop by the remark that the point whose coordinates
erties are explained
X=
=
xy y is the pole of the tangent y Y 2 the point (x, y) with respect to the parabola x denote the pole of the tangent at general, if respect to a directing conic 2, then the locus of are
,
M
to the curve c at
2y
m to
= 0.
But, in
a curve
c
with
M
the point is a, is precisely the polar of the point curve C whose tangent at with respect to 2. The relation between the two curves c and C is therefore a reciprocal one and, further, if we replace the curve c by
M
m
;
tangent to c at the point m, the reciprocal curve will be tangent to the curve C at the point M. another curve
c
,
C
from a fixed point O in the plane of a curve c, a perpen upon the tangent to the curve at the point m, the locus of the foot of this perpendicular is a curve (7, which is called the pedal of the It would be easy to obtain, by a direct calculation, the coordinates given curve. of the point Jlf, and to show that the trans Pedal curves. be let
dicular
OM
If,
fall
M
formation thus defined
is
a contact transfor
mation, but it is simpler to proceed as follows. Let us consider a circle 7 of radius E, de scribed about the point
be a point on
as center;
and let ?MI
OM such that Om\ x OM= E
The point mi
the pole of with respect to the circle transformation which carries is
;
2 .
the tangent mt and hence the c into
C
the
is
result of a transformation of reciprocal po-
followed by an inversion. When the point m describes the curve c, the point mi, the pole of mt, describes a curve Ci tangent lars,
m
with respect to to the polar of the point the circle 7, that is, tangent to the straight line miti, a perpendicular let fall Tto the curve C and the tangent m\ti to the from mi upon Om. The tangent
M
Hence, if we draw angles with the radius vector OmiM. are equal, since they are the comple the normal MA, the angles and It is the middle point of the line Om. ments of equal angles, and the point follows that the normal to the pedal is found by joining the point Mto the center of the line Om.
curve
Ci
make equal
AMO
AOM
A
y"
37. Projective transformations. Every function y which satisfies the equation = is a linear function of x, and conversely. But, if we subject x and y to
the projective transformation *
That
is,
two successive applications TRANS.
original coordinates.
of the transformation lead us
back to the
FUNCTIONAL RELATIONS
70
aX + bY+c
_
X+
a"
b"
Y+
a
_
c"
a"
[II,
X+ b Y + X Y+ -f
38
c c"
b"
= should Hence the equation In order to verify this we will first remark that the general projective transformation may be resolved into a sequence of particular transformations of simple form. If the two coefficients and are not both a straight line goes over into a straight line.
become d ^Y/dX"2
y"
0.
a"
zero,
we
will set
X\ =
X
a"
= and a on the supposition that a time
ab"
ba"
-\-
b"
b
b"
Y+ a"
b
b"
c"
=
0,
is
a"
;
X
,
The preceding formulae may
not zero.
X and Y by their values
then be written, replacing
b"
and since we cannot have at the same we will also set YI a + b Y+ c in terms of
Xi and
F
l5
in
the form
a Xi + /3 FI + 7 Xi
YI
A!
YI
7
AI
Xi
It follows that the general projective transformation can be reduced to a succession of integral transformations of the form
aX + bY +
x
y
c,
-
X+ b Y+
a
c
,
combined with the particular transformation x
Performing
=
1
X
y
,
this latter transformation,
~ dy dx
we
= Y
X
.
find
_-.-_ ~~ ~
and y"
=
~y~
= -
XY"(-
dx
X*)
=
X
s
Y".
Likewise, performing an integral projective transformation,
y
-
dy
_ ~
dt/ dx
+b Y a + bY _ (ab -ba (a + bY
we have
a
)Y"
3 )
In each case the equation = 0. goes over into We shall now consider functions of several independent variables, and, for definiteness, we shall give the argument for a function of two variables. Y"
y"
38. Problem III.
ent variables
x and
Let w y,
= f(x,
and
let
y) be a function of the two independ be two new variables connected
u and v
with the old ones by the relations
It is required to express the partial derivatives
and y in terms of u, u and v.
variables x respect to
v,
and
of u with respect to the the partial derivatives of with u>
TRANSFORMATIONS
II,
Let w
=
71
which results from/(x, y) by the F(u, v) be the function Then the rule for the differentiation of composite
substitution.
functions gives do
8
c oj
8
cu
dx cu
Cd)
C
dv
dx cv
a)
d\ff
<J>
whence we may D(,
w
C
d ta
(jt
C\}/
dy dv
and du/dy; change vanished,
find
d
the
\ji)/D(u, v)
dw
=A
o~ ex (25)
d(a a cu
Cd)
Cu) __
c
cy
D
the determinant
for, if
variables
of
performed
Hence we obtain the equations
would have no meaning.
where A, B, C,
dy du
-^
cu
r
B
d
d>
~z~
+u
cv
Coi ~^~~>
cv
u and v
are determinate functions of
formulae solve the problem for derivatives of the show that the derivative of a function with respect
;
and these
first order.
x
to
is
the
They sum of
the two products formed by multiplying the two derivatives with respect The derivative with respect to to u and v by A and B, respectively.
obtained in like manner, using C and D instead of A and B, In order to calculate the second derivatives we need respectively. first derivatives the rule expressed by the preced to the only apply
y
is
ing formulae
;
doing
2
d
575 C/X
==
or,
/8u>\
2~\\
Q
=A
so,
\
we
d I
==
"-~/
C
d&
2\p
GOT/
t/JC / /
(31
find
t
\\
- +B o>
(A cu\ Cu
i
r
C/it
v
to - \ cv /
+B
w ry- (A cu dv\ .-
_
_
_
+ B -5.
performing the operations indicated,
^= ^-^ 2
+ A[A-z+Br-%:+-^-CU CU CU CV v
^"^
-B t-^
My
P
+
cy
-z
CU CV
0v cv
2 2 find 8 2 (a/dxdy, 3
and the following derivatives which are to be carried out we need only replace the operations d /dx and d /dy by the operations
and we could in like
manner.
In
all differentiations
d
8
A-Z-+B-Z-* Cv du
C
d
cu
+
d
D^-, cv
FUNCTIONAL RELATIONS
72
[II,
Hence everything depends upon the calculation
respectively.
38
of the
coefficients A, B, C, D.
Example
Let us consider the equation
I.
a
(26)
C
+26 -- +
CO
ex
where the
coefficients a,
tion to as simple a
cy* ;
and
let
v,
us try to reduce this equa
We observe first that if a = c = 0, it would the equation. We may then suppose that c,
form as
possible.
be superfluous to try to simplify Let us take two for example, does not vanish.
and
= 0,
c
ci/
are constants
6, c
dJ
(i3
,
new independent
variables u
defined by the equations
u where a and
/3
=
are constants.
+
x
v
ay,
=
x
+
py,
Then we have u
c
<
cu
<jj
cx
-
d
du
a
and hence,
in this case,
A=B=
+
du
8y
C=
1,
a,
w
I
P
,
cv
D = p.
The general formulae then
give "
dx?
2
cucv
~dii?
"ai?
eu 2
ducv
and the given equation becomes
+ 26a + ca2 )^ +
(a
2 [a
+
b(a
+
It
au a
(a
-(-
CB
remains to distinguish several cases. First case.
a
+ ca/3]-^- +
ft)
au-^
.)-
2 6r
+
cr2
Let
=
0,
62
ac>
0.
Taking
for
a and
the two roots of the equation
the given equation takes the simple form
=
0.
cudv Since this
may be
written
Let see that dta/Su must be a function of the single variable, w, say/(w). F(u) denote a function of u such that F (u) =f(u). Then, since the derivative of w F(u) with respect to u is zero, this difference must be independent of w, The converse is apparent. Returning to and, accordingly, u = F(u) + *().
we
the variables x and y, (26) are of the form
it
follows that
all
the functions
w which
satisfy the equation
TRANSFORMATIONS
38]
II,
where
F
and $ are arbitrary functions.
73
For example, the general integral of
the equation c2 w
cy
which occurs
O c
=
2
a2
w ,
2
dx 2
in the theory of the stretched string, is
w =f(x ac
=
and
|3
+
+
ay)
(x
-
ay).
0. Taking a equal to the double root of the equa some other number, the coefficient of d^w/dudv becomes zero, for it is equal to a + ba + p(b + car). Hence the given equation 2 = It is evident that w must be a linear function of 0. reduces to 5 2 w = vf(u) + (u), where f(u) and (u) are arbitrary functions. Returning to the variables x and y, the expression for w becomes
Second
tion a -f
Let b 1
case.
2 -f cr
26r
=
0,
w/cz>
t>,
=
w
(x
+ Py)f(x +
[x
+ ay +
+
ay)
(x
+
ay),
which may be written
- a)y]f(x +
w
=
w
= yF(x + ay) +
(p
ay)
+
(x
+
ay),
or. finally,
+
(x
ay).
ac Third case. If 62 0, the preceding transformation cannot be applied without the introduction of imaginary variables. The quantities a and /3 may then be determined by the equations <
a
+ 26a + c a 2 = a
+
+
b(a
p)
which give
The equation
2& 2
2b
a + /3=
a/3=
,
of the second degree,
26
r2 H
whose roots are a and becomes
c2
importance in
Example x
= p cos
0,
which
is
many branches
II. <,
=
y
0,
has, in fact, real roots.
,
Aw = This equation Aw
-ac =
262 r H
c
set
-ac c2
c
a2
w
du 2
c2 w H
=
The given equation then
0.
c 2
known
of
as Laplace s Equation, is of fundamental mathematics and mathematical physics.
Let us see what form the preceding equation assumes when we = p sin 0. For the first derivatives we find 8
u
= du cos^
8p i
u
Su H
dx
=
dy du
("ui
p sin
-\
p cos *
,
FUNCTIONAL RELATIONS
74 or,
[H,a9
dw/dx and du/cy,
solving for
du -
COS
du -
du
du
sin
dp
p
dip
cosrf)
du
p
dip
rf>
i
dx
du
=sm0
dy
+
dp
Hence
(
=
u --
1
2 a/?
sin 2
0a 2 w
2
2
39.
But
---- du
du
t (
p
8/
S2 w
cos
2 sin
dp
d\
simp p
2sin0cos
1
p
d2
u/dy 2
is
analogous to
p
dip
this.
sin 2
0cw
p
dp
1
2
d
Adding the two, we
i
find
The preceding method
Another method. the function
dp
p
dtf>
and the expression for
when
8
sind>
------- ------p
dp
d2
sin0
)
costf>
dp\ COS 2
---- du\ --
du
I
a
whose
in certain cases it
is the most practical partial derivatives are sought is unknown. is more advantageous to use the following
method. y) be a function of the two independent variables x If x, y, and z are supposed expressed in terms of two aux iliary variables u and v, the total differentials dx, dy, dz satisfy the relation
Let and y.
z
=f(x,
O
dz
=
ex
which
is
o
/>
-^-
dx
+
/
-dy. cy
equivalent to the two distinct equations
_ du
dx du
dz
_d_fdx
dv
dx dv
dy du
dy dv
whence df/dx and df/dy may be found
as functions of u,
v,
dz/du,
But to find the succeeding dz/dv, as in the preceding method. derivatives we will continue to apply the same rule. Thus, to find d
2 f/dx and
2
d 2f/dxdy,
we
with the identity
start
dx 2
which
is
>
dxcy
equivalent to the two equations d
(dx) du
2
= d fdx dx 2 du
ay
dy
|
dx dy du
}
II,
TRANSFORMATIONS
39]
75
:
where
}
|
2
dec
dx dy dv
ov
supposed that df/dx has been replaced by its value cal 2 Likewise, we should find the values of d f/dx dy and by starting with the identity
it is
culated above. 2
a2
a
82
/
df\
?T ***
Q
I
dxdy
dy/
f
~cT~a 2
*%
dy
The work may be checked by the fact that the two values of found must agree. Derivatives of higher order may be
c 2f/8x dy
calculated in like manner.
Application to surfaces. The preceding method is used in the study of surfaces. Suppose that the coordinates of a point of a surface S
two variable parameters u and v by means
are given as functions of
of the formulae
x=f(u,v),
(27)
The equation
y
of the surface
=
$(u,v),
may
z
= f(u,v).
be found by eliminating the vari
ables u and v between the three equations (27); but we may also study the properties of the surface S directly from these equations themselves, without carrying out the elimination, which might be practically impossible.
It should be noticed that the three
D(,
D(u,
-y)
Jacobians
v)
cannot
all vanish identically, for then the elimination of w and v would lead to two distinct relations between x, y, z, and the point whose coordinates are (x, y, %) would map out a curve, and not a sur face. Let us suppose, for definiteness, that the first of these does not 0. Then the first two of equations (27) and the substitution of these values in the third would give the equation of the surface in the form z = F(x, y). In order to study this surface in the neighborhood of a point we need
vanish
may
:
D(f,
=
<j>)/D(u,
be solved for u and
v)
v,
know the partial derivatives p, q, r,s,t, of this function F(x, y) in terms of the parameters u and v. The first derivatives p and q to
are given by the equation
dz
which
is
equivalent to the
= p dx
-f-
q dy,
two equations
FUNCTIONAL RELATIONS
76
[II,
40
^ =- fpn ^4-n^ -- r~ TT
T;
du
du
Q
r q
a cv
cv
cv
du
from which p and q may be found. The equation of the tangent plane is found by substituting these values of p and q in the equation
Z-
= p(X -
z
*)
+ q(Y - y),
and doing so we find the equation
The equations (28) have a geometrical meaning which is easily They express the fact that the tangent plane to the
remembered.
surface contains the tangents to those two curves on the surface which are obtained by keeping v constant while u varies, and vice versa*
Having found p and r, s, t by means
to find
p =/i(w, v), q of the equations q,
= f (u, 2
v),
we may proceed
= r dx + s dy, dq = s dx + dy,
dp \
t
each of which
is
equivalent to two equations
and so
;
forth.
40. Problem IV. To every relation between x, y, z there corresponds by means of the equations
(30)
x =/(w,
y
w),
v,
=
<
z
(M, v, w),
= \j/(u,
v,
w),
a new relation between
It is required to express the partial u, v, w. derivatives of z with respect to the variables x and y in terms of u, v, w, and the partial derivatives of iv with respect to the variables u and v.
if
This problem can be made to depend upon the preceding. For, we suppose that w has been replaced in the formulae (30) by a
function of u and
v,
we have
x, y, z
expressed as functions of the
* The equation of the tangent plane may also be found directly. Every curve on the surface is defined by a relation between u and w, say v U (u) and the equations of the tangent to this curve are
=
X-x df fdu The elimination
+
df -f dv
n
~
Y-y + ~ n () ^ du dd>
()
d4>
dv
;
Z-z
~ d4>
-^-
du
+
d& -
dv
H
(M)
of IT(a) leads to the equation (29) of the tangent plane.
II,
TRANSFORMATIONS
41]
77
two parameters u and v; and we need only follow the preceding method, considering /, ^ as composite functions of u and v, and w as an auxiliary function of u and v. In order to calculate the first derivatives p and y, for instance, we have the two equations <,
_
P a
Q
5
T *
I
cT~
T~ ow ~o8u
i
cu
1+7 ,
--P ^--o ,
"^
_
dw
dv
dv
8w
\d
dv
dw cu
du
\8v
+ a^ 8w
a_w
The succeeding
derivatives may be calculated in a similar manner. In geometrical language the above problem may be stated as fol lows To every point of space, whose coordinates are y,
m
:
(x,
z),
there corresponds, by a given construction, another point M, whose coordinates are X, Y, Z. When the point maps out a surface S, the point maps out another surface 2, whose properties it is pro
m
M
posed to deduce from those of the given surface S. The formulae which define the transformation are of the form
x =f(x
>
y>
),
y=
(*, y, *),
Let Y)
be the equations of the two surfaces S and 2, respectively. The is to express the partial derivatives P, Q, R, S,T, of the
problem
function
$(A",
Y) in terms of
x, y, z
and the partial derivatives But this is precisely the
of the function F(x, y). above problem, except for the notation.
p,
q, r, s,
t,
The first derivatives P and Q depend only on x, y, z, p, q and hence the transformation carries tangent surfaces into tangent sur faces. But this is not the most general transformation which enjoys ;
this property, as
we
shall see in the following example.
41. Legendre s transformation. Let z =f(x, y) be the equation of a surface S, and let any point (x, ?/, z) of this surface be carried into a point M, whose coordinates are X, Y, Z, by the transformation
m
X=p, Let
Y=
q,
Z
px
+
qy
z.
Z = $ (X,
Y) be the equation of the surface 2 described by the z, p, q replaced by /, df/dx, df/dy, respec we have the three coordinates of the point expressed as
point M. tively,
If
we imagine
functions of the two independent variables x and
M
y.
FUNCTIONAL RELATIONS
78 Let P,
Q, R, S,
Y).
$>(X,
T
Then the
41
denote the partial derivatives of the function relation
dZ =
PdX+ QdY
becomes
p dx +
[II,
q dy
+ x dp + y dq
dz
= P dp +
Q,
dq,
or
+ y dq = P dp -f
x dp
Q dq.
Let us suppose that p and q, for the surface S, are not functions of each other, in which case there exists no identity of the form \dp + p.dq=. 0, unless X
=
fi
Then, from the preceding equation,
0.
In order to find R,
S,
T we may dP dQ
when X,
which,
Q
Y, P,
dx
dy
start
= RdX + = SdX+
follows that
with the analogous relations
SdY, TdY,
are replaced by their values,
= R (r dx + s dy) + = S (r dx + s dy~) +
it
S (s dx
T(sdx
+ +
t
dy)
,
t
dy}
;
become
whence
and consequently
From x
s
2
rt
the preceding formulae
= P,
r
s
t
rt
y=Q,
z
T
we
s
2
2
find, conversely,
p=
= PX+QY-Z, -S
RT-
RT- ^
s
rt
2 .S
X,
q
=
Y,
R t
RT -
S2
which proves that the transformation is
is involutory. Moreover, it a contact transformation, since X, Y, Z, P, Q depend only on x,
These properties become self-explanatory, if we notice that the formulae define a transformation of reciprocal polars with respect to the paraboloid y, z, p, q.
x2 Note. rt
s
point
2
=
+
y
2
-
2z
=
0.
for R, S, T become infinite, if the relation In this case the holds at every point of the surface S.
The expressions
M describes a curve, and not a surface, for we have
II,
TRANSFORMATIONS
42]
=
,,
79
=
D*,y
*,y)
and likewise
D(X, Z) D(*,y) This
let
is
= Jfo
+ gy - *) =
;>*
(*,
_
^=
?/)
which we had not considered.
precisely the case
42. Ampfere s transformation. Retaining the notation of the preceding article, us consider the transformation
X The
Y=
x,
relation
becomes
dZ =
Z
q,
qy
z.
PdX + QdY
qdy + ydq - dz = Pdx + Qdq,
or
p dx = Pdx + Qdq.
y dq
Hence
Q=y
P=-P, and conversely we x
= X,
It follows that this
The
transformation also
dP = r
R -f
dx
Ss
s
=-
dy r,
q
=
Y.
an involutory contact transformation.
is
relation
is,
p = - P,
z=QY-Z,
y=Q,
next becomes that
;
find
EdX+ SdY
= R dx + S (s dx + St = - s,
t
dy)
;
whence
Starting with the relation
dQ = SdX + TdY, we
find, in like
manner,
r=l. t
As an
application of these formulae, let
=
vis
try to find
all
the functions /(x, y)
Let S be the surface represented by the Y) the equation equation z =f(x, y), S the transformed surface, and Z = of S. From the formulae for R it is clear that we must have
which
satisfy the equation rt
s2
0.
4>(X,
R~ and * must be a linear function of
where
X
^ -~
S~ :
and ^ are arbitrary functions
of F.
It
follows that
FUNCTIONAL RELATIONS
80
and, conversely, the coordinates as functions of the two variables
x
= X,
y
= Xt (Y) +
y, z) of
(a;,
X and z
y(Y),
=
[II,
a point of the surface S are given
Y by
the formulae
(Y)
Y[X
+
f (Y)]
X
X
The equation of the surface may be obtained by eliminating and Y amounts to the same thing, by eliminating a between the equations z
The
= ay = y
^
(a)
;
-
t(Y).
or,
what
(a),
-x(a)-t
x
43
(a).
moving plane which depends upon the found by differentiating the first with respect to this parameter. The surfaces defined by the two equations are the so-called developable surfaces, which we shall study later. first
of these equations represents a
parameter a, while the second
is
43. The potential equation in curvilinear coordinates.
The
calculation to which
a change of variable leads may be simplified in very many cases by various devices. We shall take as an example the potential equation in orthogonal curvilinear coordinates.* Let
F (x,
y, z)
- p,
FI(X, y, Z)=PI, F*(X, y,
)=P2i
be the equations of three families of surfaces which form a triply orthogonal system, such that any two surfaces belonging to two different families intersect at right angles. Solving these equations for x, y, z as functions of the parame ters p, pi, PQ,
^6
obtain equations of the form fx-(p,
(
= j l*=
31 )
y
and we may take
02
PI, pa), Pi, Pa),
(p, Pi, pa);
p, p t p^ as a system of orthogonal curvilinear coordinates. Since the three given surfaces are orthogonal, the taagents to their curves of intersection must form a trirectangular trihedron. It follows that the equations
must be
,
where the symbol indicates that we are to replace by and add. These conditions for orthogonalism may be written in the following form, which is equivalent to the above satisfied
x>
then by
2,
:
* Lame",
TraiU des coordonnees curvilignes.
differ entiel, Vol.
I,
p. 181.
See also Bertrand, Traitt de Calcul
H,
TRANSFORMATIONS
43]
81
Let us then see what form the potential equation
ax2
assumes
in the variables p, p 1? p 2
az2
dy* First of
.
we
all,
find
dv _ dv dp
8V
dpi
aF apa
ax
dpi
dx
dp z dx
dp dx
and then a2
F_~ a^F/aA
ax
2
2
a2
2
\ax/
a/
F
a2
2
|
aF av
ap a^. ax
"aaT
"a7
ax2
a Pl a P2 |
dpidpz ax
\dx/
p^
F
apapi
ax
-\ax/
a/>ap
2
api
ax2
ap a
ax2
,
ax
ax
Adding the three analogous equations, the terms containing derivatives of the a 2 V / dp dpi fall out, by reason of the relations (33), and we have
second order like
a2
F
.
.
,
a2
v
~ +A
(34)
^ +A
A 2 (p)
2 ( P1 )
op
Lam&s
where AI and A 2 denote
The
differential
calculated.
From
differential
parameters of the the equations (31)
first
api
aj^
apg
api
ax
ap 2
ax
a 01
ap
a 01 api
ap
ax
api
ax
a0 2
a_p
a0 2
api
ap
ax
api
ax
-,
-,
,
dp
pT, apa
:
order Ai(p), Ai(pi), Aj(p 2 ) are easily
a^
dp
2 (p 2 )
we have
ax
whence, multiplying by
cpi
parameters
a^ ap ap
F
a 01 ap 2
_ _
ax
apa
a0 2 ap 2 _
. _
ax
ap 2
respectively,
and adding, we
dp
a0 ap ax
_ ~ /a0\ 2
\dp)
+
/a0A 2 /a0 2 + \dp/ \dpj
Then, calculating dp/dy and dp/dz in like manner, 2 i
*/
\dy/
/ap\ 2
\dz/
it is
easy to see that
1 (d\
,
...
+
find
FUNCTIONAL RELATIONS
82
43
[II,
Let us now set
a= dp
$
where the symbol by 0i, then indicates, as before, that we are to replace by 02) and add. Then the preceding equation and the two analogous equations
be written
may
=
A!(P)
=
Ai(pi)
,
H
Lame* obtained the expressions for A2 pa
A!(p 2 )
,
HI
= HZ
A 2 (pi), A 2 (p2 )
(p),
by a rather long calculation, which we may condense
as functions of
p, pi,
in the following form.
In the identity (34)
A2
F=
F -+ H 1
a2
dp
let
HI
us set successively a2
i
dp*
H
.
Cfr
&V +
1
z
T dpi
V
52^
i
HI
cpj
HZ
,
HI
-TT + Cpj
d
Moreover,
d<j>i/dp,
y,
.
,
)
a0
-- =0,
2
dp
cpz
W VF +
A2 ( p )
cp2
(p),
+ **( pl -^ + iF cpi op )
A 2 (pi), A 2 (p2 ).
dfa/dp, respectively, and adding,
1
=
^(pz) -
oe>4>
~d~p
62
_ ~
Up*
1
For instance, multiplying
we
find
ag
2 lp~
differentiating the first of equations (32) with respect to pi,
V
~~
~dp ~dp\
we
find
~ 2
dpi dp dpi
we have
o0 ev _~ _
^
ap"
"ap|
i a
g
2
ap
2
and consequently
A 2 (p)= -
-
+
--
2HHi
dp
2HH*
dp
0,
cpz
we have
In like manner
i
dpz
dpi
This gives the three equations
2
.a 2
,
,
dp
V = z.
c>
*->
and
.8V ,aF + A 2 (p 2 + A 2 (pi)
.8V 2 (p)
epjj
V=
which we need only solve for Aa
by
F
,
2
x,
i
a2
1
H -^ + A
,
2Hp_
\H
1
H
2
EXERCISES
H. EXS.]
83
Setting
*=i, this
zr,=
,
formula becomes
^
A2 (p) = and
*,=
,
in like
manner we
find
A2 (p!) =
A
h\
A.) hhzj
(log
api \
Hence the formula
(34) finally
A
(log \
cpz
--
ra2 F
*
ll*z
a
fifii
A
/.
,
h
LV
az 2
2
a
A
A"
becomes
^
^
ax2
A 2 (p 2 ) =
,
log
I
\aF~l I
I
*!*/
\
<>P
J
^/
*
(35)
a
condensed form,
or, in
Fa
.
*
Let us apply
this
dV\
/ h
2
2
a ,
/
\
1
/
ftj
\
nr
aF\
/ &2
a
/
dF\~|
^T^ s
I
The formulae
formula to polar coordinates.
/
I
of transforma
tion are
x
and
where values
=
p
replace pi and p 2
y
sin0cos,
,
and the
coefficients
ft,
have the following
hq
fti,
:
=
&
1
hi
=
-
h%
=
:
p sin
p
Hence the general formula becomes
A 2 F= or,
_ _ Fa
Ip
P sine \_dp
\
/
i -
2
2 sin(?
aF\ -
J
expanding, Ao
F=
F F ----i a2
a2
1
ap
which
is
2
2
p
a*?
2
a /
+
)
2 2 p sin
a2
F 2
a^>
h
a
+
c0J
- -i
h
aF\
.
(sin^
de\
dp]
/
aF\~| -J h
i
(-
a0 \sin^ a^ /
J
cot0aF aF -----
2
1
p
2
p
ap
a
susceptible of direct verification.
EXERCISES 1.
Setting
u
=
determinant D (u, exists between M,
x2 t>,
v,
+ w>)
v = x + y + z, w = xy + + /D (, y, 2) vanishes identically.
y
2
w.
Generalize the problem.
z2
,
yz
+
zx, the functional
Find the relation which
FUNCTIONAL RELATIONS
84
pi, EXS.
Let
2.
i
=
=
1
==.
-
un
,
Vi1 _ x -r 2 l
a-
2
*n
Derive the equation W2,
i,
,
M,,)
1
.
Using the notation
3.
x3
= COS 0i, = sin 0i cos 02 = sin 0i sin 02 cos0 3
Xn
=
Xi X2
,
,
sin0 n _ 1 cos0 n
sin 0i sin 02
,
show that (Xl, Xg,
,
Xn )
_
^_
1 )n s i nn
^ lS J n n-l^ 2Sm n-2^ g
Prove directly that the function
4.
z
where
and
a. is
z
=
satisfies
.
.
8 in 2
n _ ! sin
n.
F(x, y} defined by the two equations
= ax+ yf(a) + = x + yf (a) +
an auxiliary variable,
.
0(or), (a),
the equation
s2
rt
=
where /(a)
0,
(a) are arbitrary functions.
Show
5.
in like
manner that any
implicit function z
=
F(x, y) defined by
an equation of the form
where
(z)
and ^
are arbitrary functions, satisfies the equation
(z)
rg 6.
Prove that the function z z
where
a
is
equation pq 7.
=
(a)
[y
-
-
=
(a)]
2pqs
=
-
satisfies in like
+ tp2 =
0.
F(x, y) defined by the two equations 2,
an auxiliary variable and
(
(a)
x
+
a)
(a)
=
y
-
(a),
an arbitrary function,
satisfies
the
z.
Prove that the function z [Z
2
()]
2
=
Z2
(y2
manner the
F(x, y) defined by the two equations
_ a 2),
equation pq =
[z
_
((r)]
(Q,)
=
aa.2
xy.
8*. Lagrange s formulae. Let y be an implicit function of the two variables x and a, defined by the relation y = a + and let u =f(y) be any func tion of y whatever. Show that, in general, x<j>(y);
[LAPLACE.]
II,
EXERCISES
EM.] The proof
Note.
d
da where u
is
is
based upon the two formulas
F, du~\ F(u)~ =
d
x
J
dx
|_
F F(u)\dM~l
if
a and M
M\\
\
<(,(y)
dx
cu da
,
=
y
v)
(J>(u,
~i I
,.,
da-
^dx:/o v),
r~
S"
n
x =f(u,
and the nth derivative of u with
to /(a); i
-!
ff/*\
show
i
.
da_\
of y whatever, and F(u) is an arbitrary function of u. the formula holds for any value of n, it must hold for the
Setting x = 0, y reduces to respect to x becomes
If
du
7
dx ]_
any function
shown that value n + I.
It is
9.
85
J
l
\_
are two functions which satisfy the equations
dj
d
dj
d
du
dv
dv
du
that the following equation
is satisfied
<j>
identically
:
,
10. If the function F(x, y, 2) satisfies the equation
show
that the function
satisfies
the
same equation, where
11. If V(x, y, z) and Vi(x, show that the function
U= satisfies the
12.
F(z,
y, z)
y, z)
is
A;
a constant and r2
x2
+ yz + z 2 [LORD KELVIN.]
are two solutions of the equation
+
(x
2
+
+
y2
z2)
FI
.
A^V =
0,
(x, y, z)
equation
What form
does the equation (x
-
x 8 )7/"+
(1
- Sx 2 )?/ -
assume when we make the transformation x 13.
=
What form
xy
= Vl
= t-
?
does the equation 2
= Sx2
dx
dy
assume when we make the transformation x 14*. Let 0(xi, x 2 , x n MI, MS, , variables Xi, x 2 , x n , MI, u 2 , Mn , , , with respect to the variables MI, M 2 , ;
= u,
y
= l/v?
u n ) be a function of the 2 n independent homogeneous and of the second degree
,
,.
If
we
set
FUNCTIONAL RELATIONS
86 S
and then take p\ pz ,
-=pz,
~^-=p n
,
,
,
CU n
CUz
,
Exs.
cd>
C(t>
-=pi,
CUi
[II,
p n as independent variables in the place of Ui u z ,
,
,
un
,
goes over into a function of the form
the function
i,
Derive the formulae
X2
,
:
d
N be the point of intersection of a fixed plane P with the normal MN M of a given surface S. Lay on the perpendicular to the Find the tangent plane to the plane P at the point N a length Nm = NM. Let
15.
erected at
off
any point
M
describes the surface S. surface described by the point m, as The preceding transformation is a contact transformation. Study the inverse
transformation. 16. Starting from each point of a given surface S, lay off on the normal to Find the tangent plane to the surface 2 (the the surface a constant length I. parallel surface) which is the locus of the end points.
Solve the analogous problem for a plane curve.
M
to any point of 17*. Given a surface S and a fixed point O join the point the surface S, and pass a plane and the normal to the through In this plane surface S at the point M. draw through the point O a per pendicular to the line OM, and lay off on it a length OP = OM. The point ;
OMN
MN
OM
OMN
P
describes a surface 2, which is called the apsidal surface to the given surface S. Find the tangent plane to this surface.
The transformation is a contact transformation, and the relation between the and 2 is a reciprocal one. When the given surface S is an ellipsoid and the point is its center, the surface 2 is Fresnel s wave surface. surfaces
<S
Show
18*. Halphen s differential invariants.
dx2 / da*
dx2 dx dx*
remains unchanged when the variables mation ( 37). If in the
19.
of x, y,
we
z,
expression
x,
Pdx + Qdy +
,
v,
\dx*
y undergo any projective transfor
where P, Q,
fidz,
R
are any functions
set
Z = ^(M, v, y = (u, v, w) x=/(u, B, 10), w are new variables, it goes over into an expression ,
where
that the differential equation
>),
of the
form
Pidu + Q\dv + Ridw, where PI,
Qi,
RI
are functions of
,
v,
Show
w.
satisfied identically:
gi =
^
z)
I) (M, v,
w)
-P(Si
g>
that the following equation
is
II,
EXERCISES
Exs.]
87
where
/* _ du
\dv Let
20*. Bilinear covariants.
where
JTx,
X
2,
0
be a linear differential form
-^n are functions of the n variables x lt x2
,
:
,
,
xn
.
Let us
consider the expression
where
and where there are two systems
d and
of differentials,
5.
If
we make any
transformation Xi
=
4>i(yi,
y2
,
,
2/ n ),
=
(i
1, 2,
the expression Qj goes over into an expression of the
Q d = Y dy l +
F2
,
,
Yn are functions
of yi,
y2
,
,
n),
same form
+ Yn dy n
l
where FI,
.,
yn
,
Let us also set
.
and
Show tively,
that
H=H
,
identically, provided that
we
replace dx,
and
dxt,
respec
by the expressions
Syi
The expression //
is
-\
5j/ 2
+
H
Sy n
.
called a bilinear covariant of Qj.
21*. Beltrami s differential parameters.
If in
a given expression of the form
Edx* + 2Fdxdy + Gdy*, where E, F, G are functions of the variables x and y, we make a transformation z =/(M, v), y = ), we obtain an expression of the same form: (u,
EI du 2
+ 2Fidudv+
Gj dv2 ,
FUNCTIONAL RELATIONS
88
Exs.
[II,
where EI, FI, G\ are functions of u and v. Let 0(x, y) be any function of the and y, and 0i(u, v) the transformed function. Then we have, iden
variables x tically,
-
F
2
ex dy
ax
ffj
G!
-
+*
du
\dy
dv
dv
-F
Ff I
+
where x / (ex d are arbitrary constants, show that the relation
22. Schwarzian. a, 6, c,
Setting y
(ax
b)
+<8),
is
identically satisfied, respect to the variable
23*. Let
and
let
where x
x",
x
"
",
y
,
y
T/",
and
denote the derivatives with
t.
u and v be any two functions of the two independent variables x and
y,
us set
U=
au a"
where
,
t
yy
2
y
a function of
is
a, &, c,
,
c"
+ +
bv u
6"
+c +
F=
c"
are constants.
a w + w+
a"
6
u
6"
Prove the formulae
+c +
c"
:
c*udv_G*vdu
d*U
dV _ cPV SU
dx2 dx
ax2
dx
gx2 dx
dx*
dx
(u, v)
du
d*u dv
d*v
dx2 dy
ex2 dy
/dv \dx dxdy
dx dxdy/
(u, v)
dV ~ ax2 ~dy
a2
V dU
~d& Hy
dx dxdy
ex dxdy/
(ff.F)
and the analogous formulae obtained by interchanging x and
du dv
du dv
dx dy
dy dx
r
y,
where
dU dV
dV dU
dx
dx
dy
dy
[GOURSAT and PAINLEVE, Comptes rendus,
1887.]
CHAPTER
III
ELEMENTARY APPLICATIONS MAXIMA AND MINIMA
TAYLOR
S SERIES
TAYLOR S SERIES WITH A REMAINDER TAYLOR S SERIES
I.
44. Taylor s series with a remainder. In elementary texts on the Calculus it is shown that, if f(x) is an integral polynomial of degree n, the following formula holds for all values of a and h :
This development stops of (n +
l)th vanish.
/(x) which infinite
is
If
we
itself,
not a polynomial, the second member contains an In order to find the proper value to of terms.
number
assign to this development, for the difference
f(a
since all the derivatives past the
try to apply this formula to a function
+
h)
-f(a)
-
2f
we
f>
(a )
-
will first try to find an expression
li
2
f^
___ /W(a) J)
/"(a)
n
,
with the hypotheses that the function /(#), together with its first n n) derivatives / (a:), f"(x), f^ (x), is continuous when x lies in the interval (a, a -f A), and that f (n \x) itself possesses a derivative ,
/
(
"
+ J)
(x) in the
given, let
same
interval.
The numbers a and a
-f
h being
us set
(2)
is any positive integer, and where P is a number which is defined by this equation itself. Let us then consider the auxiliary function
where p
89
TAYLOR S SERIES
90
=f(a +
A)
-/() ~ _ O+ h 1.2-
It is clear
and
from equation
=
f
,
(x)
/-A
*)"
_
"
(a
44
~
_
+ h-x J
1.2..-
"ii
(2),
[in,
which defines the number P, that
from the hypotheses regarding f(x) that the func possesses a derivative throughout the interval (a, a -f A). = must have a root Hence, by Rolle s theorem, the equation (#) a + Oh which lies in that interval, where is a positive number which lies between zero and unity. The value of (x), after some results
it
tion
<(x)
easy reductions, turns out to be
The
first
other than a
P= h
p~
+
n -p +
h.
l
(l
l
cannot vanish for any value of x Hence we must have
factor (a -f h
x~)
-
+ 0)"-*
!/(
+J
>
(a
whence, substituting this value for
P
+
Ofy,
where
in equation (2),
0<^<1;
we
find
(3) /I
where JL
= .
We
2
n .p
shall call this formula Taylor s scries with a remainder,
R n the remainder.
and
This remainder depends upon the In practice, left undetermined. we have which positive integer p, about the only values which are ever given to p are p = n + 1 and p = 1. Setting p = n + 1, we find the following expression for the remainder, which is due to Lagrange the last
term or
:
setting
p
1,
we
find
Ill,
TAYLOR S SERIES WITH A REMAINDER
44]
91
an expression for the remainder which is due to Cauchy. It is will not be the same, in general, clear, moreover, that the number in these is
two
=
continuous when x
a,
we assume
If
special formulae.
the remainder
may
further that
/
(n
+ 1) (a:)
be written in the form
where e approaches zero with h. Let us consider, for definiteness, Lagrange s form. If, in the gen eral formula (3), n be taken equal to 2, 3, 4, successively, we get a succession of distinct formulae which give closer and closer approximations for f(a -f- A) for small values of h. Thus for n = 1 ,
we
find
1.2
1
which shows that the difference
/(* is
+ *) -/(a) -*/*()
an infinitesimal of at least the second order with respect to is finite near x = a. Likewise, the difference
provided that
_//
+ ,
f(a /\
7 \
_//
A)/
/
_/l / \ -f (a) *"
\
f(a) \ /
1 V
-
\ /
1
J.
is
h,
/"
JL
an infinitesimal of the third order
A
a
+
vf(a) \ /
h)J
-f v ~1
*^
O
*
//// \
\
f (a)
*/
/
and, in general, the expression
;
(a) \ /
:
-w!
f (n) \(a)f
+
1. is an infinitesimal of order n But, in order to have an exact idea of the approximation obtained by neglecting R, we need to an know an upper limit of this remainder. Let us denote by
M*
upper limit of the absolute value of of x = a, say in the interval (a a 17,
Kt provided that h |
* That defined in
which
is,
< |
y -f
i^r i<
"
+ 1)
rj).
(#) in the neighborhood
Then we evidently have
1
M)
77.
+ )(z) when |z The expression must be carefully distinguished from the expression J
3/>|/(
68,
(
I
a\
"
the
"
upper
limit,"
an upper
limit,"
is used here to denote a number greater than or equal to the absolute value of the function at any point in a certain interval. In this paragraph and in the next TRANS. a. /( + i)(a;) is supposed to have an upper limit near x
TAYLOR S SERIES
92
45
[III,
This result may be interpreted geomet that we wished to study a curve C, whose equa rically. Suppose tion is y =f(x), in the neighborhood of a point A, whose abscissa is a. Let us consider at the same time an auxiliary curve whose 45.
Application to curves.
C",
equation
A
line
x
is
=
a
-f-
h, parallel to
M
M
the axis of
y,
meets these two curves
two points and which are near A. The ordinates, by the general formula, is equal to in
This difference
is
difference of their
,
an infinitesimal of order not
less
than n
-+-
1
;
and
+
a consequently, restricting ourselves to a small interval (a 77, rj), the curve C sensibly coincides with the curve C By taking larger .
and larger values of n we may obtain in this way curves which differ less and less from the given curve C; and this gives us a more and more exact idea of the appearance of the curve near the point A. Let us
C
curve
first set
n
= \.
at the point
A
Then the curve
C"
is
the tangent to the
:
M
M
and the difference between the ordinates of the points and of the curve and its tangent, respectively, which have the same abscissa a -f h, is
Let us suppose that /"() 0, which is the case in general. preceding formula may be written in the form =
where
h.
ber
e
and
rj
c approaches zero with can be found such that
+
Since <
|
|
=
f"(a)
/"(a)
,
0,
when h
|
a positive
The
num
between rj + e will have
lies
For such values of h the quantity /"() the same sign as a an(i hence y Y will also have the same 77.
/"(
sign as
/"(a).
)>
If /"(a)
is
positive, the ordinate
y of the curve
is
Ill,
TAYLOR S SERIES WITH A REMAINDER
46]
93
greater than the ordinate F of the tangent, whatever the sign of h and the curve C lies wholly above the tangent, near the point A. ;
a ) is negative, y is less than Y, and the if below the tangent, near the point of tangency. If f"(a) = 0, let / (p) () be the first succeeding derivative which does not vanish for x = a. Then we have, as before, if f (p) (x) is continuous when x = a,
On
the other hand,
curve
/"(
lies entirely
and it can be shown, as above, that in a sufficiently small interval a ^ a -f- 17) the difference y Y has the same sign as the product ( A p/ (p) (a). When p is even, this difference does not change sign with h, and the curve lies entirely on the same side of the tangent, Y near the point of tangency. But if p be odd, the difference y its at the tangent changes sign with h, and the curve C crosses point of tangency. of inflection
it
;
now
Let us
In the latter case the point
occurs, for example,
take n
= 2.
Y =f(a) + whose axis ordinates
is
if
f
The curve C
=
A
is
called a point
0.
"(a)
is
in this case a parabola
:
(x
parallel to the axis of y\
and the difference of the
is
Y has the same sign as A 8 / "() for does not vanish, y sufficiently small values of h, and the curve C crosses the parabola
If
C
/
"(a)
at the point A.
to the curve
C
This parabola is called the osculatory parabola the parabolas of the family
for, of
;
Y = mx z + nx + p, this one
point
A
comes nearest to coincidence with the curve C near the (see
213).
of development. The formula (3) affords a development of the infinitesimal f(a + K) ~f( a } But, still more generally, let according to ascending powers of h. x be a principal infinitesimal, which, to avoid any ambiguity, we
46. General
method
method
for the
TAYLOR S SERIES
94 will
suppose positive
and
;
[III,
46
y be another infinitesimal of the
let
form y
(4)
where n l} n z
=
A lX + A 2 x* i
+ ...+x P (A p +
c ),
np are ascending positive numbers, not necessarily -, A p are constants different from zero, andc is integers, another infinitesimal. The numbers HI, A l n 2 A 2 may be cal ,
,
A l} A t
,
,
,
,
culated successively by the following process. First of all, it is clear that HI is equal to the order of the infinitesimal y with respect to x, and that
x approaches
v
is
equal to the limit of the ratio
n y/x when i
Next we have
zero.
A
y
A
:
xn
^
=
ui
=A
z
x"*
+
----|
(Ap
+ c)
x"p,
which shows that n z is equal to the order of the infinitesimal M I? and A 2 to the limit of the ratio u^/x n A continuation of this the terms. It is then clear that an infini succeeding process gives tesimal y does not admit of two essentially different developments of If the developments have the same number of terms, the form (4). coincide while if one of them has p terms and the other they -i.
;
p+
q terms, the terms of the
method
first
applies, in particular, to the
occur also in the second.
development of f(a
+
This f(a)
A)
according to powers of h and it is not necessary to have obtained the general expression for the successive derivatives of the func ;
tion f(x) in
us a practical
On the contrary, this method furnishes of calculating the values of the derivatives
advance.
means
Examples. Let us consider the equation (5) F(x, y)
where
= Ax n + By +
xy<S>(x,
y)
+
Cx n +
l
+ Dy +
+
2
-
-
-
= 0,
(x, y) is an integral polynomial in x and y, and where the terms not written down consist of two polynomials P(x) and Q(y), which are divisible, respectively, by x n + and y 2 The coefficients A 4>
1
.
B are each
supposed to be different from zero. As x approaches zero there is one and only one root of the equation (5) which ap proaches zero ( 20). In order to apply Taylor s series with a
and
remainder to this
root,
we should have
to
know
the successive deriv
which could be calculated by means of the general rules. But we may proceed more directly by employing the preceding method. For this purpose we first observe that the principal part atives,
TAYLOR S SERIES WITH A REMAINDER
46]
Ill,
equal to the substitution
of the infinitesimal root tion (5)
we make
is
y
(.4
/E)x
n
For
.
if in
95 the equa
=
and then divide by x n we obtain an equation of the same form
:
,
(cc,
yi)
As x approaches zero in y lt namely By^. the equation (6) possesses an infinitesimal root in y lt and conse quently the infinitesimal root of the equation (5) has the principal which has only one term
(A/B)x
part ?/!
n ,
as stated above.
(Al B)x**] and
is
where y z
is
we may
Likewise, the principal part of
set
another infinitesimal whose principal part
may
be found
by making the substitution
in the equation (6). Continuing in this
way, we may obtain for this root y an expres
sion of the form
(ap
+
.n
e)x
+ MJ H
1-
n
which we may carry out as far as we wish. All the numbers n np are indeed positive integers, as they should be, since H!, n z we are working under conditions where the general formula (3) is ,
,
In fact the development thus obtained is precisely the applicable. same as that which we should find by applying Taylor s series with
=
=
x. and h a remainder, where a Let us consider a second example where the exponents are not Let us set necessarily positive integers.
y
TAYLOR S SERIES
96
[III,
46
where
and fa, y u are two ascending series of positive a, ft, y, numbers, and the coefficient A is not zero. It is clear that the prin a and that we have cipal part of y is Ax ,
which
J]_
+
_
tt
y
00
-
- A x a (B x^ + Cix* + + B^ + dx* H----
+
Cx*
1
l
)
-
y
an expression of the same form as the original, and whose is simply the term of least degree in the numerator. It is evident that we might go on to find by the same process as many terms of the development as we wished. is
principal part
Let
/ (x)
be a function which possesses n
replacing a by x in the formula
f(x
+
-/(x) + -f(x) +
h)
we
(3),
~f"(x) l
+
*"
+ ..-+ 1
fi
.
Then
successive derivatives.
1
find
.
+
[/00(x)
n
2
e]
,
where e approaches zero with h. Let us suppose, on the other hand, that we had obtained by any process whatever another expression of the same form for
f(x
+
h)
= /(x) +
hfa
(x)
+
A2 02
(X )
+
.
.
.
+
hn
faty +
e
/]
.
These two developments must coincide term by term, and hence the coefficients are equal, save for certain numerical factors, to the successive 0i>
,
02>
n
derivatives of /(x)
:
This remark
sometimes useful
is
-_.
,
1.2
-n
1. 2-
in the calculation of the derivatives of certain
functions.
Suppose, for instance, that of a function of a function
we wished
to calculate the nth derivative
:
u=
where
y=f(u),
0(x).
Neglecting the terms of order higher than n with respect to
*
=
0(X
+
h)
-
0(X)
=
+
(X)
J
-^ 1
.
0"(X)
+
+
we have
** ,
1
2i
h,
.
n
2
and likewise neglecting terms of order higher than n with respect f(u
+
k)
~f(u)
=
f(u)
If in the right-hand side k
+
to k,
"
i
^
0W(Z );
.
*
1
.
2
n
be replaced by the expression
*"(
x)
+
+
T
i
.
&
n
and the resulting expression arranged according
to ascending powers of evident that the terms omitted will not affect the terms in h, h2 hn ,
,
A, it is .
The
HI,
TAYLOR S SERIES WITH A REMAINDER
47]
coefficient of
divided by
1
/i",
.
for instance, will be equal to the nth derivative of
n
2
coefficient of h n in the
For greater detail concerning Cours d Analyse (p. 59).
this
development of
method, the reader
for the
is
referred to Hermite s
Let f(x) and (x) be two functions same value of the variable x = a. Let us try
47. Indeterminate forms.*
approached by the ratio
to find the limit
f(a (a
h approaches
/[()]
and hence we may write
;
where Ai denotes the
which vanish
97
This
+ K) + h)
merely a special case of the problem of finding the limit approached by the ratio of two infinitesimals The limit in question may be determined immediately if the prin as
zero.
is
cipal part of each of the infinitesimals is known, which is the case whenever the formula (3) is applicable to each of the functions Let us suppose /(cc) and (x) in the neighborhood of the point a. that the first derivative of f(x) which does not vanish for x = a is that of order p, f (p \a) and that likewise the first derivative of (s) for x = a is that of order q, which does not vanish (a). (cc) and the the functions formula to each of Applying f(x) (x) (3)
;
<
<
<
and dividing, we
where
c
and
e
find
are
two
It is clear
infinitesimals.
from this result
that the given ratio increases indefinitely when h approaches zero, if q is greater than p and that it approaches zero if q is less than p. (7) (p) a If q a as ^ s p, however, the given ratio approaches / ( )/^ ( ) ;
=
limit,
and
this limit is different
from
zero.
Indeterminate forms of this sort are sometimes encountered in finding the Let
tangent to a curve.
x
=/(<),
y
=
z
*(t),
* See also
7.
=
*(l)
TAYLOR S SERIES
98
[m,
J
48
be the equations of a curve C in terms of a parameter t. The equations of the tangent to this curve at a point M, which corresponds to a value t of the param eter, are, as
we saw
in
5,
Z -
f
*
(tv)
f
(*o)
(o)
These equations reduce to identities if the three derivatives / (), (t), $ (t) all In order to avoid this difficulty, let us review the reasoning vanish for t = t be a point of the by which we found the equations of the tangent. Let
.
M
+
curve
C
Then
the equations of the secant
near to M, and
For the sake
let to
h be the corresponding value of the parameter.
MM
are
of generality let us suppose that all the derivatives of order less
= t but that at least the functions /(), (t), (t) vanish for t one of the derivatives of order p, say ( ), is not zero. Dividing each of the denominators in the preceding equations by hp and applying the general for mula (3), we may then write these equations in the form
than
p
(p>
1) of
\f/
,
/<*>>
to)
where
e, e
,
e"
+
J)
e
(to)
&>
are three infinitesimals.
+ If
f
e
(
(to)
+
let
h approach zero, these
">
we now
e"
equations become in the limit
which form
all indetermination has disappeared. points of a curve C where this happens are, in general, singular points the curve has some peculiarity of form. Thus the plane curve whose equations are
in
The where
X passes through the origin, and is the axis of x, and the origin
=
,
y
=
<3
= dy / dt = at that point. a cusp of the first kind.
dx/dt is
The tangent
If the sequence of derivatives of the function in unlimited the interval (a, a -f h), the number n in the f(x) formula (3) may be taken as large as we please. If the remainder 48. Taylor s series. is
Rn
approaches zero when n increases indefinitely, the following formula
down
we
are led to write
:
which expresses that the
/() +
\
series
/ () +
+ iT^r^ ^"W +
Ill,
TAYLOR S SERIES WITH A REMAINDER
48]
convergent, and that
is
formula (7)
is
fiable unless
n
is infinite,
ence of the (7)
may
"
sum
"
*
is
the quantity f(a
But
+
it is
h).
This
not justi
properly speaking. the remainder R n approaches zero when whereas the general formula (3) assumes only the exist
Taylor
we can show that
first
+
n
Replacing a by
1 derivatives.
x,
the equation
be written in the form
Or, again, replacing
(8)
its
s series,
99
/(*) =/(0)
This latter form
is
h by x and setting
+ / (O) +
-
+
a,
=
0,
--
we
find the
formula
/
often called Maclaurin s series; but
it
should
these different forms are essentially equivalent. The equation (8) gives the development of a function of x accord ing to powers of x the formula (7) gives the development of a func
be noticed that
all
;
tion of all
that
h according is
to
powers of h
:
necessary in order to pass
a simple change of notation is from one to the other of these
forms. It is only in rather specialized cases that
we
are able to show
Rn
approaches zero when n increases indefinitely. value of any derivative whatever is less the absolute for If, instance, when x lies between a and a than a fixed number h, it follows,
that the remainder
M
from Lagrange
s
+
form for the remainder, that + I
I*
1
h\"
^l. 2- ..( +
!)
an inequality whose right-hand member convergent e
x ,
is the general term of a the case, for instance, for the functions All the derivatives of ex are themselves equal to
series. f
sin x, cos x.
Such
is
x
and have, therefore, the same maximum in the interval con In the case of sin x and cos x the absolute values never exceed unity. Hence the formula (7) is applicable to these three Let us restrict ourselves to functions for all values of a and h. We find the form (8) and apply it first to the function f(x) = ex e
,
sidered.
.
* That is to say, the limit of the sum of the first n terms as n becomes infinite. For a definition of the meaning of the technical phrase the sum of a series," see 157. TRANS. This is essential to the con t The order of choice is a, h, M, n, not a, h, n, M. TRANS. vergence of the series in question. "
TAYLOR S SERIES
100
[III,
49
and consequently we have the formula
which applies to all values, positive or negative, of x. If a is any x rl Ka and the preceding formula positive number, we have a = e becomes ,
Let us
now
take f(x)
= sin x.
The
successive derivatives form a
recurrent sequence of four terms cos x, sin x, cos a;, sin x form another recurrent sequence 1, 0, their values for x
=
Hence
for
any positive or negative value
of
;
and 1, 0.
x we have
(11)
and, similarly,
(12)
.o.,-l-
J!L_
+
Let us return to the general case. The discussion of the remain R n is seldom so easy as in the preceding examples; but the problem is somewhat simplified by the remark that if the remain der approaches zero the series der
In general it is better, before examining whether this series converges. If for the given values of
necessarily converges.
Rn
,
to see
a and h the further
;
series diverges,
we can say
it
at once that
is
Rn
useless to carry the discussion does not approach zero when n
increases indefinitely.
than
all its derivatives, 1.
+
The function log(l x), together continuous provided that x is greater The successive derivatives are as follows
49. Development of log(l 4- x).
with
is
:
HI,
TAYLOR S SERIES WITH A REMAINDER
49]
_
Let us see for what values of
101
i
Maclaurin
a?
applied to this fu action. Writing have, under any circumstances,
s
formula
(8)
may
be
the series with a remainder,
first
we
*)
= -
+
+ (- !)-
+
The remainder R n does not approach
+ *,
zero unless the series
1 and of x between converges, which it does only for the values interval in this x lies When limit -f 1. 1, including the upper
+
the remainder
may
be written in the Cauchy form as follows
:
iyl.2-.-n
_ ~ 1.2 or
1
(9
(
The first factor x 1. Let us consider first the case where x factor second the and with zero 6)/(l + Ox) is (1 x, approaches less than unity, whether x be positive or negative, for the numer The last factor remains ator is always less than the denominator. Hence the remainder than less is for it |x|). finite, always 1/(1 <
|
R n actually approaches zero when n increases indefinitely. This form of the remainder gives us no information as to what happens when x = 1 but if we write the remainder in Lagrange s form, ;
it is
An
approaches zero when n increases indefinitely. 1 would be useless, examination of the remainder for x = evident that
Rn
TAYLOR S SERIES
1:02
[III,
We
sin f e the series diverges for that value of x. .
x
between
lies
1
and
49
have then, when
the formula
-f 1,
3
log(l+*)^-f + |
(13)
This formula
still
-
.
when x
holds
=
+ ....
+ (-l)-i
which gives the curious
1,
relation
Iog2l-|+|-j+. + (-1)-^ + .
(14)
....
The formula (13), not holding except when x is less than or equal to unity, cannot be used for the calculation of logarithms of whole numbers.
still
Let us replace x by
The new formula
x.
holds for values of x between
the corresponding sides,
we
+
2 1
When
x]
x varies from
\1
3
+
and
1
find the
o
+
1
;
and, subtracting
formula
f---
+2n
+T-T1
to 1 the rational fraction (1
+
obtained,
+#)/(!
a:)
and hence we may now easily cal steadily increases from 1 to still more rapidly con culate the logarithms of all integers. series be obtained, however, by forming the difference verging may <x>,
A
of the logarithms of let
two consecutive
integers.
For
this purpose
us set
1+ x 1-x
N+1 N
x
=
an equation whose right-hand member
is
or
1
2N+1
Then the preceding formula becomes
a series which converges
very rapidly, especially for large values of N. Note.
a
=
this
0,
h
Let us apply the general formula (3) to the function log (1 + x), setting We find in x, n = 1, and taking Lagrange s form for the remainder.
=
way log(l
+
x)
=
x
---x2
Ill,
If
TAYLOR S SERIES WITH A REMAINDER
49]
we now
replace x by the reciprocal of an integer n, this
Some
n is a positive number less than unity. be deduced from this equation.
where
1)
The harmonic
be written
may
2n2
n
may
103
series being divergent, the
interesting consequences
sum
-
v-r
\\/
\
r 1
1
+
1
n
I
-log
n (1
which shows that series 2(1
+
1 /
this
is
p)
limit,
which
decimals, 2)
is
is
1\
n
/
)
Hence the
n+1 = log n
n+ log
/n
1
(
1
n
1\ +-
V
difference
called Euler s constant.
C=
+
is smaller than the general term of the convergent increases indefinitely the expression
*/
under consideration approaches a finite Its exact value, to twenty places of
0.57721566490153286060.
Consider the expression
n+1 where n and p are two 7?e
+
the general term of a convergent series, for by the
log
approaches zero.
vw
n
term
When n
/p2 ).
log
For, let us write this difference in the form
finite limit.
Now 1 / p log equation above
\
difference
2n approaches a
.
\
n ,
But the
increases indefinitely with n.
%
31
may
n
positive integers
+
n+p
2
which are
to increase indefinitely.
write
n
2 1
+
1
+
2 1
2
+
+ p/
1
n
=
log (n
Then
TAYLOR S SERIES
104
[III,
C when n and p
where p n + P and p n approach the same value Hence we have also
50
increase indefi
nitely.
Now no
Hence the sum 2 approaches approaches a limit. If this ratio does approach a S approaches the limit log (1 + a).
the difference p n +p
limit unless the ratio
limit a, the
Setting
sum
p
p n approaches zero.
p/n
we
n, for instance,
n+
see that the
sum
n+2
1
2n
approaches the limit log 2. 50.
Development
continuous, and tions of x,
when
m m The function x) (1 x) is denned and derivatives all exist and are continuous func
of (1
its
1
+
+x
derivatives are of the
+1
positive, for
is
same form
= m(m = m(m )(x)
fM(x) /(
+
.
-
-
-
1)
1)
m
for the
;
:
- n + 1) (1 + - n) (1 +
(m (m
Applying the general formula (3),
any value of
as the given function
a)"
we
find
-...1.2and, in order that the remainder R n should approach zero, of all necessary that the series whose general term is
(m
1)
n
(m
"
it is first
+ 1)
1.2-.-W should converge.
But the
ratio of
m which approaches ing the case where
n
any term to the preceding -\-
1
is
x
x as n increases indefinitely. Hence, exclud is a positive integer, which leads to the ele
m
mentary binomial theorem, the series in question cannot converge unless \x 1. Let us restrict ourselves to the case in which a; 1. <
I
I
<
Ill,
TAYLOR S SERIES WITH A REMAINDER
50]
To show
105
that the remainder approaches zero, let us write
Cauchy form
it
in the
:
1.2...
The
first
factor
m(m
(m
1)
n~)
n
+l
1.2..-W approaches zero since it The second factor series. finally, the last factor (1 if
(1
m-1 + Ox) m ~
>
1
and
0,
we have
l
1 <
+
1
(1
the
is
m~l
Ox)
(1
+
is less 1 <
ftc)"
Hence
1 .
ici)"
term of a convergent than unity; and, than a fixed limit. For,
-f &c) is less
#)/(!
(1
+
"
general
2
"-
1
while
;
if
m-
for every value of x
1<
0,
between
we have the development
-.shall postpone the discussion of the case where x = 1. In the same way we might establish the following formulae
We
a
arcsm*
1 x = * + -+ ,
,
1
3 x6
^- + .
,
.5...(2r?
-1)
2.4.6--.2w 8 X -
which we
+
iC
5
-
iC
...
x2
"
+
1 "
2w-|-l
7 CC
2fl
+1
... y + + (-l)^
TI +---,
by a simpler process, and which hold 1 and + 1. Aside from these examples and a few others, the discussion of the remainder presents great difficulty on account of the increas It would therefore ing complication of the successive derivatives. seem from this first examination as if the application of Taylor s series for the development of a function in an infinite series were of shall prove later
for all values of x
between
Such an impression would, however, be utterly for these developments, quite to the contrary, play a funda mental role in modern Mathematical Analysis. In order to appre ciate their importance it is necessary to take another point of
limited usefulness. false
;
view and to study the properties of power series for their own
TAYLOR S SERIES
106
We
sake, irrespective cf their origin.
[III,
shall
5J
do this in several of
the following chapters.
Just
now we
will merely
remark that the
series
very well be convergent without representing the function f(x) from which it was derived. The following example is due to = (2/x*)eand, in Cauchy. Let /(*) = e~ *. Then / (*) general, the nth derivative is of the form
may
l
1
">
;
zero with x.*
All these derivatives vanish for x = 0, e~ l/3* by any positive power of x approaches Indeed, setting x = 1/z, we may write
and
known
where
P
is
a polynomial.
for the quotient of
it is
matter
well
how
large
that ez */z m increases indefinitely with z, no Again, let (x) be a function to which
m may be.
the formula (8) applies
=
Setting F(x)
F(0)
. <*>
F
(0),
+
)
(0)
:
e~ llx\ we
-
*
find
(0),
-
,
F<">(0)
^->(0),
,
and hence the development of F(x) by Maclaurin s series would The sum of the series thus obtained coincide with the preceding. function from that from which the different an represents entirely series
was obtained. if two
In general,
distinct functions f(x) are equal for x 0,
and
=
all their derivatives,
is
it
together with evident that the
(a;),
assumed that /(O) = 0, which is the only assignment which would = 0. But it should be noticed that no further assignment necessary for / (a:), etc., at x = 0. For *It
is tacitly
render/(:c) continuous at x is
,,,
which defines /
(x) at
x
m=
lim /(a) -/(O) x = ^~~
and makes /
(z)
_-
ft
continuous at
a:
=
0, etc.
TRANS.
Ill,
TAYLOR S SERIES WITH A REMAINDER
61]
107
Maclaurin series developments for the two functions cannot both be valid, for the coefficients of the two developments coincide. Let us consider, for
Extension to functions of several variables.
51.
= f(x, y, z) of the three independent vari definiteness, a function and let us ables x, y, z, try to develop f(x -f h, y -f k, z I) accord of to h, k, I, grouping together the terms of the same ing powers o>
+
Cauchy reduced
degree.
this
Let us give
lowing device.
to the preceding
problem
x, y, z, h, k,
by the fol and let
definite values
I
us set
where alone
;
t is
if
=f(x +
()
ht,y
to
it
Taylor
+
z
kt,
ft),
The function
an auxiliary variable.
we apply
+
depends on with a remainder, we find
s series
<()
t
(17)
f where (0), and its derivatives, for <
<(0),
(n) <
,
t,
0; and where 1 for the value
$() =/(w, u
v,
(
+
"
1)
<
+
But we may consider
zero and one.
r^
?
-T;
*<-+
>(*),
of the function (0) are the values
t
the derivative of order n
+
*<">(0)
<
(f)
&t,
is
(0)
where
(f)
the value of lies
between
as a composite function of
w), the auxiliary functions
=x
4- ht,
v
=
y
+
w=
kt,
K -f
It
According to a previous remark, the being linear functions of t. m is the same as if t/, expression for the differential of order m, d Hence we have the symbolic v, w were the independent variables. ,
equation
which may be written, after dividing by dtm
,
A For
t
=
0, u, v,
w
+ |CV
t
in the
form
+ LC CW
reduce, respectively, to x, y,
equation in the same symbolism becomes
z,
and the above
TAYLOR S SERIES
108
[III,
52
Similarly, (n
where
cc,
y, z are
to be replaced, after the expression
x If
respectively.
+ l)
+
developed, by
z + Bit, + 6kt, = 1 in (17), it becomes
y
Oht,
we now
is
set
t
,dx
(18)1
n Id \cx
1f\2 .
The remainder R n may be written
in the
-!/->
-
cy
I
*-;
cz
form n+1)
where
x, y, z
are to be replaced
+ 6h, y +
by x
6k, z
+
01 after
the
is expanded.* This formula (18) is exactly analogous to the general formula If for a. given set of values of cc, y, z, h, k, I the remainder R n
expression
(3).
we have a develop approaches zero when n increases indefinitely, of whose terms is a in each a series z ment of f(x + h, y + k, + I) it is But I. in very difficult, in gen h, k, homogeneous polynomial or not this remainder whether for R n eral, to see from the expression approaches zero. the formula (18) it is easy to draw certain conclusions analogous to those obtained from the general formula (3) in the For instance, let z =f(x, y) case of a single independent variable. function f(x, y), together If S. the surface of a be the equation order n, is continuous a certain to derivatives with all its partial up 52.
in the
/(
From
neighborhood of a point (X Q y ,
+
h,
y
+
k)
= f(x
,
y
~)
+
/ I
1
),
h df
^-
.
the formula (18) gives ,
-f
fc 7
2
Restricting ourselves, in the second member, to the first two terms, then to the first three, etc., we obtain the equation of a plane, then * It
is
assumed here that
all
the derivatives used exist and are continuous.
TRANS.
Ill,
TAYLOR S SERIES WITH A REMAINDER
52]
109
that of a paraboloid, etc., which differ very little from the given sur face near the point (x 0) y )- The plane in question is precisely the
tangent plane *
;
and the paraboloid
= Ax + 2 Bxy + 2
that one of the family
is
Cy*
which most nearly coincides with the given surface S. The formula (18) is also used to determine the limiting value of Let f(x, y) and a function which is given in indeterminate form. = a, y = b, but vanish for x which both functions be two (#, T/) <
which, together with their partial derivatives up to a certain order, Let us try to find the limit are continuous near the point (a, ). ratio the approached by
when x and y approach a and
b, respectively. derivatives df/da, df/8b, vanish simultaneously, we may write
the four
k[T*-+<}
<(>(a
+ h,b +
e,
,
c,,
e{
that all
8/db
+ k(%-
_
k)
i
h
/d
+ \,+ .
\Ta c
first,
do not
d/8a,
K)
where
Supposing,
first
, k
^)
approach zero with
\db
h and
When
&.
h and k approach zero
the point
and we will sup
(x, y) approaches (a, b~), pose that the ratio k/h approaches a certain limit a, i.e. that the point (x, y) describes a curve which has a tangent at the point (a, b~). Dividing each of the terms of the preceding ratio by A, it appears
that the fraction f(x,
y)/<$>(x,
o^~
;
y) approaches the limit
"+"
ca
a
~oT
do
--P
-Z
ca
OL
-^r-
cb
This limit depends, in general, upon a, which x and y approach their limits a and that this limit should be independent of relation
i.e. b,
a
upon the manner
respectively. it is
_ da db should hold
;
and such
is
db da
not the case in general.
in
In order
necessary that the
TAYLOR S SERIES
110
[HI,
53
vanish If the four first derivatives df/Sa, df/Sb, we should take the terms of the second order in the d/da,
8/db
simultaneously,
formula (18) and write
f(a
+
+ K) _
h, b
*
where e, e c u e/, e / are infinitesimals. Then, if a be given the same meaning as above, the limit of the left-hand side is seen to be ,
c",
a H- %-? a + 2 f-fj V? a cb co
2
which depends,
in general,
upon
a.
MAXIMA AND MINIMA
SINGULAR POINTS
II.
Let (x equation
53. Singular points.
of a curve
C whose
,
?/
is
M
be the coordinates of a point If the two first par 0. F(x, y~) )
=
derivatives 8F/dx, 8F/dy do not vanish simultaneously at this point, we have seen ( 22) that a single branch of the curve C passes through the point, and that the equation of the tangent at that tial
point
is
where the symbol d p + q F /dx$ dyl denotes the value of the derivative + vp p If dF/dx Q and dF /dy both van fip /8x di/* for x = x y = y ,
.
Let us suppose ish, the point (x y ) is, in general, a singular point* that the three second derivatives do not all vanish simultaneously x y for x y and that these derivatives, together with the third ,
=
,
,
derivatives, are continuous near that point. the curve may be written in the form.
Then the equation
* That is, the appearance of the curve is, in general, peculiar at that point. TRANS. exact analytic definition of a singular point, see 192.
of
For an
Ill,
MAXIMA AND MINIMA
SINGULAR POINTS
53]
;_ g \24-2
111
^
(19) 1
[OF I
cF
.
/
y*
where x and y are to be replaced in the third derivatives by - y ), respectively. We may assume X() + Q(x- B ) and ?/ + 0(y that the derivative d*F/dy$ does not vanish; for, at any rate, we Then, setting could always bring this about by a change of axes. 2 ic the = and t (x ar (x equation dividing by ) (19) y ) y ,
becomes
-
(20)1
0,
fyo
P (x
where
approaches x
x .
,
t*)
is
Now
a function which remains finite
let
If these roots are real
ti
and
when x
be the two roots of the equation
#2
and unequal,
i.e. if
2
the equation (20)
may
be written in the form
-
(t
,)
(t
*2
)
+ (x - x,) P =
0.
=
=x
=
t l9 t the above quadratic has two distinct roots t 2 which roots two As x approaches x that equation has approach # t and 2 respectively. The proof of this is merely a repetition of
For x
.
,
Let us set the argument for the existence of implicit functions. the write down t l -\- u, for example, and t equation connecting x
=
and u:
where Q (x, u) remains
finite,
while x approaches x and u approaches
M
and let tz Let us suppose, for definiteness, that t of value denote an upper limit of the absolute Q(x, ), and ra a h and x + h, x between t 2 + u, when x lies lower limit of t t zero.
l
>
;
TAYLOR S SERIES
112
[III,
53
and u between
h and -f h, where h is a positive number less than be a positive number less than h, and rj another positive number which satisfies the two inequalities t2
ti
Now
.
let c
m <
77
If
a;
h,
rj
e.
<
x
be given such a value that x
is
less
than
77,
\
the left-hand
side of the above equation will have different signs if e and then Hence that equation has a root which -+- c be substituted for u.
approaches zero as x approaches x
,
and the equation (19) has a
root of the form
=
V
!/o
+
- *o) (*i +
(a
a),
where a approaches zero with x x It follows that there branch of the curve C which is tangent to the straight line .
y
-
= y<>
*i
(*
is
one
z<>)
at the point (x , T/ O ). In like manner it
is easy to see that another branch of the curve passes through this same point tangent to the straight line x ). The point is called a double point; and y yQ = t 2 (x
M
the equation of the system of tangents at this point may be found x ), (y by setting the terms of the second degree in (x y ) in to zero. equal (19) If
Inside a suffi T/ O ) is called an isolated double point. (cc ciently small circle about the point Af as center the first member F(x, ?/) of the equation (19) does not vanish except at the point the point
,
M
itself.
For, let us take
x
= x + p cos
y
,
as the coordinates of a point near ,2
?/
+ p sin
Then we
.
~ cos
-
"
M
=
<^>
sin 4
find
+
-
sin 2
^
+ PL
H
where L remains finite when p approaches zero. Let be an upper limit of the absolute value of L when p is less than a certain posi tive
number
r.
For
all
values of
<
between
and 2?r the expression c2
cos A sin
d>
F
+ -T-T- sin
2
Ill,
MAXIMA AND MINIMA
SINGULAR POINTS
53]
has the same sign, since
its roots
113
m
be a lower are imaginary. Let it is clear that the coefficient
Then
limit of its absolute value. of p 2 cannot vanish for any point inside a circle of radius p<m/H. has no root other than p Hence the equation F(x, y) 0, i.e.
=
=
x
=x
,
y= y
,
inside this circle.
In case we have dx 8y the two tangents at the double point coincide, and there are, in gen to the same line, thus eral, two branches of the given curve tangent forming a cusp. The exhaustive study of this case is somewhat
and will be left until later. Just now we will merely remark that the variety of cases which may arise is much greater than in the two cases which we have just discussed, as will be seen
intricate
from the following examples. The curve y 2 = x s has a cusp of the
first
kind at the origin, both
branches of the curve being tangent to the axis of x and lying on The different sides of this tangent, to the right of the y axis.
2x 2 y + x* x 5 = has a cusp of the second kind, both curve y 2 branches of the curve being tangent to the axis of x and lying on the same side of this tangent for the equation may be written ;
y
=x
2
z%
and the two values of y have the same sign when x but are not real unless x is positive. The curve
is
very small,
has two branches tangent to the x axis at the origin, which do not becomes possess any other peculiarity for, solving for y, the equation ;
y~
x2
3 x2
V8 -
1+x
x2
2
and neither of the two branches corresponding to the two signs before the radical has any singularity whatever at the origin. It may also happen that a curve is composed of two coincident branches.
Such
is
the
case
for
the curve
represented by the
equation
When the point (x, y) passes
across the curve the first
vanishes without changing sign.
member F(x, y)
TAYLOR S SERIES
11-4
Finally, the point (cc y ) the case for the curve y 2
may
,
is
[III,
54
be an isolated double point. Such 4 0, on which the origin is an
+x +y = 4
isolated double point. 54.
F (x
>
three
In
first
that point
M
manner a point
of a surface S, whose equation is a general, singular point of that surface if the partial derivatives vanish for the coordinates x , y , z of
like
= )
y>
0, is, in
:
dF _ w,
5 CX Q
ZF -
_
CF = 7
0,
0.
CZ Q
CIJ^
The equation
of the tangent plane found above ( 22) then reduces an identity and if the six second partial derivatives do not all vanish at the same point, the locus of the tangents to all curves on to
;
M
the surface S through the point second order. For, let
is,
in general, a cone of the
be the equations of a curve C on the surface functions f(), $(), \l/(t) satisfy the equation the
first
and second
differentials satisfy the
dF -^
ex
y dx
+
cF
cF
+ 7T-
+
dy
cy
cF -5
dz
cz
=
F(x,
=
y, z)
=
0,
and
0,
dF Y ()F cF + ~dz) +~d*x + ^-d cz ex I Cy
=
Then the three
two relations
2)
,
d,j
cy
T
dx
S.
2
dF
y J
= 0. + ~d*z cz
=
For the point x x y z the first of these equations y z reduces to an identity, and the second becomes
+
2
,
,
r=dx dz = dxdy + 2 ^- dy dz + 2 -fcx dx oz dy
0y
The equation dx, dy, dz line
0.
d*t
of the locus of the tangents is given by eliminating between the latter equation and the equation of a tangent
dx
dz
dy
which leads to the equation of a cone
T
of the second degree
:
MAXIMA AND MINIMA
SINGULAR POINTS
in,M]
2
(21)
c
rV a%;
(V ^A
o
3
c2
F
-
=-
2=
-I-r-
2
8
F(Y
(A \
x nO/} (\ Y
C^-
?/o)
r
(Z
V 4-
11 y
, v,*
-
115
;
2s
)
+2^
F ^~
the other hand, applying Taylor s series with a remainder and carrying the development to terms of the third order, the equa
On
tion of the surface becomes
(22)
1.
^
7
CF
,
in the ter)ns of the third order are to be replaced- by where x, y, x + 0(x-x ), y + 8(y-yd, z + 0(z-s ), respectively. The the terms of equation of the cone T may be obtained by setting in z z the x in x the second degree equation (22) equal y y ,
,
to zero.
Let us then,
first,
suppose that the equation (21) represents a real Let the surface 5 and the cone T be cut by a
non-degenerate cone.
distinct generators G and G of plane P which passes through two find the to In order the cone. equation of the section of the sur us let this face 5 by imagine a transformation of coordinates plane, the which carried out changes plane P into a plane parallel to the
then sufficient to substitute z evident that for this curve the point
xy plane. It is
It is
=
z<>
M
from what we have just seen,
is
in the equation (22).
a double point with
this section is
composed two branches tangent, respectively, to the two generators G, G therefore resembles the two nappes The surface S near the point Hence the point n its vertex. near of a cone of the second degree
real tangents
;
of
.
M
is
M
called a conical point.
When
the equation (21) represents an imaginary non-degenerate fi. is an isolated singular point of the surface the cone, point Inside a sufficiently small sphere about such a point there exists no x other than x set of solutions of the equation F(x, y, z)
M
=
=
y =y
,
z
=z
.
For, let
M
be a point in space near
M
,
,
p the
TAYLOR S SERIES
116
MM
distance
Then
we
if
and
,
a,
(3,
[m,
y the direction cosines of the line
55
M M.
substitute
X
=X +
the function F(x,
y
pa,
=
+
y
p
K
}
= Z + py,
becomes
y, 2)
where L remains finite when p approaches zero. Since the equation (21) represents an imaginary cone, the expression or H-----h 2)
cannot vanish when the point 2
Let and
(a,
/?,
+ yS +
y) describes the sphere
2
2
y
=
1.
w
be a lower limit of the absolute value of this polynomial, be an upper limit of the absolute value of L near the Af If a sphere of radius be drawn about as center, point 2 it is evident that the coefficient of in the expression for F(x, y, z) p let
H
M
m/H
.
cannot vanish inside this sphere.
Hence the equation
F(x, y,z) = has no root except p = 0. When the equation (21) represents two distinct real planes, two nappes of the given surface pass through the point A/ , each of
which
is
tangent to one of the planes.
Certain surfaces have a
line of double points, at each of which the tangent cone degenerates into two planes. This line is a double curve on the surface along
which two distinct nappes cross each other. For example, the circle whose equations are z = 0, x 1 + y 2 = 1 is a double line on the surface whose equation is 4
+
2z 2( x *
+ ,f) -
(r*
+ ,f-
l)^
=
0.
When
the equation (21) represents a system of two conjugate imaginary planes or a double real plane, a special investigation is necessary in each particular case to determine the form of the sur face near the point The above discussion will be renewed in
M
.
the paragraphs on extrema. 55.
Extrema
of functions of a single variable.
be continuous in the interval
(a, 6),
and
let
Let the function f(x) c
be a point of that
SINGULAR POINTS
55]
Ill,
interval.
maximum
The function /(#) is minimum) for x
or a
MAXIMA AND MINIMA said to have an
=c
if
a positive
117
extremum
number
77
(i.e.
a
can be
found such that the difference f(c -\- A) f(c), which vanishes for h = 0, has the same sign for all other values of h between rj
and
+
If this difference is positive, the function f(x) has a c than for any value of x near c it is said
i).
smaller value for x to
have a minimum
ence f(c
-f A)
/(c)
=
;
is
On
the contrary, if the differ function is said to have a the negative,
at that point.
maximum. If the function f(x) possesses a derivative for tive
must vanish.
x
=
c,
that deriva
For the two quotients
-h
h
each of which approaches the limit / (c) when h approaches zero, have hence their common limit / (c) must be zero. Con
different signs
;
=
which lies between (#) versely, let c be a root of the equation a and b, and let us suppose, for the sake of generality, that the
/
first derivative which does not vanish for x = c is that of order n, and that this derivative is continuous when x = c. Then Taylor s series with a remainder, if we stop with n terms, gives
which may be written
/(
in the
form
+ A) -/() =
Let rj be a positive number such c approaches zero with h. and c that |/(n) ( c ) is greater than e when x lies between c 77. 77 For such values of x, /(n) (c) -f c has the same sign as f*- n) (c), and n (n) If consequently /(c -f A) /(c) has the same sign as A / (c).
where
+
|
n
is
there
odd, it is clear that this difference changes sign with A, and c. If n is even, is neither a maximum nor a minimum at x
=
f(c + A)
(n) /(c) has the same sign as/ (c), whether A be positive or negative hence the function is a maximum if / (c) is negative, (
;
and a minimum if f(n) (c) is positive. It follows that the necessary and sufficient condition that the function f(x) should have a maximum or a minimum f or x = c is that the first derivative which does not vanish for x = c should be of even order.
TAYLOR S SERIES
118
[III,
56
Geometrically, the preceding conditions mean that the tangent to the curve y =f(x) at the point A whose abscissa is c must be par allel to the axis of x, and moreover that the point A must not be a point of inflection. Notes.
When the hypotheses which we
have made are not
satisfied
the function f(x) may have a maximum or a minimum, although the derivative / (#) does not vanish. If, for instance, the derivative is infinite
for
=
x
c,
the function will have a
maximum
or a mini
Thus the function y = a^ is at the derivative changes sign. and the a minimum for x 0, corresponding curve has a cusp at the the the tangent being y axis. origin,
mum
if
=
When,
as in the statement of the problem, the variable which lie between two limits a and b, it
restricted to values
happen that the function has
its
absolute
maxima and minima
cisely at these limiting points, although the derivative
/ (x)
x
is
may pre does
not vanish there. Suppose, for instance, that we wished to find the shortest distance from a point P whose coordinates are (a, 0) z 2 R 2 0. Choosing for our to a circle C whose equation is x y
+
independent variable the abscissa of a point
M of
the circle C,
we
find
d2 or,
= PM = (x Z
making use
+ y = x + y - 2 ax + a 2
2
2
2 ,
of the equation of the circle,
d2
The general
2
a)
rule
=R + 2
a*
would lead us to try
-
2 ax.
to find the roots of the derived
But the paradox is explained if equation 2 a = 0, which is absurd. we observe that by the very nature of the problem the variable x If a is positive, d 2 has a minimum R and R. must lie between
+
for
x
=R
and a
maximum
for x
=
R.
56. Extrema of functions of two variables. Let f(x, y) be a con tinuous function of x and y when the point M, whose coordinates The are x and y, lies inside a region ft bounded by a contour C. function f(x, y) is said to have an extremum at the point (x 0) ?/ ) of the region O if a positive number rj can be found such that the
M
difference
which vanishes
for h
=k=
0,
of values of the increments h
keeps the same sign for all other sets and k which are each less than T in
Ill,
SINGULAR POINTS
50]
MAXIMA AND MINIMA
119
Considering y for the moment as constant and becomes a function of the single variable x and, by
absolute value.
equal to
;
?/o,
the above, the difference
cannot keep the same sign for small values of h unless the deriva Likewise, the derivative df/dy tive df/dx vanishes at the point must vanish at Q and it is apparent that the only possible sets of values of x and y which can render the function f(x, y) an extremum are to be found among the solutions of the two simultaneous
M
M
.
;
equations
*=o, tix
Let x
=x
,
y
=y
Q
f=o. cy
be a set of solutions of these two equations.
shall suppose that the second partial derivatives of f(x, y) do whose coordinates not all vanish simultaneously at the point
We
M
are y ), and that they, together with the third derivatives, Then we have, from Taylor s expansion, continuous near
are (x all
,
M
A
.
= 1.2
(23)
(3)
+ We
6
can foresee that the expression
will, in general,
dominate the whole discussion.
M
Q it is necessary and In order that there be an extremum at sufficient that the difference A should have the same sign when the inside a sufficiently small square point (X Q + h, y + k) lies anywhere as center, except at the center, where drawn about the point A = 0. Hence A must also have the same sign when the point
M
inside a sufficiently small circle whose k) lies anywhere for such a square may always be replaced by its center is A/ Then let C be a circle of radius inscribed circle, and conversely. All the points inside this r drawn about the point Q as center. (x
+ h,
y
+
;
M
circle are given
by
TAYLOR S SERIES
120 where
is
<
to vary from
to 2
[111,
and p from
TT,
+
r to
not to introduce this restriction.
A becomes
expression for
A=
what follows
substitution, the
this
Making
We might,
r.
indeed, restrict p to positive values, but it is better in
56
2
S
2 (A cos
+ 2 B sin
<
+ C sin
cos
2 <)
+ ^- Z,
where
and where Z
is
a function whose extended expression
it
would be
useless to write out, but which remains finite near the point (X Q y ). It now becomes necessary to distinguish several cases according to ,
the sign of
B 2 - A C.
First case.
Let
A
Bz
cos
AC <
has two real roots in tan of
>
Then the equation
0.
+ 2 B sin
2
<,
cos
<
and the
+ C sin
first
=
2
member
is
the difference
Hence we may write
two squares. 2
A o"
8
E
a ( a cos ^
+ b sin ^)
2 ""
P(
a>
cos
^
+
b>
sin ^) 2 ]
+^L
>
where
a If
>
0,
be given a value which
a cos
<
fta
aft
0,
>
satisfies the
=jfc
0.
equation
+ b sin = 0, <
A
will be negative for sufficiently small values of p be while, if = such that a 6 A will be for infinitesimal + 0, positive <
;
sin<
cos
Hence no number r can be found such that the differ has the same sign for any value of when p is less than r. It follows that the function f(x, y) has neither a maximum nor a minimum for x = x y = y values of
ence
p.
A
<
.
,
Second
case.
Let
A
B2
AC
cos 2
<
The expression
0.
+ 2.Bcos
cannot vanish for any value of
<.
H
Let
m
be a lower limit of
its
absolute value, and, moreover, let be an upper limit of the abso lute value of the function L in a circle of radius R about (z , y ) as
121
Finally, let r denote a positive number less than R and less H. Then inside a circle of radius r the difference A will
center.
than
MAXIMA AND MINIMA
SINGULAR POINTS
57]
Ill,
3m/
2 have the same sign as the coefficient of p i.e. the same sign as A Hence the function f(x, y~) has either a maximum or a mini or C. ,
mum
for x
To
= XQ
,
y
=
?/o-
recapitulate, if at the point (x
y
,
we have
)
* *">(>,
^dx dy
there
is
neither a
there
is
either a
maximum
maximum
nor a minimum.
or a
The ambiguous
The
case.
if
minimum, depending on the
2 the two derivatives c f/dx%, o^f/dyl. derivatives are negative, a minimum
57.
But
There if
maximum
a
is
sign of these
if
they are positive.
case where
B2
AC
=
is
not cov
The geometrical interpretation ered by the preceding discussion. be the shows why there should be difficulty in this case. Let surface represented by the equation z
= f(x,
?/).
If the function
f(x, y) has a maximum or a minimum at the point (X Q , y ), n ear which the function and its derivatives are continuous, we must have
which shows that the tangent plane whose coordinates are (x 0) y
M
,
,
to the surface ),
must be
In order that there should be a
S
at the point
parallel to the xy or a minimum it
maximum
plane. should lie is also necessary that the surface S, near the point , led we to study hence are of the on one side tangent plane entirely
M
;
the behavior of a surface with respect to point of tangency.
its
tangent plane near the
Let us suppose that the point of tangency has been moved to the and that the tangent plane is the xy plane. Then the equa
origin
tion of the surface is of the
(24)
where
z
=
a, b, c
ax 2
+
is
+
+ ax + s
cy*
3 /3x*y
are constants, and where a,
and y which remain tion
2 bxy
form
/8,
y, 8
+ 3 yxy + Sy 2
,
are functions of
x
when x and y approach zero. This equa same as equation (19), where x and y have
finite
essentially the
8
been replaced by zeros, and h and k by x and
y, respectively.
TAYLOR S SERIES
122
[III,
57
In order to see whether or not the surface S lies entirely on one side of the xy plane near the origin, it is sufficient to study the This section is given by the section of the surface by that plane.
equation
ax*
(25)
cy
2
+
ax*
= 0;
+
2
ac has a double point at the origin of coordinates. If b the negative, the origin is an isolated double point ( 53), and
hence is
+ 2bxy +
it
= y = 0, when the point equation (25) has no solution except x lies inside a circle C of sufficiently small radius r drawn (x, y) about the origin as center. The left-hand side of the equation (25)
keeps the same sign as long as the point (x, y) remains inside this S which project into the circle, and all the points of the surface interior of the circle
C
same side of the xy plane except case there is an extremum, and the por
are on the
In this the origin itself. tion of the surface S near the origin resembles a portion of a sphere
or an ellipsoid. the intersection of the surface S by its tangent If b 2 ac> 0, plane has two distinct branches C lf C z which pass through the the origin, and the tangents to these two branches are given by
equation
ax*
+ 2bxy +
cy
2
=
0.
Let the point (x, y) be allowed to move about in the neighborhood As it crosses either of the two branches C x C 2 the of the origin. left-hand side of the equation (25) vanishes and changes sign. ,
,
Hence, assigning to each region of the plane in the neighborhood of the origin the sign of the left-hand side of the equation (25), we find a configuration similar to Fig. 7. Among the points of the surface which project into points inside a circle about the origin in the xy plane there are always some which
below and some which lie above the xy plane, no matter how small the circle lie
The general aspect of the sur face at this point with respect to its tan gent plane resembles that of an imparted
be taken.
hyperboloid or an hyperbolic paraboloid. The function f(x, y) has neither a maxiFlG
The
-
7
case where b 2
ac
mum nor a minimum at the origin. = is the case in which the curve
of
intersection of the surface by its tangent plane has a cusp at the will postpone the detailed discussion of this case. If the origin.
We
Ill,
MAXIMA AND MINIMA
SINGULAR POINTS
58]
intersection
is
123
composed of two distinct branches through the
origin,
there can be no extremum, for the surface again cuts the tangent plane. If the origin is an isolated double point, the function f(x, y~)
has an extremum for x
=
It
0.
y
may
also
happen that the inter composed of two
section of the surface with its tangent plane is For example, the surface K coincident branches.
2 x*y -f x* y* 2 all along the parabola y = x The tangent to the plane z = 2 4 2 2 x y -}- x is zero at every point on this parabola, but is function ?/ positive for all points near the origin which are not on the parabola. is
.
58. In order to see which of these cases holds in a given example it is neces sary to take into account the derivatives of the third and fourth orders, and some times derivatives of still higher order. The following discussion, which is usually
When is applicable only in the most general cases. the equation of the surface may be written in the following form by using Taylor s development to terms of the fourth order: sufficient in practice,
62
ac
z
(26)
- f(x,
y)
= A(xsinu -y cos w) 2 +
fa
(x,
+
y)
--[x
24 \ dx
+
iW y
, ftr
dy /
Let us suppose, for definiteness, that A is positive. In order that the surface S lie entirely on one side of the xy plane near the origin, it is necessary that all the curves of intersection of the surface by planes through the z axis should But if the surface be cut lie on the same side of the xy plane near the origin. should
by the secant plane
=
y
xtan
the equation of the curve of intersection
x
=
p cos 0,
0,
found by making the substitution
is
= p sin
y
in the equation (26), the
new axes being the
plane on the xy plane.
Performing
z
where
K
sin
w
-f
trace of the secant
find
Kp
3
+
Lp*,
independent of p. If tan w ^ tan 0, z is positive for sufficiently small hence all the corresponding sections lie above the xy plane near the Let us now cut the surface by the plane ;
=
y If the
cos w sin 0) 2
and the
we
this operation,
is
values of p origin.
= A p 2 (cos
<j>
old z axis
corresponding value of
K
is
x tan
u.
not zero, the development of z
is
of the form
Hence the section of the surface by this plane has a p. It follows that the point of inflection at the origin and crosses the xy plane. Such is function /(x, y) has neither a maximum nor a minimum at the origin. the case when the section of the surface by its tangent plane has a cusp of the and changes sign with
first
kind, for instance, for the surface z
= w2
x8
.
TAYLOR S SERIES
124
[HI,
58
K
= for the latter substitution, we would carry the development out to If terms of the fourth order, and we would obtain an expression of the form where K\
a constant which
is
We
the fourth order.
be readily calculated from the derivatives of K\ is not zero. For infinitesimal val
may
shall suppose that
ues of p, z has the same sign as K\ if K\ is negative, the section in question lies beneath the xy plane near the origin, and again there is neither a maximum nor x4 whose a minimum. Such is the case, for example, for the surface z = y 2 2 = consists of the two x intersection with the xy plane Hence, parabolas y ;
,
.
K=
at the same time, it is evidently useless to carry the and K\ investigation farther, for we may conclude at once that the surface crosses its tangent plane near the origin.
unless
But
if
>
K=
and KI
at the
>
same time,
made by
the sections
all
planes
above the xy plane near the origin. But that does not show conclusively that the surface does not cross its tangent plane, as is seen
through the z axis
lie
by considering the particular surface z
-
(y
-
x 2 ) (y
-
2 x 2 ),
its tangent plane in two parabolas, one of which lies inside the other. In order that the surface should not cross its tangent plane it is also necessary
which cuts
that the section of the surface
through the z axis should
made by any
cylinder whatever which passes Let y = (x) be the
wholly above the xy plane.
lie
equation of the trace of this cylinder upon the xy plane, where (x) vanishes for x = 0. The function F(x) =/[x, 0(x)] must be at a minimum for x = 0, what
ever be the function In order to simplify the calculation we will suppose (x). that the axes have been so chosen that the equation of the surface is of the form z
where
A
is
positive.
With
= Ay 2 + this
+
3 (x, !/)
system of axes
,
we have
=0 at the origin. The derivatives of the function F(x) are given
F
"
x
^>0
dx dy
dx*
dyQ
8xo
by the formulae
=
+
3
L
$"
(
X)
+
3
dxdy
dy
+4 -*-* 6
d2
f
dx dy
fart
4.
6
(x)
+
12
2
W *
*(x)
53 f
^3 f -^-
dx 2 dy
^ ^
0"
,
d
f
dy
2
$
"(),
cy
|
1
^ ^(z)
f + 6 -^3 3
^-
dxdy2 2
^
+
2
4>"
dy
,
,
//2
8/ dy
|
g*^
*
Ill,
MAXIMA AND MINIMA
SINGULAR POINTS
59]
from which, for x
=
y
= 0, we
125
obtain
c !/0 If
tf>
(0)
does not vanish, the function F(x) has a minimum, as is also apparent But if (()) = 0, we find the formulas
from the previous discussion.
<
Hence, in order that F(x) be at a minimum,
and
that the following quadratic
r dx* be positive for
necessary that d*f/x% vanish
tf>"(0),
cx2 T~ dy
values of
it is
in
dy
2
>
0"(0).
easy to show that these conditions are not satisfied for the above function 3x 2 y + 2z4 but that they are satisfied for the function z = y2 + x*.
It is
z
all
form
= y2
,
that the latter surface lies entirely above the xy plane. attempt to carry the discussion farther, for it requires extremely nice reasoning to render it absolutely rigorous. The reader who wishes to exam ine the subject in greater detail is referred to an important memoir by Ludwig It is evident, in fact,
We
shall not
Scheffer, in Vol.
XXXV of the Mathematische Annalen.
59. Functions of three variables.
function of the three variables tion
is
said to have an
of values x
,
y
,
z
Let u
x, y, z.
= f(x, y, z) be
a continuous
Then, as before, this func
extremum (maximum or minimum) number rj can be found
a positive
if
for a set
so small
that the difference
which vanishes
for
h
=
sets of values of h, k,
k I,
= = 0, I
has the same sign for
each of which
is
all
other
less in absolute
value
than
If only one of the variables *, y, z is given an increment, i]. while the other two are regarded as constants, we find, as above,
that u cannot be at an
extremum unless the equations
are all satisfied, provided, of course, that these derivatives are con tinuous near the point (or y0) z Let us now suppose that x y z are a set of solutions of these equations, and let be the point ,
~).
whose coordinates are xw yQ sphere can be drawn about
,
z
M
.
,
,
M
There will be an extremum
so small that f(x, y, z)
f(x
0)
y
if ,
a
z)
TAYLOR S SERIES
126
[III,
59
M
inside the sphere. has the same sign for all points (x, y, z) except Let the coordinates of a neighboring point be represented by the
equations
x
= x + pa,
y
= z + py, = 1 and let us replace
= y + pft,
z
where a, ft, y satisfy the relation a 2 + /3 2 + y2 z in Taylor s expansion of f(x, x y x y z ;
y, ) by pa, This gives the following expression for A
,
,
py, respectively.
A= where
ft,
(a,
pft,
:
2
p
[>O,
+.--],
y)
ft,
y) denotes a quadratic
form in
ft, y whose coeffi and where Z is a
a,
cients are the second derivatives of f(x, y, z),
M
function which remains finite near the point The quadratic Q form may be expressed as the sum of the squares of three distinct linear functions of a, ft, y, say P, P multiplied by certain con stant factors a, a except in the particular case when the dis criminant of the form is zero. Hence we may write, in general, ,
,
.
P",
a",
= aP + a P + 2
2
*(a,
ft,
y)
2 a"P"
,
where a, a are all different from zero. If the coefficients a, a have the same sign, the absolute value of the quadratic form will remain greater than a certain lower limit when the point a, ft, y ,
a"
,
a"
describes the sphere 2
+
2
ft
+y = 2
1,
and accordingly A has the same sign as a, a when p is less than a certain number. Hence the f imction f(x, y, z) has an extremum. If the three coefficients a, a do not all have the same sign, a"
,
,
there will be neither a
example, that a
>
0,
a
<
a"
maximum 0,
and
P =
let
nor a minimum. us take values of
=
Suppose, for a,
ft,
y which
P
0. These values cannot cause 0, satisfy the equations to vanish, and A will be positive for small values of p. But if, on the other hand, values be taken for a, ft, y which satisfy the equa
P=
P"
= 0, A
will be negative for small values of p. the same for any number of independent variables the discussion of a certain quadratic form always plays the prin In the case of a function u cipal role. f(x, y, z) of only three
tions
0,
P"
The method
is
:
=
independent variables it may be noticed that the discussion is equivalent to the discussion of the nature of a surface near a singu lar point.
For consider a surface 2 whose equation F(*>
y,
)
=f(x
>
y>
*)
-f(*o, y
,
is
O = 0;
Ill,
MAXIMA AND MINIMA
SINGULAR POINTS
60]
127
M
this surface evidently passes through the point n whose coordi if the function f(x, y, z) has an extremum and nates are (x y ), is a singular point of 2Hence, if the cone of there, the point ,
,
M
M
is imaginary, it is clear that F(x, y, z) will keep the tangents at as center, and same sign inside a sufficiently small sphere about
/(#
)
2/j
M
w iH
maximum
surely have a
cone of tangents
or
is real,
is
minimum. But if the two real distinct planes,
or a
composed of
several nappes of the surface pass through A/ , and F(x, y, z) changes sign as the point (x, y, z) crosses one of these nappes. 60. Distance from a point to a surface. Let us try to find the maximum and the values of the distance from a fixed point (a, b, c) to a surface S whose
minimum equation
is
F(x,
y, z)
=
u
=
The square
0.
=
d*
(x
-
a)2
of this distance,
+
(y
-
+
6)
-
(
c),
x and y, for example, if z be a function of two independent variables only, = 0. In order considered as a function of x and y defined by the equation is
F
that
u be
at
an extremum for a point
(x, y, z)
of the surface,
we must
have, for
the coordinates of that point, 1
du
=
2 dx 1
du _
=
(x
y
,
a)
+
(z
&
+
(z
_
2 dy
We find,
in addition,
dFdz =
-\
dx
-
.
dz ix dz
c)
= 0, =
0.
cy
F=
from the equation
dF
c)
the relations
dFdz =
dF U,
1
dx
dz
0,
dz
dy
rt
U,
dy
whence the preceding equations take the form a
x
_ ~~
djr dx
This shows that the normal
b
y
_
c
z
"
d_F_
c_F_
dz
dy
to the surface
S
at the point
(x, y, z) passes through Hence, omitting the singular points of the surface S, the the points sought for are the feet of normals let fall from the point (a, 6, c) upon In order to see whether such a point actually corresponds to a maxi surface S. mum or to a minimum, let us take the point as origin and the tangent plane as the xy plane, so that the given point shall lie upon the axis of z. Then the func
the point
(a, 6, c).
tion to be studied has the
form
u where z
is
= x* +
y2
+
(z
-
c)2,
a function of x and y which, together with both its first derivatives, = y = 0. Denoting the second partial derivatives of z by r, s, t,
vanishes for x
we
have, at the origin,
^= dx*
2(1 -or),
-fiL dxdy
1*
^ = 2(l-cO, dy
2
TAYLOR S SERIES
128 and
it
(r
=
_
C2 2
_
(1
_
cr) (1
roots of the equation A (c) 2 + 4 (s2 rt) = 4 s2 + (r )
+
61
only remains to study the polynomial A(C)
The
[III,
= t)
2.
ct )
=
C 2( S 2
_
rt )
+
(r
+
t
)
c
_L
are always real by virtue of the identity There are now several cases which must
be distinguished according to the sign of
s2
rt.
0. First case. Let s 2 rt The two roots Ci and c 2 of the equation A (c) = have the same sign, and we may write A(c) = (s 2 Let us rt) (c Ci) (c Cj). <
now mark the two points A\ and A%
of the z axis
whose coordinates are
c\
and
c2
.
These two points lie on the same side of the origin and if we suppose, as is always allowable, that r and t are positive, they lie on the positive part of the If the given point A (0, 0, c) lies outside the segment AiA z z axis. A(c) is In order to see negative, and the distance OA is a maximum or a minimum. which of the two it is we must consider the sign of 1 cr. This coefficient does not vanish except when c = 1 /r and this value of c lies between Ci and c 2 ;
,
;
,
cr is positive hence 1 cr is posi But, for c = 0, 1 (1/r) = s /r tive, and the distance OA is a minimum if the point A and the origin lie on the same side of the segment A\A%. On the other hand, the distance OA i& a
A
since
2
2
.
;
maximum When the
if
A
nor a maximum.
A
and the origin lie on different sides of that segment. between AI and A 2 the distance is neither a minimum The case where A lies at one of the points AI, A 2 is left in
the point
point
lies
doubt. of the two roots c\ and c 2 of A (c) = is and the origin lies between the two points If the point A does not lie between A\ and A 2 A\ and J. a A(c) is positive and there is neither a maximum nor a minimum. If A lies between AI and A 2 A (c) is negative, 1 cr is positive, and hence the distance OA is a minimum.
Second
positive
s2
Let
case.
and the other
rt
is
>
0.
One
negative,
.
,
,
Third case.
Let
s2
-
rt
=
0.
seen, as above, that the distance
Then A(c) =
OA
a
(r
+
minimum
t) (c
-
cj),
and
it
is
easily
A
the point and the origin lie on the same side of the point AI, whose coordinates are (0, 0, Ci), and that there is neither a maximum nor a minimum if the point AI lies between the point
A
and the
is
if
origin.
A 2 are of fundamental importance in the study of curva they are the principal centers of curvature of the surface S at the point 0.
The points AI and ture
;
61.
the
Maxima and minima
maxima and minima
are connected
We often need to find
of implicit functions.
of a function of several variables
which Let us consider, for
by one or more relations.
example, a function to = f(x, y, z, tt) of the four variables which themselves satisfy the two equations /i (*, y,
*,
)
=
0,
/,(*, y,
z,
M)
x, y, z, u,
= 0.
For definiteness, let us think of x and y as the independent vari and of z and u as functions of x and y defined by these equa
ables, tions.
Then
the necessary conditions that
u>
have an extremum are
HI,
MAXIMA AND MINIMA
SINGULAR POINTS
61]
2
dx
and the
^^
+ dz
dx
+
129
= !/+? + 3?!? du dy dy dy
=
dz
du dx
partial derivatives dz/dx, du/dx, dz/dy,
are given
du/dy
by the relations
=o M_i.^^4.M^ dz dx^ du dx --P 0/i0"_n --,0/10*, o P ~^T~ "TT~
^z/
The elimination
0/,0*
,
P
"a
o
"o~
oz cy
cy
o
du dx
0/,0u
,
P a cu
U-
"a~
cy
du/dy leads
of dz/dx, du/dx, dz/dy,
=
?!f
4.
dz dx
0/,
",
du dy
"5
^* dy
4.
dx
0/i "^
a*
2
dx
to the
new
equations of condition
.p
_
ft
/)(*, *, u)
= 0, /2 = 0, determine the val which, together with the relations /x But the equa ues of x, y, z, u, which may correspond to extrema. of \ and p. find values we can that tions (27) express the condition which
satisfy the equations -
-t-
A
c
3 02
-^-
-I- w.
OX
IT-
=
OX
~
a
"j
l
d
Jl
^=o
3s
ds
A
"
"o
I*
dy
dy
"a~
=w
i
oy
^ ^ = + X^4 du du dw 2
2
0-
hence the two equations (27) may be replaced by the four equations unknown auxiliary functions. (28), where X and p. are The proof of the general theorem is self-evident, and we may state the following practical rule
:
Given a function /m ^2 iCjj Atm
.
,
,
*
*F *, \
n)
of n variables, connected by h distinct relations
in order to find the values of
x^
a; 2
of the auxiliary function
regarding \ 1} X 2
,
,
XA as constants.
,
,
function an extremum we must equate
to
xn which
may
render this
zero the partial derivatives
TAYLOR S SERIES
130 62. Another example.
mum
[III,
We shall now take up another example,
62
where the mini
not necessarily given by equating the partial derivatives to zero. Given a triangle ABC; let us try to find a point of the plane for which the sum is
P
PA + PB + PC
of the distances
to the vertices of the triangle is a Let (01, 61), (a 2 , 62 ), (a 3 63) be respectively the coordinates of the B, C referred to a system of rectangular coordinates. Then the func
minimum.
,
vertices
A, tion whose minimum (29)
z
P
from
is
sought
= V(x - oi)a + (y-
where each
is
+ V(x - a2 2 + (y - 62 2 + V(x -
a
6i)
of the three radicals
)
is
)
to be taken with the positive sign. This equa is evidently entirely above the xy plane,
S which
tion (29) represents a surface
and the whole question reduces to that of finding the point on is nearest the xy plane. From the relation (29) we find
a V(x - m) +
s*
dz
_
y
V(z _
ft
(y
-
H
x
--
+
a-2
,
V(x -
&!)2
-
y
-
(y
1
V(x -
&!)2
+
a 2) 2
-bi
ai )2
+ (y -
a,)*
2
as)
(y
x
+
-
this surface
V(z -
6 2 )2
+
as
as )
62
y (y
-
V(x - a 3 )
btf
which
2
+
(y
-
-b3 2 + (y -
63 ) 2
and it is evident that these derivatives are continuous, except in the neighbor hood of the points A, B, C, where they become indeterminate. The surface S, therefore, has three singular points which project into the vertices of the given The minimum of z is given by a point on the surface where the tan triangle. gent plane is parallel to the xy plane, or else by one of these singular points. In order to solve the equations cz/cx
=
=
cz/dy
0,
0, let
us write them in the
form x
V(x
-
i
+ (y-
fll )2
+
(y
V(x -
a2 ) 2 y
-
x
:
2
&i)
y-E>i
V(X -
a2
x
0,1
ai)
V(X -
&!)
Then squaring and adding, we
aa)
+
2
(y
-
V(x -
62 ) 2
_
&2
a3
as)2
y
-
+
(y
-
b3
+(y- 62 )2
find the condition
_ V(x - a2 ) 2 + The geometrical interpretation of this result cosines of the angles which the direction
(y
-
is easy denoting by a and /3 the makes with the axes of x and j/, the angles which PB makes with the :
PA
respectively,
same
axes,
and by a and
we may
the cosines of
/3
write this last condition in the form 1
+ 2 (aa +
APE by
or, denoting the angle
0,
u
+
1
=
0.
segment AB subtends an For the same reason each of the angles BPC and clear that the point P must lie inside the triangle
in question expresses that the
angle of 120 at the point P.
CPA
=
o>,
2 cos
Hence the condition
)
7
flS
must be 120.*
It is
*
The reader
is
urged to draw the figure.
MAXIMA AND MINIMA
SINGULAR POINTS
63]
Ill,
131
ABC, and that there is no point which possesses the required property if any In case none of the is equal to or greater than 120. angle of the triangle angles is as great as 120, the point P is uniquely determined by an easy con In this case the minimum is given struction, as the intersection of two circles.
ABC
P or by one of the vertices of the triangle. But is easy to show PA + PB + PC is less than the sum of two of the sides of the tri For, since the angles APB and APC are each 120, we find, from the angle. two triangles PAG and PBA, the formulae AC = Va 2 + c 2 + oc, AB = Vi + b + ab, by the point that the
it
sum
2
where
PA = a, PB = Vo2 +
PC =
6,
62
+
06
>
c.
But
evident that
it is
Va 2
6+-,
-)-
+
c2
ac>
c
2
and hence
AB + AC
>
a
+
+
b
+
~, 2
c.
P therefore actually corresponds to a minimum. one of the angles of the triangle ABC is equal to or greater than 120 there exists no point at which each of the sides of the triangle ABC subtends an angle of 120, and hence the surface S has no tangent plane which is parallel to In this case the minimum must be given by one of the vertices of the xy plane. The point
When
and
the triangle, It is
evident, in fact, that this
it is
is
the vertex of the obtuse angle.
easy to verify this fact geometrically.
63. D Alembert s theorem. Let F(x, y) be a polynomial in the two variables x and y arranged into homogeneous groups of ascending order
=
F(x, y)
where
H+
P
(x,
y)
+
p+i
H
(x, y)
+ = 0,
+
m (x,
y),
considered as an equation If the equation p (x, y) is a constant. has a simple root, the function F(x, y) cannot have a maximum or a mini for x = y = 0. For it results from the discussion above that there exist sec
in y/Xj
mum
H
+ = F(x, y) made by planes through the z axis, some which lie above the xy plane and others below it near the origin. From this remark a demonstration of d Alembert s theorem may be deduced. For, let/(z) be an integral polynomial of degree m, tions of the surface z
of
/(z)
where the
f(x
where
+
iy)
OQ, &o
let
= i,
a
us write this in the form
+
&ii
ib
+
(ai
+
t&i) (x
P and Q have
and hence,
finally,
+
iy)
We
m, &m are real.
f(z)
where
,
In order to separate the real and
coefficients are entirely arbitrary.
imaginary parts
+ A m zm
= A 9 + AIZ + A 2 z* +
o
Q=
&
+
-
+ etix + bix +
:
biy
a^y
(a m
have then
= P+iQ,
the following meanings
P=
+
----
-\
,
+
;
+
ib m ) (x
+
iy)
m ,
TAYLOR S SERIES
132
We
will first
show that
|/(z)
or,
|,
63
[III,
what amounts
to the
same
thing, that
Q2 cannot be at a minimum for z = y = except when a = 60 = 0. For this purpose we shall introduce polar coordinates p and 0, and we shall suppose,
ps
-j.
f
A
for the sake of generality, that the first coefficient after Then we may write the equations vanish is p
A
P=o + P
2
Q= + Q2 =
which does not
.
b of,
+ +
(ap cos (bp
cos
6g +
p
bp sin
p0 +
2/>p
dp sin
[(aoOp
+
p)
p<j>)
pp pp
+ +
b bp ) cosptf.
,
,
+
- a
ap
(b
bp ) sinp0]
+
where the terms not written down are of degree higher than p with respect But the equation (aoap
+
b bp)
cosp<j>
+
ap
(b
a bp )
smp<J>
,
to
p.
=
= K, which determines p straight lines which are separated by It is therefore impossible by the above remark that angles each equal to 2 n /p. 2 + Q2 should have a minimum for z = y = unless the quantities gives tan p$
P
aoap both vanish. that
-f
&o&p
boa,p
(tobp
But, since a + ft is not zero, this would require that a = 60 = ; that the real and the imaginary parts of /(z) should both vanish at the 2
is,
2
origin. If |/(z) has a minimum for z = a, y /3, the discussion may be reduced to the preceding by setting z = a + i/3 + z It follows that \f(z) cannot be at a minimum unless and Q vanish separately for a, y = p. The absolute value of /(z) must pass through a minimum for at least one |
.
\
P
value of
z,
nitely.
In
for
it
fact,
x=
increases indefinitely as the absolute value of z increases indefi
we have
where the terms omitted are of degree written in the form
where
t
less
than 2 m in
approaches zero as p increases indefinitely.
drawn whose radius
VP
R is
p.
This equation
Hence a
circle
may
be
may be
2 so large that the value of + Q2 is greater at every It follows that point of the circumference than it is at the origin, for example. there is at least one point
x
a,
y
=
Vp + Q 2 is at a minimum. By the above it fol y = /3 is a point of intersection of the two curves which amounts to saying that z = a + /3i is a root of the equation
inside this circle for
lows that the point x
P = 0, Q = 0,
=
which
=
a,
/(*)=. In this example, as in the preceding, we have assumed that a function of the two variables x and y which is continuous in the interior of a limited region actually assumes a minimum value inside or on the boundary of that region. This is a statement which will be readily granted, and, moreover, it will be rigorously demonstrated a
little later
(Chapter VI).
Ill,
EXERCISES
EM.]
133
EXERCISES 1.
Show
that the
number
which occurs in Lagrange s form of the re + 2) as A approaches zero, provided that
0,
mainder, approaches the limit l/(n /( + 2J(a) i s not zero. 2.
Let F(x) be a determinant of order n,
Show
of x.
that the derivative
F
the
(x) is
all
of
sum
whose elements are functions n determinants obtained
of the
all of the elements of a single line by their deriva State the corresponding theorem for derivatives of higher order.
by replacing, successively, tives. 3.
maximum and
Find the
the
minimum
values of the distance from a fixed
point to a plane or a skew curve between two variable points between two variable points on two surfaces. ;
on two curves
;
4. The points of a surface S for which the sum of the squares of the dis tances from n fixed points is an extremum are the feet of the normals let fall
upon the surface from the center
mean
of
distances of the given
n
fixed points.
the quadrilaterals which can be formed from four given
5. Of all which is inscriptible in a circle has the greatest area. theorem for polygons of n sides. 6.
an
Find the
maximum volume
sides, that
State the analogous
of a rectangular parallelepiped inscribed in
ellipsoid.
7. Find the axes of a central quadric from the consideration that the vertices are the points from which the distance to the center is an extremum. 8.
Solve the analogous problem for the axes of a central section of an ellipsoid.
9. Find the ellipse of minimum area which passes through the three vertices of a given triangle, and the ellipsoid of minimum volume which passes through the four vertices of a given tetrahedron.
Find the point from which the sum of the distances to two given straight and the distance to a given point is a minimum. [JOSEPH BERTRAND.]
10. lines
11.
log (3
Prove the following formulae
+
2)
=
2 log(z
+
1)
:
- 2 log (x-l) +
log(x
-3z
2)
-
_
-
z3
-
3Vx
3
-3z/
6\z 3
-3z
[BORDA log(x
+
6)
=
log(x
-f 4)
+ +
log(x
log(z
of |_x*
+ -
3) 3)
+
2 logx
log(z
72
- 25z2 +
-
4)
-
If 72
3 \z*
S
Series.]
-
log(x 5) 72
- 26z 2 +
y 72/
[HARO
S
-1
J
Series.]
CHAPTER IV DEFINITE INTEGRALS I.
SPECIAL METHODS OF QUADRATURE
64. Quadrature of the parabola.
The determination
the area
of
a problem which has always engaged the genius of geometricians. Among the examples which have come down to us from the ancients one of the most celebrated is
bounded by a plane curve
Archimedes
is
quadrature of the parabola.
We
shall
proceed to
indicate his method.
Let us try to find the area bounded by the arc A CB of a parabola Draw the diameter CD, joining the middle
and the chord A 13. point
D
Connect
of
AB
AC
to the point C,
and BC, and
let
where the tangent is parallel to AB. E and E be the points where the tangent is parallel to .BC and
We
shall
AC,
respectively.
first
compare the area of the
triangle BEC, for instance, with that of the triangle ABC.
Draw
the tangent
ET, which
cuts
CD at
eter
EF, which cuts
T.
Draw the diam
CB
EK
at
and and, finally, draw AB. to the chord parallel
F;
FH By
an elementary property of the parabola TC = CK. Moreover, CT EF = KH, and hence
EF= CH/2 = CD/ 4.
The
areas of the two triangles BCE and BCD, since they have the
FIG. 8
same base BC, are to each other as their altitudes, or as EF is CD. Hence the area of the triangle BCE is one fourth the area of the triangle BCD, or one eighth of the area 5 of the triangle ABC. The area of the triangle A CE is evidently the same. Carrying out the same process upon each of the chords BE, CE, CE E A, we
to
,
134
IV,
SPECIAL METHODS
65]
obtain four
new
2 triangles, the area of each of which is S/8 , and so triangles, each having the operation gives rise to
The nih n The area S/8
forth.
135
2"
of the segment of the parabola is evidently the limit approached by the sum of the areas of all these triangles as n increases indefinitely that is, the sum of the following descend
area
.
;
ing geometrical progression
:
It follows that the required area is equal to this sum is 4 5/3. two thirds of the area of a parallelogram whose sides are AB and CD. Although this method possesses admirable ingenuity, it must be
and
its success depends essentially upon certain special The of the parabola, and that it is lacking in generality. properties other examples of quadratures which we might quote from ancient
admitted that
writers would only go to corroborate this remark each new curve But whatever the device, the area to be required some new device. :
evaluated was always split up into elements the number of which to increase indefinitely, and it was necessary to evaluate
was made
sum cases,* we
the limit of the particular of subdivision,
which
of these partial areas. Omitting any further will proceed at once to give a general method will lead us naturally to the Integral Calculus.
65. General method. For the sake of definiteness, let us try to evaluate the area 5 bounded by a curvilinear arc A MB, an axis xx which does not cut that arc, and two perpendiculars AA and BB let fall
upon xx from
the points A and B.
We
will
suppose
further that a par allel to these lines
AA
,
BB
cannot
cut the arc in
more
than one point, as
FIG. 9
indicated in Fig. 9. Let us divide the segment
A B
into a certain
unequal parts by the points P l9 P 2 points let us draw lines PiQi, P 2 Q 2
and meeting the arc
AB
,
,
,
,
in the points Qi,
number
of equal or
P n .\, and through these Pn _iQ H _i parallel to AA Q2
>
j
Qn
-i>
respectively.
*A large number of examples of determinations of areas, arcs, and volumes by the methods of ancient writers are to be found in Duhamel s TraiM.
DEFINITE INTEGRALS
136
65
[IV,
Now draw through A a line parallel to xx cutting P t Q t at q through Qi a parallel to xx cutting P 2 Q 2 at q 2 and so on. We ,
;
,
obtain in this
;
R a sequence of rectangles RI, R 2 , lie entirely inside the contour
way
,
Each of these rectangles may but some of them may lie
Rn
,
t,
AB A
.
,
partially outside that contour, as is indicated in the figure. Let a ( denote the area of the rectangle R and /^ the area bounded In the first place, each of the ratios by the contour P _ Pi Q Qi _ l approaches unity as the number of fii/ a D ^2/ a A/ a { ,
i
2>
l
.
i
i>
>
points of division increases indefinitely, of the distances A P l} PiP 2 P^P,,
the ratio
L /P i
i
_
/?,/<*,,
for example, evidently lies
at the
same time each
approaches zero. For between ,-/ /*,-_! Q,-_i and
l f and L { are respectively the minimum and the distances from a point of the arc Q i _ l Q i to the axis xx clear that these two fractions each approach unity as the
Q
l
if
,
,
_
i
l
where
,
maximum But
it is
distance
.
P
t
_
l
P
i
It therefore follows that the ratio
approaches zero.
+ H-----h an A+&+ +& a-!
which a
or 2
between the largest and the least of the ratios tfi//3i, a n/ Pm w iH a ^ so approach unity as the number of the But the denominator of rectangles is thus indefinitely increased. this ratio is constant and is equal to the required area S. Hence this area is also equal to the limit of the sum a x + a 2 + + an as 2//?2>
lies
>
,
the
number
of rectangles
n
is
indefinitely increased in the
manner
specified above.
In order to deduce from this result an analytical expression for the area, let the curve AB be referred to a system of rectangular axes, the x axis Ox coinciding with xx and let y =f(x) be the The function f(x) is, by hypothesis, a equation of the curve AB. ,
continuous function of x between the limits a and
b,
the abscissae
of the points A and B. x n _ l the abscissas Denoting by x 1} x 2 of the points of division P 1} P 2 Pn _j, the bases of the above ,
,
,
,
x ly x b xn _ l and their rectangles are x a, x.2 , altitudes are, in like manner, f(a) t f(x^ -, /(<_,), /(_i). Hence the area S is equal to the limit of the following- sum ,
x^^
t
,
t
,
:
(1) as the
( Xl
-
a)f(a)
number n
the differences x l
+ (x 2
*,)/(*!)
+
-
-
+ (b - *_,)/(*_,),
increases indefinitely in such a a,
x2
x l}
approaches
way
zero.
that each of
SPECIAL METHODS
137
66. Examples. If the base AB be divided into n equal parts, each a of length h (b nh), all the rectangles have the same base h, and their altitudes are, respectively,
=
/(a), /(a It only
h
+
h),
f(a
+ 2 A),
-,
/[
+( -
1) A].
remains to find the limit of the sum
!/() +/(
+
+/( +
7
2
7
+
where
as the integer if
easy
w increases
we know how
This calculation becomes indefinitely. sum of a set of values f(x) corre
to find the
sponding to a set of values of x which form an arithmetic progres such is the case if f(x) is simply an integral power of x, or, again, if /(o;)= s mmx or /"(#)= cosmx, etc. Let us reconsider, for example, the parabola x* = 2py, and let us try
sion
;
t
by an arc OA of this parabola, the axis of x, and the straight line x = a which passes through the extremity A. The length being divided into n equal parts of length h (nh = a), we must try to find by the above the limit of the sum to find the area enclosed
The quantity first
inside the parenthesis 1) integers, that is, n(n
(n
foregoing
sum
is
is
the
sum
1) (2 n
of the squares of the
l)/6; and hence the
equal to
increases indefinitely this sum evidently approaches the limit 2 a*/6p (1/3) (a. a /2p), or one third of the rectangle constructed
As n
=
upon the two coordinates of the point A, which the result found above.
is
in
harmony with
In other cases, as in the following example, which is due to Fermat, it is better to choose as points of division points whose abscissae are in geometric progression.
Let us try to find the area enclosed by the curve y a I (0 axis of x, and the two straight lines x a, x
=
<
= Ax*, <
b),
the
where
DEFINITE INTEGRALS
138
66
[IV,
the exponent /* is arbitrary. In order to do so let us insert between a and b, n 1 geometric means so as to obtain the sequence
where the number a satisfies the condition a (1 4ing this set of numbers as the abscissae of the points of
=
b.
a)"
Tak
division, the
corresponding ordinates have, respectively, the following values
+
ay, Aa* (I and the area of the pth rectangle [a (1
+
a (1
a)"
Hence the sum
If
/i
+
1
a)*"
]
Aa*(l
2 *1
a)
,
is
+ a)
<-*
= Aa
+l
of the areas of all the rectangles
1 is not zero, as
parenthesis
or,
+
we
:
shall suppose
first,
is
the
sum
inside the
equal to
is
replacing a (1
+
a)"
by
i,
sum may be written
the original
in the
form .
\^
I
v
M+1 zero the quotient [(1 a) approaches !]/<* 1 + with as its limit the derivative of (1 0, respect to a for a a)^ 1 hence the required area is that is, /i
+
As a approaches
=
+
+
If
p.
=
;
1,
this calculation
no longer applies.
The sum
of the
areas of the inscribed rectangles is equal to nAa, and we have to find the limit of the product na where n and a are connected by the relation
a(l
From
this
it
-I-
= b. a)"
follows that
na
=
,
log &
a b r alog(l +
= )
.
log h
1
b -
a
,
,4
log(l
IV,
SPECIAL .METHODS
67]
139
where the symbol log denotes the Naperian logarithm. As a approaches zero, (1 + a) /* approaches the number e, and the prod 1
uct vl
na approaches
log
Hence
log (b fa).
the required area
is
equal to
(&/)
The invention of the Integral Calculus of reduced the problem evaluating a plane area to the problem of Let y =f(xj be the derivative is known. function whose finding a 67. Primitive functions.
equation of a curve referred to two rectangular axes, where the function f(x) is continuous. Let us consider the area enclosed by
MP
and a variable this curve, the axis of x, a fixed ordinate , ordinate MP, as a function of the abscissa x of the variable ordinate. In order to include
all
pos
sible cases let us agree to
denote by A the
sum of the
areas enclosed by the given curve, the x axis, and the
M
P , MP, straight lines each of the portions of this area being affected by a certain
+
sign
:
the
for the portions to sign the right of and above Ox, the sign
MP
M
FIG. 10
for the portions to the
P and below Ox, and the opposite convention for por right of P we tions to the left of M^Pg. Thus, if were in the position
M
MP
would take A equal
JI/
and likewise,
With
if
MP
,
to the difference
P C-
were at
these conventions
M"P",
we
shall
M P C; A = M"P"D - M P D. now show that the derivative
of
the continuous function A, defined in this way, is precisely /(#). As in the figure, let us take two neighboring ordinates MP, NQ, whose
x and x -f Ax. The increment of the area A.4 evidently between the areas of the two rectangles which have the same base PQ, and whose altitudes are, respectively, the greatest and the abscissae are
lies
least ordinates of the arc
H and
the
minimum by
MN.
h,
Denoting the
we may
AAz
<
<\A
maximum
ordinate by
therefore write <
7/Ax,
//. As Ax approaches zero, // and A/l /Ax dividing by Ax, h h approach the same limit MP, or /(x), since /(x) is continuous.
or,
<
<
DEFINITE INTEGRALS
140
[IV,
<;8
Hence the derivative of A is f(x). The proof that the same result holds for any position of the point .17 is left to the reader. If we already know a primitive function of f(x), that is, a function F(x) whose derivative is/(z), the difference A F(x) is a constant, since its derivative stant,
x
=
we need only
a of the line
is
zero
(
8).
In order to determine this con
notice that the area
MP.
A
is
zero for the abscissa
Hence
A =F(x)-F(a). It follows
from the above reasoning,
of a plane area
first,
that the determination
be reduced to the discovery of a primitive func tion; and, secondly (and this is of far greater importance for us), that every continuous function f(x) is the derivative of some other
may
This fundamental theorem is proved here by means of a somewhat vague geometrical concept, that of the area under a plane curve. This demonstration was regarded as satisfactory for a long time, but it can no longer be accepted. In order to have a stable
function.
foundation for the Integral Calculus
imperative that this theo
it is
rem should be given a purely
analytic demonstration which does not intuition whatever. In giving the above rely upon any geometrical the motive was not its historical interest, geometrical proof wholly
however, for it furnishes us with the essential analytic argument of the new proof. It is, in fact, the study of precisely such sums as and sums of a slightly more general character which will be (1) of preponderant importance. Before taking up this study we must first consider certain questions regarding the general properties of functions and in particular of continuous functions.*
II.
DEFINITE INTEGRALS ALLIED GEOMETRICAL CONCEPTS
68. Upper and lower limits. An assemblage of numbers is said to have an upper limit (see ftn., p. 91) if there exists a number so large that no member of the assemblage exceeds N. Likewise, an exists than assemblage is said to have a lower limit if a number which no member of the assemblage is smaller. Thus the assem
N
N
blage of
all
positive integers has a lower limit, but no upper limit
;
* Among the most important works on the general notion of the definite integral there should be mentioned the memoir by Riemann fiber die Mb glichkeit, eine Func tion durch eine trigonometrische Reihe darzustellen (Werke, 2d ed., Leipzig, 1892, p. 239 and also French translation by Laugel, p. 225) and the memoir by Darboux, to which we have already referred Sur les fonctions discontinues (Annales de VEcole :
;
;
:
Normals Suptrieure, 2d
series, Vol. IV).
IV,
ALLIED GEOMETRICAL CONCEPTS
68]
141
the assemblage of all integers, positive and negative, has neither and 1 has and the assemblage of all rational numbers between ;
both a lower and an upper limit. Let (E) be an assemblage which has an upper limit. With respect to this assemblage all numbers may be divided into two
We
shall say that a number a belongs to the first class if of the assemblage (7?) which are greater than a, members there are it that and belongs to the second class if there is no member of the classes.
assemblage
(7?)
Since the assemblage (7?) has an greater than a. clear that numbers of each class exist. If A be
it is
upper limit, a number of the is
which
lie
first class
and
B there between A and B, A
evident that
<
;
B
exist
number members
a
but there
is
of the second class,
it
of the assemblage (7?) no member of the assem
The number C = (A -f jB)/2 greater than B. blage (7?) or first to the second class. In the former case to the may belong we should replace the interval (A, B*) by the interval (C, 7?), in the which
is
by the interval (A, C). The new interval (.4^ 7^) is half and has the same properties there exists at least (^4, B) one member of the assemblage (7) which is greater than A 1} bnt none which is greater than B. Operating upon (A l} B^) in the same way that we operated upon (A, B}, and so on indefinitely, we obtain an latter case
the interval
:
unlimited sequence of intervals (A, 73), (A lf 7^), (A 2 , 7?2 ), each j which is half the preceding and possesses the same property as (A, B} with respect to the assemblage (/?). Since the numbers A A never decrease and are n A, AI, z , always less than B, they of
,
approach a limit A ( 1). Likewise, since the numbers B, B 1} B 2 never increase and are always greater than A, they approach a limit X A n = (B A ) Moreover, since the difference Bn approaches zero as n increases indefinitely, these limits must be equal, i.e. A = A. Let L be this common limit then L is called the iqjper limit of the ,
.
/2"
;
assemblage
(7?).
clear that
it is
From
L has
the
manner
in
which we have obtained
the following two properties
it,
:
No member
of the assemblage (7i) is greater than L. There always exists a member of the assemblage (7?) which is e, where c is any arbitrarily small positive number. greater than L 1) 2)
For
let
us suppose that there were a
+
member
h (Ji Since greater than L, say L 0). increases indefinitely, B n will be less than value of
On
n.
But
>
this is impossible since
the other hand, let
e
Bn
Bn
of the assemblage approaches L as n
L
-f
is
of the second class.
h
be any positive number.
after a certain
Then, after a
DEFINITE INTEGRALS
142
[IV,
A n will be greater than L and since there are e of (E) greater than A n these numbers will also be greater It is evident that the two properties stated above can e.
certain value of n,
members than L
69
;
,
not apply to any other number than L. The upper limit may or may not belong to the assemblage (). In the assemblage of all rational numbers which do not exceed 2,
number 2
is precisely the upper limit, and it belongs the other hand, the assemblage of all irra tional numbers which do not exceed 2 has the upper limit 2, but
for instance, the
On
to the assemblage.
upper limit is not a member of the assemblage. It should be particularly noted that if the upper limit L does not belong to the assemblage, there are always an infinite number of members of () this
which are greater than L For if c, no matter how small e be taken. there were only a finite number, the upper limit would be the largest of these and not L. When the assemblage consists of n different
numbers the upper limit is simply the largest of these n numbers. It may be shown in like manner that there exists a number L\ in case the assemblage has a lower limit, which has the following two properties :
L
No member of the
assemblage is less than L a member There exists of the assemblage which 2) where e is an arbitrary positive number.* -\1)
.
is less
than
(.,
This number L 69.
Oscillation.
is
called the lower limit of the assemblage.
Let/()
be a function of x defined in the closed f
interval (a, ) that is, to each value of x between a and b and to each of the limits a and b themselves there corresponds a uniquely deter ;
mined value of f(x}. interval
if all
The function
the values which
it
is
said to be finite in this closed lie between two fixed
assumes
numbers A and B. Then the assemblage of values of the function Let and m be the upper and has an upper and a lower limit.
M
lower limits of this assemblage, respectively *
Whenever
all
numbers can be separated
into
two
;
then the difference
classes
A
and B, according to
any characteristic property, in such a way that any number of the class A is less than any number of the class B, the upper limit L of the numbers of the class A is at the same time the lower limit of the numbers of the class B. It is clear, first of all, that any number greater than L belongs to the class B. And if there were a number L belonging to the class B, then every number greater than L would belong to the class B. Hence every number less than L belongs to the class A, every number greater than L belongs to the class B, and L itself may belong to either of the two classes. is used merely for emphasis. TRANS. closed See 2. t The word
"
"
IV,
ALLIED GEOMETRICAL CONCEPTS
70J
A=
M m
is
143
called the oscillation of the function f(x) in the
interval (a, b). These definitions lead to several remarks.
In order that a func
tion be finite in a closed interval (a, b~) it is not sufficient that it should have a finite value for every value of x. Thus the function
defined in the closed interval
=
0,
(0, 1) as
/(aj)
=
follows for
l/aj
has a finite value for each value of x
;
biit
:
x
>
0,
nevertheless
not
is
it
A defined the word, for/(ce) in closed finite the which is a function if we take x
we have
the sense in which
>
M
or from the lower limit m and still never from the upper limit assume these values themselves. For instance, the function /(#),
defined in the closed interval (0, 1) by the relations
=
f(x)
0,
has the upper limit
M=
1,
= l-x
for
0<x
but never reaches that limit.
70. Properties of continuous functions.
We
shall
now
turn to the
study of continuous functions in particular.
THEOREM
A.
interval (a, b)
Letf(x) be a function which
and
e
an arbitrary
is
continuous in the closed
Then we can positive number. into a certain number of partial
always break up the interval (a, ft) intervals in such a way that for any two values of the variable which belong to the same partial interval, we whatever, x and x",
always have \f(x
)
f( x
<
")
\
Suppose that this were not
true.
Then
let
c=(a +
ft)/2;
at
would have the same prop erty as (a, ft); that is, it would be impossible to break it up into partial intervals which would satisfy the statement of the theorem. Substituting it for the given interval (a, ft) and carrying out the reasoning as above ( 68), we could form an infinite sequence of which is half the preced intervals (a, ft), (a l} b^, (a 2 2 ), -, each of For ing and has the same property as the original interval (a, ft). any value of n we could always find in the interval ( n n ) two
least one of the intervals (a,
,
c),
(c, ft)
ft
,
ft
would be larger than e. numbers x and such that |/(V)~/( X Now let X be the common limit of the two sequences of numbers Since the function /(#) is continuous an ^ b, b^ b 2 a, a 1} a 2 = for x X, we can find a number rj such that |/(.r) e/2 /(X)| x"
")I
"
>
,
.
<
144
DEFINITE INTEGRALS
whenever a n and b n
A| is less differ from A
ja;
than
than
Then the
77.
-
will lie wholly within the interval A (A rj, are any two values whatever in the interval
and hence
|/
-/(.x")
+
77)
(a n
,
;
Z>
interval (a n , b n }
and
if
* and
B ),
hence the theorem
;
a;"
we must have
It follows that the hypothesis
.
< |
above leads to a contradiction
70
Let us choose n so large that both
rj.
less
by
[IV,
is
made
proved.
xp _ 1} b be a method of subdivision Corollary I. Let a, x lt x 2 , of the interval (a, i) into p subintervals, which satisfies the con ditions of the theorem. In the interval (a, a^) we shall have !/(*) and, in particular, I/O) e Like Xl ,
+
<
\
I
\f(
5
wise, in the interval (x l}
and, a fortiori,
|/(0
< I
I
\f(x)
/() +
2
e
|
;
we
a-,)
shall
+
2 |/(a) and so forth. <
\
|
c
;
)
<
\
|/(a)
have [/(*)
|
+
.
j/(*i) in particular, for x
For the
I
<
last interval
we
+ = xj,
e,
shall
have I/(*)|<|/(P-I)
+
|
<|/(a)
Hence the absolute value of f(x)
1
+ pe.
the interval (a, b~) always than |/(a) + pe. It follows that every function which continuous in a closed interval (a, b) is finite in that interval.
remains
in
less
|
is
Corollary II. Let us suppose the interval (a, b) split up into 7? subintervals (a, x^, (x lt x2 ), ..., e (xp _v b) such that \f(x ) /2 f(x")\< for any two values of x which belong to the same closed subinterval.
Let x2
77
be a positive number less than any of the differences a, b xp _ l Then let us take any two numbers whatever
^
Xi
.
,
in the interval (a, b) for
an upper limit for (/(* fall in
it is
do
not,
x
and
-
\x
x"
easy to see that
we must
|/
x"
^
If the
shall
x"
f(x")\<
lie in
c
=
2 ( e /2) c Hence cor another positive number rj can be
-f(x")
responding to any positive number found such that
and let us try to find two numbers x and have \f(x ) e /2. two consecutive intervals, <
) -/(*")).
the same subinterval,
If they
and
which
<
.
\
I/W-/CO.I<*
where x and ,
[**
*"!<
>?
function /(x)
are
x"
any two numbers of
This property is
the interval (a, b) for which
also expressed by saying that the uniformly continuous in the interval (a, b).
A
is
THEOREM B. function f(x) which is continuous in a closed interval (a, b) takes on every value between /(a) and f(b) at least once for some value of x which lies between a and b.
IV,
ALLIED GEOMETRICAL CONCEPTS
70]
Let us
145
consider a particular case.
Suppose that f(a) and 0, for instance. and/(6) least one value of x between is negative near a and posi Let us consider the assemblage of values of x between tive near b. a and b for which /(#) is positive, and let \ be the lower limit of A this assemblage (a By the very definition of a lower b*). first
that /(a) /() have opposite signs, We shall then show that there exists at a and b for which f(x) = 0. Now/(x)
<
<
>
<
negative or zero for every positive value of h. A) /(A Hence /(A.), which is the limit of /(A A), is also negative or zero. But /(A) cannot be negative. For suppose that /(A) m, where limit
is
=
m
a positive number. Since the function /(x ) is continuous for x = A, a number rj can be found such that )/(#) m when /(A) ever \x and the function would A, be rj, negative for all f(x) is
<
<
|
values of x between
A.
limit of the values of
/(A)
-
Now
0.
=
N
let
function
x
and A + rj. Hence A could not be the lower x for which /(ic) is positive. Consequently
<(#)
a and x
be any number between /(a) and Then the is continuous and has =f(x) opposite signs for b. Hence, by the particular case just treated, it /(>).
N
vanishes at least once in the interval
THEOREM val (a, b)
(a,
&).
Every function which is continuous in a closed inter actually assumes the value of its upper and of its lower C.
limit at least once.
In the first place, every continuous function, since we have and a lower already proved that it is finite, has an upper limit limit m. Let us show, for instance, that f(x) for at least one
M
M
value of x in the interval
=
(a, 5).
M
+
for (a b)/2, the upper limit of f(x) is equal to at least one of the intervals (a, e), (c, b). Let us replace (a, b) by this new interval, repeat the process upon it, and so forth.
Taking
c
Reasoning as we have already done several times, we could form an infinite sequence of intervals (a, b), (a u & t ), (o 2 & 2 ), each of which is half the preceding and in each of which the upper limit of ,
,
Then, if A is the common limit of the sequences a, a if and b, b 1} bn /(A) is equal to M. For suppose that We can find a positive number h, where h is positive. /(A) = such that f(x) remains between /(A) + h/2 and /(A) rj h/ 2, and therefore less than h/2 as long as x remains between A rj and A -f rj- Let us now choose n so great that a n and b n differ from their common limit A by less than 77. Then the interval (a,,, &) lies
f(x) ,
is
an
M.
,
,
M
M
,
,
DEFINITE INTEGRALS
146
wholly inside the interval
(A.
A.
77,
4-
*;),
[IV,
and
follows at once
it
that the upper limit of f(x) in the interval (a n
,
71
b n)
could not be
M.
equal to
theorem with the preceding, we see that any func continuous in a closed interval (a, ft) assumes, at least
this
Combining tion which
is
upper and its lower limit. Moreover Given a function which is be stated as follows
once, every value between its
A may
theorem
:
continuous in a closed interval (a, ft), it is possible to divide the inter val into such small subreyions that the oscillation of the function in
any one of them will be less than an arbitrarily assigned positive For the oscillation of a continuous function is equal to of the values of /(x) for two particular values of the difference the
number.
variable.
The sums S and
71.
Let /(#) be a
s.
or discontinuous, in the interval (a, ft), pose the interval (a, b) divided into a
x lt x 2
,
,
limits of
# p _i
is
a;
b.
2 ),
,
(xp _ l
,
b),
Let us sup
of smaller partial
M
{
in the interval (a^-i,
= M, (x, s = m (x
S
and
#,-),
- a) + M
l
l
number
<
M
in
f(x)
where a
where each of the numbers Let and m be the greater than the preceding. and m the limits the original interval, and
(a; u
intervals (a, o^),
continuous
finite function,
2
a)
-f-
let
(x 2
77*2(3-2
i
us set
-x )+--- + Mp (b - zp _ X P -I)x H mP (*
t ),
1
i)-\
To every method of division of (a, b) into smaller intervals there corresponds a sum S and a smaller sum s. It is evident that none of the numbers of the sums 5 are less than m(b a), for none
M
i
m
hence these sums S have a lower limit /.* Like none of which exceed M(b sums the s, a) have an upper wise, limit / We proceed to show that / is at most equal to I. For this are less than
;
.
purpose S, s
it is
and S
,
evidently sufficient to show that s^S and s 5j S, where are the two sets of sums which correspond to any
s
two given methods of subdivision of the interval In the (#1,
a 2 ),
division
* If f(x)
(a, b).
us suppose each of the subintervals (a, a^), redivided into still smaller intervals by new points of
first place, let
and
is
let
a constant, S
become equations.
=
TRANS.
s,
M = m, and, in general,
all
the inequalities mentioned
IV,
be the is
ALLIED GEOMETRICAL CONCEPTS
72]
new
This new method of subdivision
suite thus obtained.
called consecutive to the
first.
147
Let 2 and
cr
denote the sums anal
with respect to this new method of division of the Let us interval (a, b), and let us compare S and s with 2 and a. 5 two sums and which of the the for 2 portions example, compare, ogous to
S and
s
from the interval
arise
Let M[ and m[ be the limits of
(a, a^).
the interval (a, y^, M[ and m^ the limits in the interval f(x) Then k the limits in the interval (y t k an(i %i)1/2)9 the portion of 2 which comes from (a, a^) is in
M
"
G/i>
m
-i>
M
cannot exceed and since the numbers M{, M%, lt it is clear -, M/. sum is at most equal to 3/j (x l Likewise, the a). a: most equal is at from the interval of which arises 2 (x l} 2 ) portion
that the above
M (x
Adding all these inequalities, we find in like manner that a- ^ s. show 2 = S, Let us now consider any two methods of subdivision whatever, and let S, s and S s be the corresponding sums. Superimposing the points of division of these two methods of subdivision, we get a third method of subdivision, which may be considered as consecu be the sums Let 2 and tive to either of the two given methods. to
2
2
,),
and
that
and so
it is
on.
easy to
,
with respect to this auxiliary division.
By
the above
we have
the
relations 2<S,
2<S
(r>s,
,
o->s
;
2 is not less than a-, it follows that s ^ S and s^ S Since none of the sums S are less than any of the sums s, the limit 7 and, since
.
cannot be less than the limit / 72. Integrable functions.
A
;
that
is,
/
function which
^/
.
is finite
in an inter
said to be integrable in that interval if the two sums b~) S and 5 approach the same limit when the number of the partial
val (a,
is
is indefinitely increased in such a intervals partial approaches zero.
intervals
way
that each of those
The necessary and sufficient condition that a function be integrable an interval is that corresponding to any positive number e another number rj exists such that S s is less than c whenever each of the in
partial intervals
This condition limit
7,
we can
is
less
is,
find a
than
first,
-r\.
necessary, for if S so small that
number ^
and S |
s T\
have the same
and
js
7|
are
DEFINITE INTEGRALS
148
72
[iv,
each less than e/2 whenever each of the partial intervals s is less than e. Then, a fortiori, S 77. Moreover the condition is sufficient, for we may write *
is
less
than
s-s = s-i + i- r + r -
s,
I I and since none of the numbers S s can be negative, I, I each of them must be less than e if their sum is to be less than e. I is a fixed number and e is an arbitrary positive But since I it follows that we must have / = 7. Moreover S I e number, s and / e whenever each of the partial intervals is less than 77, which is equivalent to saying that S and s have the same limit 7. ,
<
<
The function /(#) (a,
),
is
and the limit 7
is
then said to be integrable in the interval called a definite integral. It is represented
by the symbol
=
ff(x)dx, *J a
which suggests its origin, and which is read the definite integral from a to b of f(x) By its very definition 7 always lies between the two sums S and s for any method of subdivision whatever. If any number between S and s be taken as an approximate value of 7, the error never exceeds S s. "
dx."
Every continuous function
is
integrable.
The difference S s is less than or equal to (b a), where w denotes the upper limit of the oscillation of f(x~) in the partial intervals. But 77 may be so chosen that the oscillation is less than a preassigned positive number in any interval less than 77 ( If then 77 be so chosen that the oscillation is less than (./(b the difference S
s will
be
than
less
70). a),
e.
monotonically increasing or monotonically decreasing function interval is integrable in that interval.
Any in
an
A f unction
/"(
x)
is
said to increase monotonically in a given interval
in that interval /(# ) when any two values x ever x The function may be constant in certain portions of the interval, but if it is not constant it must increase with x. Dividing the interval (a, b) into n subintervals, each less than 77, we may write (a, 6) if for >
S =/<X>
S
,
x"
>/(")
x".
(*!
=/(a)(ar
1
-
a)
+/(* 2 ) (x _ aJt) +
a)
+/(.T )(* 2
2
*For the proof that I and /
1
exist, see
- xj
73,
-\-
+/() (b - *_!), +f(x n _,} (b - *_,),
which may be read before
72.
TRANS.
IV,
ALLIED GEOMETRICAL CONCEPTS
72]
149
upper limit of f(x) in the interval (a, a^), for instance, and so on for the other /"();
for the
precisely f(x\), the lower limit subintervals. Hence, subtracting, is
-s =
(x 1
-
-/(a)]
a) [/(zj)
+ (* 2
-/(*._,)].
None
of the differences
which occur
equation are negative, and are less than
t]
all
in the right-hand side of this
of the differences x l
a,
x2
x l}
consequently
;
or -<*[/(*)
-/()],
and we need only take
in order to
make 5
<s
<
e.
The reasoning
tonically decreasing function. Let us return to the general case.
gral the sums S and
(2)
In the definition of the inte
s
:
i*
(z, _ i
the same for a mono-
may be replaced by more general expres Given any method of subdivision of the interval (a, i)
sions.
let
is
4-
be values belonging to these intervals in order
*,(>
s>
=
Then the sum
= x i)
.
evidently lies between the sums S and s, for we always have 7w f If the function is integrable, this new sum has the ^/(,.) 5 Af,-. limit /. In particular, if we suppose that 1? 2 ^n coincide , ,
with
a,
ajj,
,
a;
n
_
1 ,
(1) considered above
respectively, the (
sum
(2) reduces to the
sum
65).
There are several propositions which result immediately from the definition of the integral. We have supposed that a b if we now x _! these two limits a and b, each of the factors x interchange hence changes sign; <
;
t
Cf(x)dx
Ja
=-
Cf(x)dx.
Jb
{
DEFINITE INTEGRALS
150
[IV,
72
from the definition that
It also evidently follows
C
f f(x)dx
=
f(x)dx
+
f(x}dx,
Jo.
if c lies between a and b; the same formula still holds when between a and c, for instance, provided that the function f(x) integrable between a and c, for it may be written in the form
at least lies
I)
is
C
= Cf(x)dx~ f f(x}dx = ff(x)dx Ja
Jo.
Jc
If f(x)
= A(x)
Cf(x)dx
Ja
where A and
-f B\j/(x),
B
are
+ f f(x)dx. Jb
any two constants,
we have />
I
Ja
b
s*b
/->h
f(x)dx
=A
J
and a similar formula holds
The expression eral
expression. intervals (a, a^),
+B
I (x)dx
I
ij/(x*)dx,
\J a
a.
for the
sum
of any
number
of functions.
/(,-) in (2) may be replaced by a still more gen The interval (-, I) being divided into n sub,
(#/_!,
a;,-),
,
let
us associate with each of the
subintervals a quantity ,, which approaches zero with the length shall say that x x i _ l of the subinterval in question. approaches zero uniformly if corresponding to every positive num
We
t
ber
c
another positive number
such that
j,-
<
e
whenever
a\
-rj
,
can be found independent of i and x f-1 is less than 77. We shall now
proceed to show that the sum
approaches the definite integral j^ffx^dx as its limit provided that approaches zero uniformly. For suppose that rj is a number ,-
so small that the
two inequalities
are satisfied whenever each of the subintervals x i
than
17.
Then we may write
/(*-i)(i-*-i)- f
x _1 i
is
less
ALLIED GEOMETRICAL CONCEPTS
IV,
73]
and
it is
clear that
S
we
shall
151
have
- C f(x)dx
<
+
e
c(b
a)
\J a
whenever each of the subintervals is
than
is less
77.
Thus the theorem
proved.*
73. Darboux s theorem. Given any function f(x) which is finite in an inter val (a, 6); the sums S and s approach their limits I and / , respectively, when the number of subintervals increases indefinitely in such a way that each of them approaches zero. Let us prove this for the sum S, for instance.
We
suppose that a<6, and that/(x) is positive in the interval (a, 6), which can be brought about by adding a suitable constant to/(x), which, in turn, amounts shall
to adding a constant to each of the sums S. Then, since the number / is the lower limit of all the sums S, we can find a particular method of subdivision, say a, zi,
x2
,
-,
Xp-i,
6,
which the sum S is less than I + e/2, where e is a preassigned positive num ber. Let us now consider a division of (a, 6) into intervals less than r;, and let us try to find an upper limit of the corresponding sum S Taking first those inter vals which do not include any of the points x lt x 2 Xp_i, and recalling the for
.
,
,
reasoning of 71, it is clear that the portion of S which comes from these inter vals will be less than the original sum S, that is, less than I + e/2. On the other hand, the number of intervals which include a point of the set Xi, x2 Xp_j cannot exceed p 1, and hence their contribution to the sum S cannot exceed ,
(p
1) Mil,
where
M
is
the upper limit of /(x).
S
,
Hence
+ (p-l)Mr,,
M
- 1) in order to make S less than and we need only choose r) less than e/2 (p I + f. Hence the lower limit I of all the sums /S is also the limit of any sequence of s which corresponds to uniformly infinitesimal subintervals. <S
be shown in a similar manner that the sums s have the limit / function /(x) is any function whatever, these two limits I and 7 are in In order that the function be integrable it is general different. necessary and sufficient that 7 = I. It
may
.
If the
74.
First law of the
mean
for integrals.
From now on we
shall
assume, unless something is explicitly said to the contrary, that the functions under the integral are continuous. sign
*
The above theorem can be extended without
grals
;
we
shall
make
use of
it
in several places
(
difficulty to
double and triple inte
80, 95, 97, 131, 144, etc.).
The proposition is essentially only an application of a theorem of Duhamel s according to which the limit of a sum of infinitesimals remains unchanged when each of the infinitesimals is replaced by another infinitesimal which differs from the given infinitesimal by an infinitesimal of higher order. (See an article by W. F. Osgood, Annals of Mathematics, 2d series, Vol. IV, pp. 161-178 The Integral as the Limit of a Sum and a Theorem of Duhamel s.) :
P
DEFINITE INTEGRALS
152 Let f(x) and
<
one of which, say
throughout the interval. And b sake of definiteness, that a <
i>
74
two functions which are each continuous has the same sign
(x) be
in the interval (a, b),
[IV,
we
<(:*),
shall suppose further, for the
and
(x)
0.
>
Suppose the interval (a, b~) divided into subintervals, and let be values of x which belong to each of these o 2, >
All the quantities /(,)
smaller intervals in order. limits
M and m of f(x)
in the interval (a, b)
lie
between the
:
Let us multiply each of these inequalities by the factors
respectively,
which
are all positive
by hypothesis, and then add
z _,) evidently lies The sum S/(&) fa between the two sums ra 2 ,._,). fa o^) and 3/2 fa Hence, as the number of subintervals increases indefinitely, we
them
together.
f
<(&)
<()
<()
have, in the limit,
m f $ (x} dx Ja
C /(
<,
$ (x} dx
<
M
C $ (x) dx,
Ja
Jo.
which may be written s*b
Xb
f(x}$(x}dx
where
/x
tinuous,
which
=
p.
$(x)dx,
\
J
u.
between m and M. Since the function f(x) is con of the variable assumes the value /t for some value between a and b and hence we may write the preceding
lies it
lies
;
equation in the form
f /(*)
(3)
Ja
<(>
(x) dx
= /(*) C $ (x) dx, /
between a and b.* If, in particular, =1, the of an inte definition the dx to reduces very integral JH* (b a) by becomes the formula and gral,
where
(4)
lies
<(#)
f
/(*)<&
(*-a)/tf).
= k.
* The lower sign holds in the preceding relations only when / (a-) b in any case. that the formula still holds, however, and that a<
<
It is
evident
TRANS.
IV,
ALLIED GEOMETRICAL CONCEPTS
75]
153
75. Second law of the mean for integrals. There is a second formula, due to Bonnet, which he deduced from an important lemma of Abel s.
Lemma. Let and MO MI If A and B are ties,
s\
=
,
UQ
+
i
,
e be a set of monotonically p decreasing positive quanti Up the same number of arbitrary positive or negative quantities. respectively the greatest and the least of all of the sums s = u
e
,
ei,
,
,
,
,
i
=
p
+
w
MI
+
e
W
S
Ae Q and
will lie between
-Be
i-e.
)
+ +
u t>
eiWi
Aeo
+
+
S^
>
sum
the
,
Bf
p up
e
.
For we have UQ
Ui=Si
SO,
whence the sum S
S
+
i)
e2
Si (ei
Since none of the differences
)
e
+
are given by replacing s by their lower limit B. In this way
and
it is
Now
<
A (e
ei
+
and <j>
(x)
Sp-i e2
,
we
,
Up
=
-i
-
fp
~i
Sp
Sp ^i,
(ep
,
,
sp
+
t
p)
Sp ep
.
two and then
p are negative,
f
by their upper limit
A
find
+
e2
ei
i
,
-f
ep_i
e
p
S ^ Be be two continuous functions
likewise evident that
let/(x)
+
ei, ei
S
S
,
equal to
is
So (*0
limits for
,
+
= At
e
v)
,
.
of x, one of which,
a positive monotonically decreasing function in the interval a the integral f^f(x) is the limit of the sum is
<
x
<
(a;),
Then
b.
<j>(x)dx
/(a)
(a) (xi
-
a)
+ f(xi)
<j>
(xj) (x 2
-
xa)
+
.
.
. .
The numbers 0(a), 0(xi), form a set of monotonically decreasing positive and numbers; hence the above sum, by the lemma, lies between where A and B are respectively the greatest and the least among the following sums A(a)
B(a),
:
/(a)
(X!
/(a)
(xi
/(a)
(X!
-
-
a)
,
a)
+/(xi)
(x 2
a)
+/(xj)
(x 2
Passing to the limit, Ai
-
xi)
,
Xi)
+
+f(x n -
l)
(b
-
x n ^).
must lie between where AI and BI denote the maximum and the minimum, the integral f^f(x)dx, as c varies from a to b. Since this inte
and #10
respectively, of
-
it
is
clear that the integral in question
(a),
gral is evidently a continuous function of its upper limit c the following formula
(
76),
we may
write
:
(5)
fa
J
/(x)0(x)cZx
=
0(a)
T Ja
When the function 0(x) is a monotonically decreasing function, without being always positive, there exists a more general formula, due to Weierstrass. In such a case let us set = (b) + (x). Then f (x) is a positive monoton (x)
ically decreasing function.
\fs
Applying the formula
(5) to it,
we
Jfa f(x)dx.
find
DEFINITE INTEGRALS
154 From
this
it is
76
[IV,
easy to derive the formula
C f(x)+(x)dx = C Ja
c/a
f i/a
/(x) 0(z) dx
f(x)+(b)dx
=
+
[0(o)
-
0(6)]
fV(x) Ja
b
f %s)
0(a)
+
<*z
0(6)
f
f(x) dx
.
"u
Similar formulae exist for the case
when
the function 0(x)
is
increasing.
We
76. Return to primitive functions. are now in a position to give a purely analytic proof of the fundamental existence theorem ( 67). Let/(x) be any continuous function. Then the definite integral
where the limit a limit x.
isf(x).
is regarded as fixed, is a function of the upper proceed to show that the derivative of this function
We
In the
first place,
we have >x
= or,
applying the
where
lies
first
+
f Jx
law of the mean
A
f(t)dt,
(4),
+ h. As h approaches zero, hence the derivative of the function F(x) is /(x),
between x and x
approaches /(x) to be proved. ;
which was
All other functions which have this same derivative are given by adding an arbitrary constant C to F(x). There is one such function, and only one, which assumes a preassigned value x a, namely, the function
T/O
for
=
When
there
is
no reason
to fear
ambiguity the same
letter
x
is
used to denote the upper limit and the variable of integration, and dx i g written in place of dt. But it is evident that /*/"(*) f*f(t) a definite integral depends only upon the limits of integration and The letter the form of the function under the sign of integration.
which denotes the variable of integration is absolutely immaterial. Every function whose derivative is /(x) is called an indefinite and is represented integral of /(x), or & primitive function of /"(x),
by the symbol
r f(x)dx,
IV,
ALLIED GEOMETRICAL CONCEPTS
70]
the limits not being indicated.
By
we
the above
155
evidently have
Conversely, if a function F(x) whose derivative is /(x) can be discovered by any method whatever, we may write
f(x)dx
=
F(aj)+ C.
C we need only note that Hence C = F(a), and
In order to determine the constant left-hand side vanishes for x
=
a.
the the
fundamental formula becomes
= F(*)-F(a).
f(x)dx
(6) C/ U
If in this formula /(a- ) be replaced
by
F (x),
it
becomes
F (x)dx, F(a)-F(a)*= *Jf a or,
applying the
first
law of the mean for integrals,
where lies between a and x. This constitutes a new proof of the law of the mean for derivatives but it is less general than the one given in section 8, for it is assumed here that the derivative F (a:) is ;
continuous.
We
shall consider in the next chapter the simpler classes of func Just now we will merely state
whose primitives are known. a few of those which are apparent tions
A(x
a a) dx
=
A
at once
(-
(
:
a-fl^O;
,
dx
I
cos x dx
=
sin
x
x
dx
-f
=
C
I
;
h C,
sin
x dx
=
m
0;
=
cos x
-f-
C
;
DEFINITE INTEGRALS
150
[IV,
= log/(x) +
76
C.
of the fundamental formula (6) was based upon the the function f(x) was continuous in the closed inter that assumption If this condition be disregarded, results may be obtained val (a, b). 2 which are paradoxical. Taking f(x) == I/a; for instance, the for
The proof
,
mula
(6) gives f"^
The
= i_i.
x2
\J} a
a
I
left-hand side of this equality has no unless a and b have the same sign
in our present the but right-hand side
meaning
system has a perfectly determinate value, even when a and b have different We shall find the explanation of this paradox later in the signs. taken between imaginary limits. study of definite integrals leads to the equation Similarly, the formula (6)
^
r
1
;
/()
/(*>
between a and b, opposite signs, f(x) vanishes and neither side of the above equality has any meaning for us at shall find later the signification which it is convenient
If /(a)
and/() have
We
present. to give them.
Again, the formula (6) 2 /() = !/(!+ * ), we find
may
=
lead
arc tan b
to
ambiguity.
arc tan
Thus,
if
a.
Here the left-hand side is perfectly determinate, while the righthand side has an infinite number of determinations. To avoid this ambiguity,
let
us consider the function
This function F(x) is continuous in the whole interval and van Let us denote by arc tan x, on the other hand, an ishes with x. between - Tr/2 and + Tr/2. These two functions have the angle
iv,
ALLIED GEOMETRICAL CONCEPTS
77]
same derivative and they both vanish for x = 0. they are equal, and we may write the equality
r b dx
Jm
= r + x* J
1
b
r a dx
dx
I
:
\
;
1
|
+ x* J
-.
1
+
157
It follows that
= arc tan b
arc tan a,
x*
where the value to be assigned the arctangent always -7T/2 and +7T/2. In a similar manner we may derive the formula
dx
>b
f
=
lies
between
arc sin a,
arc sin b
I/O
where the radical is to be taken positive, where a and b each lie 1 and + 1, and where arc sin x denotes an angle which between lies between Tr/2 and + Tr/2. 77. Indices.
of this branch as
the primitive F(x) is multiply determinate, we values F(a) and follow the continuous variation
when
In general,
should choose one of the
initial
x varies from a
to
Let us consider, for instance, the integral
b.
fobzar*. J.f-^w+/
J.
i
^+
(*)
where
and where P and Q are two functions which are both continuous in the interval If Q does not vanish time. (a, b) and which do not both vanish at the same between a and 6, /(x) does not become infinite, and arc tan/(x) remains between But this is no longer true, in general, if the equation Q = 7f/2 and + rt/2. has roots in this interval. In order to see how the formula must be modified, let us retain the convention that arc tan signifies an angle between if/2 and + if/2, and let us suppose, in the first place, that Q vanishes just once between a and b We may write the integral in the form for a value x = c.
r
b
f
(x)dx
Ja where
e
infinite
and e are two very small positive numbers. Since /(x) does not become c + e and 6, this may again be written between a and c e, nor between f dx
=
arc tan/(c
+ Several cases
arc
-
e)
tan/ (6)
may now present
-
arc tan/(a)
arc tan/(c
themselves.
+
e
)
+
Suppose, for the sake of definite-
ness, that/(x)
+
be positive
will be very
becomes infinite by passing from and very large, and arc tan /(c e)
fC+t I
Jc-e
oo
to
oo.
Then/(c
e)
will
near to ir/2; while
DEFINITE INTEGRALS
158 /(c
+
e
)
and very
will be negative
large,
[IV,
and arc tan/(c
+
7
e ) will
78
be very near
Also, the integral J^lV wil1 be verv small in absolute value; and, passing to the limit, we obtain the formula 7T/2.
f(x)dx
f
=
+
TT
arc tan/(6)
arctan/(a).
show
that it would be necessary to subtract n if /(x) In the general case we would divide the interval into subintervals in such a way that /(x) would become infinite just once
Similarly,
it is
passed from (a, 6)
easy to
+
co to
.
in each of them.
Treating each of these subintervals in the above manner and adding the results obtained, we should find the formula f (x) dx .
*s
a
i
T
+j
=
-
arc tan/(6)
arc tan/(o)
+ (K -
K
)
x,
(x)
K denotes the number of times that /(x) becomes infinite by passing from co, and K the number of times that /(x) passes from to + The number K K called the index of the function /(x) between a and where
+
oo to
oo
is
When/(x) reduces
oo.
6.
to a rational function
Vi/V,
this
index
may
be calculated
by elementary processes without knowing the roots of V. It is clear that we may suppose Vi prime to and of less degree than V, for the removal of a poly nomial does not affect the index. Let us then consider the series of divisions necessary to determine the greatest common divisor of Fand FI, the sign of the remainder being changed each time. First, we would divide V by FI, obtaining a quotient Qi and a remainder F2 Then we would divide FI by F2 obtaining a quotient Q2 and a remainder Vs and so on. Finally we should obtain a con stant remainder F + 1. These operations give the following set of equations .
,
;
:
= FiQi - F2 = F2 Q2 - F8
F F!
The sequence F,
(7)
FI,
,
,
of polynomials
F
a,
Vr.it
-,
r
r,
Fr +
1
,
-..,
Fn
,
Fn +
1
has the essential characteristics of a Sturm sequence 1) two consecutive poly nomials of the sequence cannot vanish simultaneously, for if they did, it could be shown successively that this value of x would cause all the other polynomials to vanish, in particular Vn + \; 2) when one of the intermediate polynomials FI, :
,Vn vanishes,
the
number
of changes of sign in the series (7) is not altered, c. _i and Vr + \ have different signs for x It follows that the number of changes of sign in the series (7) remains the same, = 0. If Fi/F passes from + oo to oo, except when x passes through a root of
Pai for
if
Fr
vanishes for x
=
c,
F
r
F
number
by one, but it diminishes by one on the other hand V\/V passes from co to +00. Hence the index is equal to the difference the number of changes of sign in the series (7) for x = 6 and x = a. this
increases
We
if
of
78. Area of a curve. can now give a purely analytic definition of the area bounded by a continuous plane curve, the area of the For this purpose we need rectangle only being considered known.
IV,
ALLIED GEOMETRICAL CONCEPTS
78]
159
72. Let the results of only translate into geometrical language interval in the closed continuous is which function (a, b), f(x) be a in b and that f(x) and let us suppose for definiteness that a <
>
Let us consider, as above (Fig. the plane bounded by the contour AMBB A the interval.
,
9, 65), the portion of composed of the seg
axis, the straight lines AA and BB parallel to the y axis, and having the abscissae a and b, and the arc of the curve whose equation is y =f(x). Let us mark off on A B Q a certain A
ment A B
x
of the
MB
whose abscissae P,_i, Pi} and through these points let us draw meet the arc A MB in the points parallels to the y axis which Let us then consider, in respectively. Qi-D Qi, bounded of the the by the contour plane portion particular,
number
of points of division
are x 1} x 2 ,
,
#;_!, x i}
,
#2>
0>i>
P l} P2
,
,
,
,
)
the highest Qi-iQiPiPi-iQi-i, an(i l et us m ark upon the arc Q.-^Q,and the lowest points, that is, the points which correspond to the and to the minimum m of f(x) in the interval maximum the figure the lowest point coincides with
M
{
{
(#,_!,
(In
a;,-).
<2,-_j.)
Let Rf be the area of the rectangle P base Pi-i-P, with the altitude JJ/ and i
f,
rectangle Pf-iP^Q,--! erected
tude
m
t
P
erected upon the let r{ be the area of the
l
i
si si _
1
upon the base P,-^^ with the
alti
Then we have
.
and the results found above ( 72) may now be stated as follows whatever be the points of division, there exists a fixed number / which is always less than 2A\ and greater than 2r,., and the two sums 2Ri and 2r approach / as the number of sabintervals P ^ P increases in such away that each of them approaches zero. We shall call this common limit I of the two sums 2-Ri and 2r the area of Thus the portion of the plane bounded by the contour AMBB A A. :
i
f
1
i
{
the area under consideration integral
I
j
is
defined to be equal to the definite
ax, (<)
This definition agrees with the ordinary notion of the area of a plane curve. For one of the clearest points of this rather vague notion is that the area bounded by the contour P -iPiQ n Q _ P _ l i
i
i
i
]
i
between the two areas R and r of the two rectangles Pj_iP,-*,- s -i and Pi-iPiiiQi-i; hence the total area bounded by the contour AMBB A A must surely be a quantity which lies between the two lies
f
(
sums 2/? and 2r,-. But the definite integral / is the only fixed quan sums for any mode of tity which always lies between these two subdivision of A Q B since it is the common limit of 2-R, and f
,
2r<.
160
DEFINITE INTEGRALS
[IV,
79
The given area may also be defined in an infinite number of other ways as the limit of a sum of rectangles. Thus we have seen that the definite integral /
where
&
is
is
also the limit of the
any value whatever
sum
in the interval (x _ 1} t
a;,).
But the
element
of this
sum
represents the area of a rectangle whose base is P.-.jPj is the ordinate of any point of the arc Q _ l n Q It should be noticed also that the definite integral / represents the area, whatever be the position of the arc with respect to the x axis, provided that we adopt the convention made in 67.
and whose altitude
i
i
i
.
AMR
Every
definite integral therefore represents an area an integral is called a quadrature.
;
hence the calcu
lation of such
The notion
of area thus having been
there remains no reason
arguments which
why
it
made rigorous once for all, should not be used in certain
renders nearly intuitive. For instance, it is perfectly clear that the area considered above lies between the areas of the two rectangles which have the common base A B and which have the least and the greatest of the ordinates of the arc A it
,
MB,
respectively, as their altitudes. a rectangle whose base is A B
It is therefore equal to the area of
and whose altitude is the ordinate of a properly chosen point which is a restate upon the arc AMB, ment of the first law of the mean for integrals.
The following remark is also important. Let f(x) be a func which is finite in the interval (a, b) and which is discontinuous
79.
tion
in the
a
n
finite
manner described below for number of values between
a and is
b. Let us suppose that /(a-) continuous from c to c + &(&>0),
and that f(c -f c ) approaches a cer which we shall denote
tain limit,
+ 0), as e approaches zero through positive values and like f( c
FIG. 11
;
wise let us
suppose that f(x) is continuous between c k and c and - c) approaches a limit that/(c - 0) as e approaches zero f(c through positive values. If the two - 0) are different, the function limits f(c + 0) and f(c f(x) is dis continuous for x = c. It is usually agreed to take for /(c) the
IV,
ALLIED GEOMETRICAL CONCEPTS
80]
161
If the function /(a;) has a certain 4- 0) +f(c 0)]/2. of points of discontinuity of this kind, it will be repre Let c sented graphically by several distinct arcs AC, C D, D B.
value [f(c
number and
example, be the abscissae of the points of discontinuity.
d, for
Then we
shall write /~>
Xb
f(x)dx
=
s*d
c
f(x)dx
I
+
i/a
s*b
f(x)dx
I
*J c
+
I
f(x)dx,
i/a*
in accordance with the definitions of
72. Geometrically, this definite BB A A. area bounded by the contour A CC integral represents the If the upper limit b now be replaced by the variable x, the definite
DD
integral
is still
tinuous
x
=
a continuous function of
we
c for
still
have
example,
we
x.
F (x )=f(x
In a point x where f(x)
con
is
For a point of discontinuity,
).
have
shall
S>C+fl
F(c
+
K)
-
F(c) =
I
f(x) dx
=
hf(c
BK),
<
$
<
1,
+
+
and the
+
ratio \_F(c 0) 0) or f(c A) F(c)]/h approaches f(c or This is an h is as negative. according example of a positive function F(x) whose derivative has two distinct values for certain
values of the variable. 80. Length of a curvilinear arc. Given a curvilinear arc AB; let us take a certain number of intermediate points on this arc, 1 2)
m m ,
,
m n -\,
and
let
us construct the broken line A?n 1
m
2
mn _ B l
by
connecting each pair of consecutive points by a straight line. If the length of the perimeter of this broken line approaches a
number of sides increases in such a way that each of them approaches zero, this limit is defined to be the length of the arc AB. Let limit as the
be the rectangular coordinates of a point of the arc AB expressed in terms of a parameter t, and let us suppose that as t varies from
a to
b (a continuous
the arc
<
AB
and are continuous and possess and that the point (x, y, z) describes without changing the sense of its motion. Let
b)
the functions /,
first
derivatives,
<,
\j/
DEFINITE INTEGRALS be the values of
Then the
line.
or,
,.,
&
77,,
lie
80
which correspond to the vertices of the broken c is given by the formula
side
t
applying the law of the mean to x
where is
t
[IV,
between
and
,._!
tt
very small the radical differs very
In order to estimate the error we
#,_!,
{
When
.
the interval
(,._!,
/,)
from the expression
little
may
-,
write
A
in the
it
.--
.)]
form
+
But we have !/
() + / (**1
!
and consequently
Hence,
if
each of the intervals be made so small that the oscillation
of each of the functions
we
interval,
/ (*),
^
(t),
(*) is less
than c/3 in any
shall have
where M<J
and the perimeter of the broken
line
The supplementary term 2e,-(,than e2(fc that is, than ._,), #,-
c(/>
small as small,
it
is
less in absolute value
is
Since
e
may
be taken as
equal to the definite integral
s=C c/
,
a).
;
AB
This definition
,-_,)
please, provided that the intervals be taken sufficiently follows that this term approaches zero hence the length S
(8)
,
therefore equal to
we
of the arc
/
is
$
may
(l
be extended to the case where the derivatives
are discontinuous in a finite
number
of points of the arc
AB,
IV,
ALLIED GEOMETRICAL CONCEPTS
80]
which occurs when the curve has one or more corners.
AB
divide the arc
into several parts for each of
163
We need only
which/
,
<
,
^ are
continuous. It results from the formula (8) that the length S of the arc between a fixed point A and a variable point M, which corresponds to a value t of the parameter, is a function of t whose derivative is
2 whence, squaring and multiplying by dt we find the formula ,
dS
(9)
= dx + 2
2
dy*
+ dz
2 ,
which does not involve the independent variable. It is also easily remembered from its geometrical meaning, for it means that dS is the diagonal of a rectangular parallelepiped whose adjacent edges are dx, dy, dz.
Note.
Applying the first law of the mean which represents the arc
correspond to the values s
where
the chord c2
t
the interval
M M^ by n
1}
of the parameter (^
^
,
= arc JUoJ/i = (t, -
lies in
for integrals to the
MM
definite integral
t
(t
)
,
V/ ^).
2
(0)
On
+ 4,
2
(0)
whose extremities * we find ), >
+
,/,"(0),
the other hand, denoting
we have
c,
= [/CO -/Co)]
2
Applying the law of the mean for derivatives to each of the we obtain the formula
ences f(ti)f(tn ),
differ
,
where the three numbers
belong to the interval ( rj, By ^). the above calculation the difference of the two radicals is less than e, ,
,
provided that the oscillation of each of the functions/ ^), (), is less than e/3 in the interval (# ^). Consequently we have <
,
or, finally,
1s
If the arc
MM Q
l
is infinitesimal,
tl
t
approaches zero; hence
and therefore also 1 c/s, approaches zero. It follows that the of an infinitesimal arc to its chord approaches unity as its limit.
c,
ratio
DEFINITE INTEGRALS
164
81
[IV,
Example. Let us find the length of an arc of a plane curve whose as independent equation in polar coordinates is p = /(w). Taking variable, the curve is represented by the three equations x = p cos w, z = hence y = p sin
;
o>,
ds or,
2
=
+
dx 2
dy
2
= (cos
to
p sin
dp
simplifying, ds 2
=
2 o>
e?a>)
+P
2
dp
2
+ (sin wdp
-}-
p cos to
By
.
= R + R cos
ds or, letting
=R
o>
2
2
dta [sin
2
vary from
is
a).
we have
the preceding formula 2
,
d^ 2
Let us consider, for instance, the cardioid, whose equation p
2
c?w)
w
+
(1
+
2
cos o>)
]
=4R
2
cos 2
^ do
2 ,
to TT only,
ds
and the length of the arc
= 2 R cos
f/o)
U
;
is f
&R sin where w and ities of
u^ are the polar angles
the arc.
The
which correspond
to the
extrem
total length of the curve is therefore 8 R.
81. Direction cosines. In studying the properties of a curve we are often led to take the arc itself as the independent variable. Let us choose a certain sense along the curve as positive, and denote by s
AM
the length of the arc between a certain fixed point A and a vari able point M, the sign being taken or lies in according as the positive or in the negative direction from A. At any point
M
+
M
of the curve let us take the direction of the tangent which coincides with the direction in which the arc is increasing, and let a, be ft,
y
the angles which this direction makes with the positive directions of the three rectangular axes Ox, Oy, OK. Then we shall have the
following relations COS
dx
To
a
_
COS
:
ft
dy
_
COS y
dz
1
~
-
Vrfz
2
+ dif + dz
1
ds
2
which sign to take, suppose that the positive direction of the tangent makes an acute angle with the x axis then x and s increase simultaneously, and the sign + should be taken. If the angle a is obtuse, cos a is negative, x decreases as s increases, dx/ds find
;
IV,
ALLIED GEOMETRICAL CONCEPTS
82]
165
negative, and the sign -f- should be taken again. case the following formulae hold
Hence
is
in
any
:
a
cos
(10)
= dx
B
cos
>
dy = ~r
cos y
as
as
dz-
=
>
ds
with respect to the same otherwise arbitrary.
dx, dy, dz, ds are differentials taken
where
independent variable, which
is
segment of a straight line. Let MM^ be a segment whose extremities describe two curves C, C x On
82. Variation of a
of a straight line
.
each of the two curves
us choose a
let
point as origin and a positive sense of motion, and let us adopt the follow ing notation s, the arc AM; s 1} the arc the two arcs being taken with l l :
A
M
,
the same sign
;
angle between
the length MM the MM^ and the positive di I,
l
MT;
rection of the tangent
between
AT,
1}
;
B,
the angle
M and the positive direction
FIG 12
We
of the tangent M^ 7\. try to find a relation
proceed to between $, 6 1 and the differentials
Let
ds,
ds l}
be the coordinates of the points M,
(x, y, z), (x 1} y lt z^ respectively, a, ft, y the direction angles of Then we have direction angles of M^ 7\.
P
= (x -
arO"
MT, and
+ (y - y^ + (z -
a^, fa,
dl.
M
ly
y l the
ztf,
from which we may derive the formula ldl
= (x- x^ (dx
dxj
+ (y -
y,)
(dy
- dyj +
(z
- z^ (dz
dzj,
which, by means for
of the formulae (10) and the analogous formulae be written in the form
C lf may dl
=
-
Ix I
\
+ But
-
/x, -L ^6 \
a
+J
?/, Jl
COS
C
x cos
H
z -j
z,
1
l>
cos ft
-f
\
(/
_
COS y ds / I
z,
11
?/,
>/i
<*!
(3
+ -*-j
z
cos
l>
y\) fl, (z z-^/l are the direction cosines of Likewise coefficient of ds is cos 0.
and consequently the
the coefficient of ds l
(10
11
C03
I
x^)/l, (y
(a;
M M,
x,
V
)
is
dl =
cos ^; hence the desired relation
ds cos 6
ds! cos
is
0,.
We shall make frequent applications of this formula proceed to discuss immediately.
;
one such
we
DEFINITE INTEGRALS
166 Theorems
83.
and
let the
of
Graves and
two tangents
E and E
Let
of Chasles.
[IV,
be two confocal
83
ellipses,
MB to the interior ellipse E be drawn
MA,
M, which
lies
difference
MA
from a point on the exterior ellipse E The arc ANB remains con + .
MB
M describes the
stant as the point
ellipse
E
.
OA
and OB, the arc O M, I and I the distances and BM, 6 the angle between MB and the positive
and
Let
denote the arcs
s
AM
M
T. direction of the tangent Since the ellipses T is and are confocal the angle between
M
MA
AM
coincides 6. equal to it Noting that FlQ 13 with the positive direction of the tangent at A, is the negative direction of the tangent at B, we find from the
BM
and that
formula (10
),
successively, dl
=
+
ds ds
dl
d
cos 6
do-
cos 6
,
whence, adding,
+ )=d (s -s)=d (arc ANB), 1
d(l
l
which proves the proposition stated above. The above theorem is due to an English geometrician, Graves. The following theorem, discovered by Chasles, may be proved in a similar manner. Given an If from a point on that ellipse and a confocal hyperbola which meets it at N. and MB the two tangents branch of the hyperbola which passes through
M
N
be drawn to the
ellipse, the difference of
difference of the tangents
III.
A
the arcs
number
NB will
be equal to the
INTEGRATION BY PARTS
which cannot be evaluated two general processes which we shall discuss
of definite integrals
directly yield to the in this section.
Change
NA
MB.
CHANGE OF VARIABLE
large
84.
MA
MA
of variable.
If in the definite integral /*/(*) dx the new independent variable t by means
variable x be replaced by a of the substitution x
=
new
a
(t),
definite integral is obtained.
continuous and possesses a and that proceeds from a to b without changing sense as t goes from a to ft. The interval (a, ft) having been broken up into subintervals by xn _ l} b tn _ let a, x l} x z the intermediate values a, t v t, ft, be the corresponding values of x == Then, by the law of the
Let us suppose that the function continuous derivative between a and
is
(t)
,
ft,
(f)
,
l ,
,
,
(t).
mean, we shall have
where B
t
value of
Let between t _ and x which lies between x _ and x
lies
i
l
?,-.
,-
i
l
<(0,-)
(
.
be the corresponding
Then the sum
IV.
CHANGE OF VARIABLE
5*1
-
(x,
a)
+ /(&) (x, -
*
t
)
+
+/(,) (6
approaches the given definite integral as may also be written
and
in this
form we see that
-
new
xn _
But
limit.
its
approaches the
it
167
this
sum
definite integral
C
Ja
This establishes the equality
as its limit.
C
(\Y\ L L \J
I
-
which
t
b
f(x~\dx v
v
,/
va
"
^
called the
formula for the change of variable. It is to new differential under the sign of integration is obtained by replacing x and dx in the differential f(x}dx by their values and (t)dt, while the new limits of integration are the values of t which correspond to the old limits. By a suitable choice is
be observed that the
<}>
(t)
of the function
the <()
new
evaluate than the old, but rules in the matter.
may
integral
it is
turn out to be easier to
impossible to lay
down any
definite
Let us take the definite integral
for
instance,
and
let
dx
/ Jo
(x
us
make
a)*
+
p*
the substitution x
=
a
-f- fit.
It
becomes
dx
1
r
dt
I
tan
or,
*
+ arc tan
a
returning to the variable x, 1 / x - a -arc tan l
;
Xot
\-
arc tan
a -
made in establishing the formula (11) were Thus it is not necessary that the function should always move in the same sense as t varies from a to f3. For defiall
the hypotheses
necessary.
<()
niteness let us suppose that as t increases from a to y (y /8), then as t increases from y to steadily increases from a to c (c i) decreases from c to I. If the f unction /(x) is continuous in /3, <
<()
>
;
<()
the interval (a, vals (a,
the formula may be applied which gives
c),
c), (c, b),
to each of the inter
DEFINITE INTEGRALS
168
or,
L*v,$84
adding,
On the other hand, it is quite necessary that the function should be uniquely denned for all values of t. If this condition be disregarded, fallacies may arise. For instance, if the formula be
(t)
applied to the integral f_ l dx, using the transformation x we should be led to write +i
r
l
=
1?
/2 ,
3
/ *-J
j
which
is evidently incorrect, since the second integral vanishes. In order to apply the formula correctly we must divide the interval In the first of 1, 0), (0, 1). 1, -f- 1) into the two intervals ( (
these
we should
take x
second half interval to
We
1.
=
we
Vr
then find a correct +
X
i
dx 1
it
let
result,
t
vary from 1 to
= ~\/t
s
and
3
I
t/O
and
ft
be replaced by x
().
and
t
in the
becomes
<()
/[<()]<
In the
0.
vary from
namely
which shows that the transformation x = F(x), whose derivative is /(#), into a function is
let t
<~i
=
If the upper limits b
Note.
formula (11),
and
should take x
carries a function
<()
whose derivative
This also follows at once from the formula for the
derivative of a function of a function.
Hence we may
write, in
general,
which
is
integrals.
the
formula for the change of variable in indefinite
ivr,
INTEGRATION BY PARTS
85]
169
by parts. Let u and v be two functions which, with their derivatives u and v are continuous between a together and b. Then we have 85. Integration
,
d(uv) _ dx
dv
du
dx
dx
whence, integrating both sides of this equation,
we
find
C dhm) = C dv C du dx u \ dx. dx+ v b
b
-----
I
dx
Ja This
may
\
dx
Ja
Ja
be written in the form f*b
/*b
u dv
/
(12)
= \_uv~\l -
\J a
we
I
v du,
*J a
where the symbol [F(x)] denotes,
If
dx
in general, the difference
replace the limit b
by a variable limit x, but keep the limit a which amounts to passing from definite to indefinite inte constant, grals, this formula becomes (13)
|
u dv
=
uv
v du.
I
Thus the calculation of the integral / u dv is reduced to the cal culation of the integral fvdu, which may be easier. Let us try, for example, to calculate the definite integral
rx m logxdx,
ra
I
+ 1^0.
\J ct
Setting u
= logic,
v
c\logx.x m
Ja
xm +
l
/(m
+ 1),
n^+ dx=\
1
+1
log X
m+1 This formula
is
not applicable
if
_
r
i
iogarr
m +1 J
L "
case
the formula (12) gives
I
in
+ 1 Ja
Xm +
b
x m dx
16
*
2
~(m+l) m+1 = in ;
a
that particular
we have
It is possible to generalize the
sive derivatives of the +1 ( u, v , -.., w
formula (12). Let the succes two functions u and v be represented by
>;
v",
v (n + \ 1
"
M",
-,
Then
the application of the
DEFINITE INTEGRALS
170
formula (12) to the integrals following equations
85
leads to the
dv<*-",
:
s*b
/>6
uv (n + 1) dx=
I
fudv, fu
[IV,
=
udv^
I
(n)
]*
[>w
Ja
Ja s*\>
u
I
=1 Ja
Ja ~b /
M
^ --
-
=
(
b
f
r>t>
v^dx
I*-
r<"
|>
wV
I
J*
/-.b
u<*>v
dx
=1 Ja.
Ja
u^do
=[
( "]a
Multiplying these equations through by and then adding, we find the formula
C
+l
+1
and
1 alternately,
(n
Ja n+l) dx which reduces the calculation of the integral fuv^ H
+
l)
to the cal
vdx.
culation of the integral In particular this formula applies when the function under the of a polynomial of at most the wth integral sign is the product function v. order of 1) of a known (n degree and the derivative (M + 1) no contains member second the integral For then w 0, and f*i<
+
=
signs.
Suppose, for instance, that
we wished
to evaluate the definite
integral
fW(*)*,
\J
a
wi n+v where /(x) is a polynomial of degree n. Setting u =/(z), v = e /u) x has been taken the formula (14) takes the following form after ,
e"
out as a factor
:
The same method,
or,
what amounts
to the
same
thing, a series of
definite integrals integrations by parts, enables us to evaluate the
I
Ja where f(x)
is
c,osmxf(x)dx,
a polynomial.
I
*/m
sinmxf(x)dx,
IV,
INTEGRATION BY PARTS
86]
171
s series with a remainder. In the formula (14) let us u replace by a function F(x) which, together with its first n + 1 n derivatives, is continuous between a and b, and let us set v = (b x) Then we have
86.
Taylor
.
v
=
= n(n - l)(b = (-!)!. 2 .--W, y + = 0, = b, vanish for v, v 1
n(b
2
v"
,
a;)"-
-re)"-
(
,
-.,
1
"
>
v<>
1
and, noticing that
,
,
v",
i/"-
)
a:
following equation from the general formula
= (-!)
n\F(b)
-n\F
n\F(a)
-jF
t
(a) (b
-a)
(a)(b
ri\
-a)
2
we obtain the
:
----
F<>
(a) (b
~]
-
a)
]
which leads
to the equation
-
ay
-n,
rb
i
^
-7 ?i
Since the factor b,
n
x)
(i
we may apply the law
/
^
i
+
I
)(a-)(&-a;)
n
^.
!i/ a
keeps the same sign as x varies from a to of the mean to the integral on the right,
which gives I
F
Ja
where
lies
+
l
\x)(b
- x)dx =
between a and
b.
+ F<
1
(f)
87. Transcendental character of e
Hermite
to
equation whose coefficients are Setting a
=
and w
=
1
:
n
dx
Substituting this value in the preced
ing equation, we find again exactly Taylor form of the remainder.
famous theorem due
f (b- x}
Ja
.
From
s
formula, with Lagrange
s
the formula (15) we can prove a e is not a root of any algebraic
The number
all integers.* in the formula (15),
it
becomes
JT * The present proof used by Hermite.
is
due to D. Hilbert,
who drew
his inspiration
from the method
DEFINITE INTEGRALS
172
87
[IV,
where F(x) =/(x)
and
this again
written in the form
may be
F(b)
(16)
Now
us suppose that
let
cients are
all
integers
=
e
= were the root of an algebraic equation whose
coeffi
:
c
Then, setting b
+/ (z)
+
0, 1, 2,
+
c\e ,
wi,
c 2 e2
+
----1-
cm em
=
0.
and adding c m we
successively, in the formula (16),
the results obtained, after multiplying them respectively by
c
,
c l5
,
,
obtain the equation e
(17)
~* dx
=
m. We proceed where the index i takes on only the integral values 0, 1, 2, which is up to if the is relation a such that polynomial /(x), show to impossible ,
the present arbitrary, be properly chosen. Let us choose it as follows :
/(X )
---
-
1
(P
where p
a prime
is
I)
number
XP~ I
-
(X
l)p(x-2)P---(x -m)P,
1 -
This polynomial
greater than m.
of degree
is
of the coefficients of its successive derivatives past the pth are integral multiples of p, since the product of p successive integers is divisible for 1) derivatives, vanishes by p!. Moreover /(x), together with its first (p are all integral mul it follows that F(l), F(2), and , F(m) x m, -, 1, 2,
mp
p
-|-
i
?
and
all
-
=
tiples of p.
It only
remains to calculate F(0), that
is,
=/(0) In the
first
place, /(O)
= /O- 2
=/(0) =
= >(0)
0,
while /Cp>(0),
have just shown. integral multiples of p, as we - 1) !, I only multiply the coefficient of XP~ in/(x) by (p
are
all
To
find
/^ + -
/
which gives
1)
1 >(0),
(0)
(1
.
we need
2
m)p.
Hence the sum c
is
F(0)
equal to an integral multiple of
p
increased by
i- c (l
.
2
m)p.
be divisible p be taken greater than either m or c the above number cannot be an will sum from zero. of the different integer first the hence (17) portion by p We shall now show that the sum If
,
;
can be made smaller than any preassigned quantity by taking p sufficiently As x varies from to i each factor of /(x) is less than m hence we have ;
large.
IV,
INTEGRATION BY PARTS
88]
u
.
f(x)e-*dx
from which
it
m mp+ P -\
e~ x dx<
|
Jo
(p-1)!
173
(P-1)I
follows that
2
/o
M
As p increases indefi c + cm where is an upper limit of + Ci + nitely the function 0(p) approaches zero, for it is the general term of a conver It gent series in which the ratio of one term to the preceding approaches zero. follows that we can find a prime number p so large that the equation (17) is impossible hence Hermite s theorem is proved. .
|
|
|
|
|
;
88. Legendre s polynomials.
Pn (x)
Let us consider the integral
a polynomial of degree n and Q is a polynomial of degree less us try to determine Pn (x) in such a way that the integral van We may consider n (x) as the nth derivative of a ishes for any polynomial Q. polynomial R of degree 2n, and this polynomial R is not completely determined, for we may add to it an arbitrary polynomial of degree (n 1) without changing
where
is
than n, and
let
P
its
We may therefore
nth derivative.
together with
its first
set
Pn = d n R/dx n
,
where the polynomial E, = a. But integrating
vanishes for x
1) derivatives,
(n
by parts we find
rQQ dnE dx--
^
Ja and
since,
by hypothesis,
B
E(o)=0,
(a)
=
0,
B(-J)(o)=0,
-,
the expression R( -
Q (6) must also vanish
if
1)
(6)
Q
n (&)>
~l
Qf-
>
B(6)
=
The polynomial R (x)
-
n
-
(b) fi(
- 2)
(6)
~
+
Qf.*
)(b)
If
a)
(x
b)
R (b) =
0, is
0,
-,
E<-i)(6)
(18)
s
=
0.
therefore equal, save for a constant factor, to the product
n
;
the limits a and 6 are
Legendre
R (b)
is to be arbitrary, the quantities are themselves arbitrary; hence we must also have
and the required polynomial save for a constant factor, in the form (x
(b)
the integral is to vanish. Q of degree n 1
Since the polynomial Q(&)i
- Q
1
and
+
1,
--
Pn (x)
X
n
=
completely determined,
respectively, the polynomials
Choosing the constant
polynomials.
is
-
2.4.6...2nax
[(x LV
C 2
Pn
are
with Legendre, we will set
-
!)].
DEFINITE INTEGRALS
174 If
we
also agree to set
y =
-AO
y-
i
1,
-<M
X
we
1,
Xr
l
JL Z
s
=
In general,
(
>
2i
X
n.
n is a polynomial of degree n, all the exponents of x being even or Leibniz formula for the nth derivative of a product of two factors
17) gives at
once the formulae
=
-T.(-l)
(19)
-Z(l)
By
the general property established above,
l,
C
(20)
+
/
where
tf>
-3x
5x3
"2i
odd with
88
have
shall
r = 3x*~
= x,
[IV,
(x) is
X,
t (x)dx
=
=
(-!)"
0,
i
less than always have
any polynomial of degree
are two different integers,
we
shall
In particular,
n.
if
m
and n
+
C
(21)
J-i
This formula enables us to establish a very simple recurrent formula between
X
three successive polynomials n Observing that any polynomial of degree n can be written as a linear function of Xi, n it is clear that we may set .
X
,
X
,
,
are constants. In order to find C3 for example, let us Ci, C2 multiply both sides of this equation by ^Tn _ 2 and then integrate between the limits 1 and + 1. By virtue of (20) and (21), all that remains is
where C
,
,
,
,
+
3
C J_1
2 ""
"~
2
and hence C3 = 0. It may be shown in the same manner that C = 0, 5 = 0, The coefficient Ci is zero also, since the product xXn does not contain Finally, to find Co and C 2 we need only equate the coefficients of x n + 1 and then equate the two sides for x = 1. Doing this, we obtain the recurrent formula .
x".
(n
(22)
+ l)Xn + l
(2n
+ l)xXn +
nX,,_i
=
0,
which affords a simple means of calculating the polynomials The relation (22) shows that the sequence of polynomials /oo\
~V
~TT
\&)
-^-Oi
-**-!}
~Y~ -"-2i
X
n successively.
~y
"
i
-"-n
1 possesses the properties of a Sturm sequence. As x varies continuously from to + 1, the number of changes of sign in this sequence is unaltered except when But the formulse (19) show that there are n x passes through a root of n = 0.
X
changes of sign in the sequence (23) f or x = has n real roots between the equation n = follows from Rolle s theorem.
X
1, 1
and none for x = 1. Hence and -f 1, which also readily
IV,
IMPROPER AND LINE INTEGRALS
89]
175
GENERALIZATIONS OF THE IDEA OF AN INTEGRAL IMPROPER INTEGRALS LINE INTEGRALS*
IV.
89. The integrand becomes infinite. Up to the present we have sup finite between the limits of inte posed that the integrand remained In certain cases, however, the definition may be extended gration. Let us first to functions which become infinite between the limits.
consider the following particular case f(x) is continuous for every value of x which lies between a and b, and for x b, but it becomes b. for definiteness that a will suppose a. infinite for x :
=
=
Then the b (e
>
We
<
0) has a definite
+e
and
be taken.
If
taken between the limits a
integral of f(x)
value, no matter how small
e
this integral approaches a limit as e approaches zero, natural to denote that limit by the symbol
Jf/(*)
it is
usual and
dx.
If a primitive of /(cc), say F(x~), be
known, we may write
C Ja +
and
limit as
examine F(a
sufficient to
is
it
c
approaches
zero.
We
Mdx
-f-
e)
for convergence
toward a
have, for example,
r
lL the term
1
increases indefinitely as e approaches zero. 1 and it is than unity, we may write l/e?~ = e Hence in this case clear that this term approaches zero with c. If
fj.
But
>
if
1, /u,
l/c^"
C"
fi
1,
we
we may
write
Mdx
I Tr X Ja (
=
"
,
the definite integral approaches a limit, and
If
1
l
is less
^~ a)
have;
/: /a4
M dx = M (b log
a
and the right-hand side increases indefinitely when
e
approaches
zero.
To sum
up, the necessary and sufficient condition that the given inte should gral approach a limit is that /x should be less than unity. *It is possible, if desired, to read the next chapter before reading the closing sec tions of this chapter.
DEFINITE INTEGRALS
176
The tion
straight line x
=
a
[IV,
89
an asymptote of the curve whose equa
is
is
U if
p.
It follows
is positive.
from the above that the area bounded by
=
the x axis, the fixed line x b, the curve, and finite value provided that //,
not known, we
is
asymptote, has a
its
may compare
the given
integral with known integrals. The above integral is usually taken as a comparison integral, which leads to certain practical rules which are sufficient in many cases. In the first place, the limit b
upper
does not enter into the reasoning, since everything depends upon the manner in which f(x} becomes infinite for x = a. We may therefore replace b by any to writing
f*+f
number whatever between a and
= fa +( + f*
.
number of roots near x the same sign between a and nite
We
will first
=
that the integral f*
Then, if \f(x)
val, the definite integral
Hf(x ) is
is
is ^
positive in the interval (a, b), dx approaches a limit as e
(x)
\<<j>(x)
fa + f(x)dx t
throughout the whole inter
also approaches
positive throughout the interval (a, For, since f(x) is less than
/
f(x)dx +
infi
that f(x) keeps
:
immediate.
*J a
which amounts
c.
prove the following lemma
zero.
approaches
we may suppose
a,
Let $(x) be a function which
and suppose
b,
In particular, unless f(x) has an
<
I
Ja +
e
limit.
the demonstration
b),
(x),
a
we have
^(x}dx. e
Moreover f* f(x)dx increases as c diminishes, since all of its ele +f ments are positive. But the above inequality shows that it is con stantly less than the second integral hence it also approaches a limit. If f(x} were always negative between a and b, it would ;
be necessary merely to change the sign of each element. Finally, the function f(x) has an infinite number of roots near x = a, we may write down the equation
if
f+
*Ja
The
f(x) dx f
|/(*)|
second <$(*).
=
f
c/a +
integral
Now
[/(*)
+
|
/(or)
\\dx-f
e
<Ja
\f(x) dx. |
+s
on the right approaches a limit, since the function f(x) \f(x)\ is either positive
+
IV,
IMPROPER AND LINE INTEGRALS
89]
or zero
between a and
and
b,
value cannot exceed 2
its
177
<(#);
hence
the integral
f i/a
+
also approaches a limit, and the lemma is proved. It follows from the above that if a function f(x) does not approach a, but always remains less than a fixed any limit whatever for x
=
integral approaches a limit. sin(l/x)dx has a perfectly definite value.
number, l
fQ
the
Thus
the
integral
Suppose that the function /(#) can be written in
Practical rule. the form
where the function
ij/(x)
remains
finite
when x approaches
a.
remains less in absolute value than 1 and the function If (x~) a fixed number M, the integral approaches a limit. But if ^ 1 and the absolute value of ty(x) is greater than a positive number ra, the fjL
<
\]/
/JL
integral approaches no limit.
The
part of the theorem
is very easy to prove, for the abso than M/(x aY, and the integral of the 1. latter function approaches a limit, since p, In order to prove the second part, let us first observe that ^(a*)
first
lute value of f(x)
is less
<
keeps the same sign near x = exceeds a positive number m.
between a and
b.
a,
since its absolute value always shall suppose that \(/ (x)
We
>
Then we may write
X
m
dx
4-
and the second integral increases indefinitely as e decreases. These rules are sufficient for all cases in which we can
find an
for ^/(.i") approaches, exponent p such that the product (x x = a, a limit A different from zero. If /* is less than unity, the limit b may be taken so near a that the inequality
holds inside the interval
(a, J),
where L
is
a positive
number
greater
DEFINITE INTEGRALS
178
K
than I
if
p^
[IV,
Hence the \.
1, b
may
integral approaches a limit. be taken so near to a that
"
I./
V"V
\
On the other hand,
n a )
/, r
I
89
c
\t>.
where I is a positive number less than K\. inside the interval (a, Moreover the function f(x), being continuous, keeps the same sign b hence the integral f f(x)dx increases indefinitely in absolute |
>),
;
value.*
Examples. Let/(x) = P/Q be a rational function. If a is a root m of order m of the denominator, the product (x a) /(x) approaches Since m is at least equal to a limit different from zero for x = a. unity, it limit as
clear that the integral f
is e
approaches
But
zero.
P and R
two polynomials and
is prime to its deriv -R(ar) /(z) approaches a limit for x = a if a Thus a root of R(x), and the integral itself approaches a limit.
ative, the is
all
consider the function
=
/(*)
where
f(x)dx increases beyond
we
if
are
product (x
a)
1/2
the integral
dx
J f -\
approaches 7r/2 as e approaches zero. l 1/2 loga; Again, consider the integral ff \ogxdx. The product # has the limit zero. Starting with a sufficiently small value of x, we
may
therefore write log x
<
Mx~ 1/2 where ,
M
is
a positive
number
Hence the
chosen at random.
Everything which
integral approaches a limit. has been stated for the lower limit a
may
be
repeated without modification for the upper limit b. If the function f (cc) is infinite for x b,we would define the integral Ja /(#) dx to be the limit of the integral / ( x ) dx as c approaches zero. If /(#) /J*"
is infinite
at each limit, b
the integral
C
"
f(x)dx
ently of each other. we may write
*The
first
we would
e
Let
c
and
part of the proposition
c
may
can be found (0 an exponent approaches a limit A as a; approaches a, a limit
if
/JL
define
f
f (x~) dx
as the limit of
both approach zero independ be any number between a and b. Then
as
e
also be stated as follows: the integral has
<
/*
<
1)
such that the product (x a)>*f(x) A = not being excluded.
the case where
IV,
IMPROPER AND LINE INTEGRALS
90]
r
. f
/(*)<&
Ua +
f
<J
a-
179
+ c
f(x)dx
*J c
+f
and each of the integrals on the right should approach a limit in this case. if f(x) becomes infinite for a value c between a and b, the integral */(%) dx as the sum of the limits of define would we e b two the integrals f~ f(x)dx, fc +f f(x)dx, and we would proceed in a similar manner if any number of discontinuities whatever lay
Finally,
between a and
b.
noted that the fundamental formula
It should be
(6),
which was
established under the assumption that f(x) was continuous between a and b, still holds when f(x) becomes infinite between these limits,
provided that the primitive function F(x~) remains continuous. For the sake of definiteness let us suppose that the function f(x) becomes Then we have infinite for just one value c between a and b. I
f(x) dx
=
lim
t) a
and
if
F(x")
e
is
a primitive of /(x), this
f(x) dx
=
Xft
lim F(c =0
-
+
f(x) dx
I
=0t/a
c
)
-
F(a)
lim I f(x) dx e = 0c/c + e
may
+
;
be written as follows
F(b)
-
lim F(c f=0
+
=
:
e).
+
Since the function F(x) is supposed continuous for x c, F(c e) have the same limit F(c), and the formula again e and F(c )
becomes
f j (xj ix I
*J a
The following example
is
+
J
i
illustrative 1
:
dx
-Jx
If the primitive function F(x) itself becomes infinite between a and for the integral on the left has as yet b, the formula ceases to hold,
no meaning
in that case.
The formulae
may
for
change of variable and for integration by parts
be extended to the
new kinds
of integrals in a similar
manner
by considering them as the limits of ordinary integrals. 90. Infinite limits of integration. is
continuous for
Then
all l
the integral
f
Let/(x) be a function of x which
values of x greater than a certain number a, has a definite value, no f(x) dx, where I .
>
DEFINITE INTEGRALS
180
[IV,
90
how large I be taken. If this integral approaches a limit increases indefinitely, that limit is represented by the symbol
matter as
I
f If a primitive of f(x) be integral approaches a limit.
f(x)dx.
known, it is easy to decide whether the For instance, in the example dx
f
=
arc tan
I
Jo
the right-hand side approaches Tr/2 as I increases indefinitely, and this is expressed by writing the equation 7T
/
2
Ja Likewise,
if
a
is
positive
I
and
kdx
//,
k
1
is
/
1
different
from
zero,
we have
1
c/a
If I
/A is
greater than unity, the right-hand side approaches a limit as
increases indefinitely,
and we may write
kdx
On
the other hand,
nitely with
if
/i
is less
The same
I.
is
k
than one, the integral increases indefi = 1, for the integral then p.
true for
results in a logarithm.
When no
primitive of /(#) is known, we again proceed by com parison, noting that the lower limit a may be taken as large as we Our work will be based upon the following lemma please. :
Let a, and suppose that (x) be a function which is positive for x l the integral dx a limit. Then the approaches integral f f(x) dx JJ (x) also approaches a limit provided that \f(x) ^ () for all values of >
<
\
x greater than
The proof
a.
of this proposition
is
exactly similar to that given above.
If the function f(x) can be put into the
/() =
form
*<
ty(x) remains finite when x is infinite, the follow theorems be can ing demonstrated, but we shall merely state them
where the function
IV,
IMPROPER AND LINE INTEGRALS
91]
181
M
p.
and If the absolute value of ^ (x~) is less than a fixed number greater than unity, the integral approaches a limit. If the absolute value of (x) is greater than a positive number m
is
(]/
and
p.
is less
than or equal
to
unity, the integral approaches no limit.
For instance, the integral /
cos ax
iT^ approaches a limit, for the integrand cos
ax
dx
may
1 cos
be written
ax
i+*-**rT| and the coefficient of 1/x 2 is less than unity in absolute value. The above rule is sufficient whenever we can find a positive num ber p, for which the product x*f(x) approaches a limit different from zero as x becomes infinite. The integral approaches a limit if p, is greater than unity, but it approaches no limit if p. is less than or equal to unity.*
For example, the necessary and
sufficient condition that the inte
gral of a rational fraction approach a limit when the upper limit increases indefinitely is that the degree of the denominator should
exceed that of the numerator by at least two units. take
Finally,
if
we
P and R are two polynomials of degree p and r, respectively, the product x r/2 ~"f(x) approaches a limit different from zero when x becomes infinite. The necessary and sufficient condition that the where
integral approach a limit is that
p
be less than r/2
The rules stated above are not always whether or not an integral approaches a ing 91.
f(x)
= (sin x)/x
is p.
less
and
1.
1.
sufficient for limit.
determin
In the example
for instance, the product x*f(x) approaches zero if than one, and can take on values greater than any given number if p, is greater than one. If p. = 1, it oscillates between + 1
None
proach a limit.
}
of the above rules apply, but the integral does ap Let us consider the slightly more general integral
* The integral also approaches a limit zero as x becomes infinite.
if
the product x^f(x) (where
M>
1)
approaches
DEFINITE INTEGRALS
182
sin -ax S1
/i e-~The integrand changes sign
for
a;
91
,
dx,
=
x
[IV,
a>0.
We
kir.
are therefore led to
study the alternating series
-
(24)
i
+
where the notation used /
a
=
I
2
is
-
as
+
+ (_ !)
...
27T
[
e~ ax
I
Jo sin
+
i<
that
is,
cc
,
+ WT
Moreover the general term a n
Hence the above
than 1/n.
sin --dx,
x
It is evident that the integrand decreases as
a n-
.,
term a n may be written
nir for x, the general
y an +
.
X
BIT
Substituting y
.
the following:
sm x
/
+
is
n increases, and hence
less than
f*(l/mr)dy,
series is convergent, since the
we proceed in the series, and the general term approaches zero. If the upper limit I lies between mr and (n + 1) TT, we shall have absolute values of the terms decrease as
-dx
=
Sn
6a n
0<9<1,
,
where Sn denotes the sum of the first n terms of the series (24). As I increases indefinitely, n does the same, a n approaches zero, and the integral approaches the sum S of the series (24). In a similar manner it may be shown that the integrals
r+*
r+ sino; 2 ^x,
I
Jo
I
Jo
which occur in the theory of diffraction, each have finite values. The curve y = sin a; 2 for example, has the undulating form of a sine curve, but the undulations become sharper and sharper as we go out, ,
since the difference ^/(n x 2 approaches zero as
sin
Remark. This indefinitely sin
last 2
+ I)TT n
^/n7^ of two consecutive roots of increases indefinitely.
example gives rise between
oscillates
approach a limit even
if
to 1
an interesting remark.
and
+
1.
Hence an
As x
increases
integral
the integrand does not approach zero, that
is,
may
even
if
IV,
IMPROPER AND LINE INTEGRALS
92J
the x axis of the
is
remains positive when x
=
The following
which the function /(x) does not change
in
+
l
f(kn)
= /(x).
not an asymptote to the curve y
same kind
is
is
an example
The
sign.
function
x6 sin 2 x
and
positive,
183
does not approach zero, since let us con
it
In order to show that the integral approaches a limit,
kit.
sider, as above, the series
+
flo
<*i
+
+
+
o
i
where
=
a
l
As x
varies
from
nit to (n
+
1)
TT,
x6
is
x6 sin 2 x
+
6 constantly greater than n
7t
6 ,
and we may
write l)rr
A primitive
function of the
new
==
integrand arc tan
f n*7t6
(
V
dx
is
1
+ n n6
tan x),
and as x varies from mt to (n + 1) TT, tan x becomes infinite just once, passing from + co to oo. Hence the new integral is equal ( 77) to 7T/V1+ ns 7f6 and we have ,
n
It follows that the series
approaches a
2a
*2 _
,
Vl + is
n
(n
-
+ -
1)
n3 it
TT"
convergent, and hence the integral J^ /(x) dx
limit.
On the other hand, it is evident that the integral cannot approach any limit For /(x) approaches a limit h different from zero when x becomes infinite. beyond a certain value of x, /(x) will be greater than h/2 in absolute value if
|
\
and will not change sign. The preceding developments bear a close analogy to the treatment of infinite series. The intimate connection which exists between these two theories is brought out by a theorem of Cauchy s which will be considered later (Chapter We shall then also find new criteria which will enable us to determine VIII). whether or not an integral approaches a limit in more general cases than those treated above.
92.
The function
The
T(a).
definite integral
+
T(a)= f
(25)
Jo
V-
has a determinate value provided that a For, let us consider the two integrals
r I
i
x a - e- x dx,
is
1
e-*dx
positive.
r I
i
-l -*
184
DEFINITE INTEGRALS
[IV,
93
where t is a very small positive number and is a very large positive number. The second integral always approaches a limit, for past a sufficiently large value a+ of x we have x a - e~ x As for the first 1/x 2 that is, e x the I
l
l
<
.
>x
,
integral, 1 - a product x f(x) approaches the limit 1 as x approaches zero, and the necessary and sufficient condition that the integral approach a limit is that 1 - a be less than unity, that is, that a be positive. Let us suppose this condition satisfied.
Then
the
sum
of these
two
limits
integral of the second kind.
is
the function T(a), which
This function T(a) becomes
is
also called
Euler
s
a approaches zero, it is positive when a is positive, and it becomes infinite with a. It has a minimum for = 1.4616321-.., and the corresponding value of T(a) is 0. 8856032-.
Let us suppose that er x
differential of
a>
1,
and integrate by
infinite as
parts, considering e~ x dx as the
This gives
.
but the product x a ~ l e- x vanishes at both limits, since a only the formula (26)
r(o)
=
(a
-
l)T(o
-
1,
>
and there remains
1).
The repeated
application of this formula reduces the calculation of Y(a) to the case in which the argument a lies between and 1. Moreover it is easy to determine the value of T(a) when a is an integer. For, in the first place,
and the foregoing formula therefore
and, in general,
if
n
is
(27)
gives, for
a
=
2, 3,
a positive integer,
r(n)
= 1.2.3...(n-l) =
(n-l)l.
Let AB be an arc of a continuous plane curve, be a continuous function of the two variables x and y) where x and denote the coordinates of a point of AB y along AB, y with respect to a set of axes in its plane. On the arc AB let us 93. Line integrals.
and
let
P (x,
m
m
take a certain number of points of division li m z whose coordinates are (x lt y^, (x 2 yz ), ., and then upon each (x i; y.), of the arcs _^rn i let us choose another point n { (,., ^.) at random. let us consider the sum Finally, ,
m
,
,
{ ,
,
,
i
(28) ,
*,*! -*,_,
-..
these partial intervals. When the number of points increased indefinitely in such a way that each of the differences x x _ l approaches zero, the above sum approaches a
extended over of division
all
is i
i
IV,
IMPROPER AND LINE INTEGRALS
93]
185
which is called the line integral of P(x, y) extended over the arc AB, and which is represented by the symbol
limit
JAB
P(x, y)dx.
In order to establish the existence of this limit, let us first sup pose that a line parallel to the y axis cannot meet the arc AB in more than one point. Let a and b be the abscissae of the points A
and
Then
and
y=
be the equation of the curve AB. a continuous function of x in the interval (a, b), by
B, respectively, is <(#)
let
(x)
hypothesis, and if we replace y by resulting function $(cc) P[x, <(X)]
(x~)
=
is
in the function P(x, y), the
Hence we
also continuous.
have
and the preceding sum may therefore be written
*(,) to
-
in the
form
a)
It follows that this surn
approaches as
its
limit the ordinary definite
integral I i/a
and we have
finally the
&(x)dx=
P[x,
I
(x)~\dx,
t/a
formula
P(x, y}dx
I JAB
=
P[x, t(x)-]dx.
I Ja
If a line parallel to the y axis can we should divide the arc
meet the arc
AB
in
more than
one point,
into several portions, each of
met
is
in but
which
one point by any line
parallel to the y axis. arc is of the form A
If the given
CDB
(Fig. 14),
where C and D are which the abscissa has an
for instance,
points at
extremum, each of the arcs A C, CD, DB satisfies the above condition, and
we may I
JACDB
But
it
FIG. 14
write
P(x,
I P(x, y)dx= J,T
y)dx
+
I
JCD
P(x, y)dx
+
f JOB
P(x,y)dx.
should be noticed that in the calculation of the three integrals
DEFINITE INTEGRALS
186
[IV,
on the right-hand side the variable y in the function P(x, must be replaced by three different functions of the variable
93
y) x,
respectively.
Curvilinear integrals of the form JAR Q(x, y)dy may be denned manner. It is clear that these integrals reduce at once
in a similar
to ordinary definite integrals, but their usefulness justifies their introduction. may also remark that the arc AB may be com
We
posed of portions of different curves, such as straight lines, arcs of circles, and so on.
A
case which occurs frequently in practice is that in which the coordinates of a point of the curve AB are given as functions of a variable parameter
and
where
together with their derivatives () and t. We shall suppose that as t varies <
\(/(t),
<j>(t)
are continuous functions of
describes the arc AB without changing ft the point (x, y) Let the interval (a, /?) be divided into a the sense of its motion.
from a to certain
and let t _ and t be two consecu which correspond, upon the arc AB, two points whose coordinates are (#,_!, y _i) and (x,-, y ), respec
number
tive values of m,...!
and
lies
0,.
a point
or,
{
i
l
f
to
t
f
Then we have
tively.
where
m
of subintervals,
t
(,-,
between
17,)
ti _
1
of the arc
and
m
i
_l
t{
To
.
m
i
;
this value
hence we
0,
may
there corresponds
write
passing to the limit,
f
P(x,
/.4
An analogous
formula for
Adding the two,
f
(29)
we
JQdy may
P<&
-f
Qdy
J^l
which course,
is if
be obtained in a similar manner.
find the formula
= f /
Of the formula for change of variable in line integrals. the arc AB is composed of several portions of different
and \fr() will not have the same form curves, the functions along the whole of AB, and the formula should be applied in that (t)
case to each portion separately.
IV,
IMPROPER AND LINE INTEGRALS
t)4]
187
We
of a closed curve. have already defined the area of a of the bounded an arc A MB, a straight line which plane portion by does not cut that arc, and the two perpendiculars AA Q BB let fall
Area
94.
,
from the points A and B upon the straight line ( 65, 78, Fig. 9). Let us now consider a continuous closed curve of any shape, by which we shall understand the locus described by a point whose
M
y =
coordinates are continuous functions x
of a
=f(), param which assume the same values for two values t and T of the parameter t. The functions f(f) and may have several distinct forms between the limits t and T; such will be the case, eter
(t)
t
<j>(t~)
for instance, if the closed contour
Let
several distinct curves.
M M
denote points upon the curve values t t l} t 2 t _ l t -, ,
,
i
,
lt
,
C
t ,
C
be composed of portions of
J/2
-,
,
M_uM
if
{
,
M_ M n
lt
corresponding, respectively, to the t n _ j, T of the parameter, which
,
increase from
t
zero, is called
the area of the closed curve C.*
Connecting these points in order by straight The limit lines, we obtain a polygon inscribed in the curve. approached by the area of this polygon, as the number of sides is indefinitely increased in such a way that each of them approaches to T.
This definition
is
seen to agree with that given in the particular case treated above. For if the polygon A A(2 1 Q 2 BB A (Fig. 9) be broken up into
small trapezoids by lines parallel to is
trapezoids
where
lies
,.
in
polygon,
(*,.
- *,._,)
[/(a-,-)
between x^-^ and this
special
case,
AA
+ f(x _ {
cc
f
.
x
,
the area of one of these
)]/2, or
(a\ -*<_,)/&),
Hence the area
of the whole
approaches the definite
ff(x)dx. Let us now consider a closed curve C which
is
integral
cut in at most two
line parallel to a certain fixed direction. Let us choose as the axis of y a line parallel to this direction, and as the
by any
points
axis of
curve
C
a;
a line perpendicular to it, in such a the quadrant xOy (Fig. 15).
way that the
entire
lies in
The points
of the contour C project into a segment ab of the axis and any line parallel to the axis of y meets the contour C in at most two points, m^ and m z Let y v = ^(cc) and ?/2 = 2 (x) be the equations of the two arcs Am v B and Am z B, respectively, and let
Ox,
.
tl/
us suppose for simplicity that the points A and B of the curve C b are taken as two of the vertices of the
which project into a and
* It is supposed, of course, that the curve under consideration has no double point, and that the sides of the polygon have been chosen so small that the polygon itself
has no double point.
DEFINITE INTEGRALS
188
94
[IV,
The area of the inscribed polygon is equal to the differ polygon. ence between the areas of the two polygons formed by the lines A a, in the two arcs Am 2 B and ab, bB with the broken lines inscribed
AmiB,
Passing
respectively.
of the curve
C
to the limit, it is clear that the area
the difference between the two areas
equal to
is
bounded by the contours Am^BbaA and Am^BbaA, respectively, that is, to the difference between the corresponding definite in tegrals /tb
X6
\l/ z
(x)dx
^
I
*J a
These two integrals represent the curvilinear integral
taken then
first
along
along
fydx
Am B
and
If
we
2
Am-^B.
agree to say that the contour FIG. 15
C
is
described in the positive
observer standing upon the plane and walking around the curve in that sense has the enclosed area constantly on his left
sense
when an
hand
(the axes being taken as usual, as in the figure), then the above may be expressed as follows the area O enclosed by the
result
:
contour
C
is
given by the formula
r (30) J(C)
where the
line integral is to be taken along the closed contour
C
in
Since this integral is unaltered when the origin is moved in any way, the axes remaining parallel to their original positions, this same formula holds whatever be
the positive sense.
the position of the contour the coordinate axes.
C with
Let us now consider a contour whatever. to
draw a
We
C
shall suppose that
finite
number
pairs of points on
C
of
respect to
of any it is
form
possible
lines
connecting in such a way that the
resulting subcontours are each met in at most two points by any line parallel to the y axis.
Such
FIG. 16
the case for the region bounded by the contour C in Fig. 16, which we may divide into three subregions bounded by the contours amba, abndcqa, cdpc, by means of the is
IV,
IMPROPER AND LINE INTEGRALS
95]
189
Applying the preceding formula to each and of these subregions adding the results thus obtained, the line from the auxiliary lines ab and cd cancel each arise which integrals transversals ab and cd.
other,
and the area bounded by the closed curve C
the line integral
is still
C
fydx taken along the contour
given by
in the positive
sense. it
Similarly,
may be shown
that this
same area
is
given by the
formula
n=
(31)
and
x dy\
I
J(C
combining these two formulae, we have
finally,
=- f 2
(32)
xdy
ydx,
J
where the integrals are always taken in the positive sense. formula is evidently independent of the choice of axes. in the form If, for instance, an ellipse be given
This
last
x its
area
=
a cos
=
y
t,
b sin t,
is
fi
=
C
1
I
2 Jo
2
"
ab(cos
2
t -{-
sm 2 f)dt =
Trab.
95. Area of a curve in polar coordinates. Let us try to find the area enclosed by the contour OAMBO (Fig. 17), which is composed of the two straight lines OA, OB, and the arc A MB, which is
met
in at
most one
point by any radius Let us take vector.
,,-/
as the pole and a straight line Ox as
/
the initial line, and be the let p =
I
\
/(o>)
equation
-^37
of the arc FIG. 17
A MB.
Inscribing a polygon in the arc A MB, with A and the vertices, the area to be evaluated is the limit of the triangles as
OMM 1 -
.
But the area of the triangle
p(p
+
Ap) sin
Aw
= Aw
P I
OMM
B
as
two of
sum
of such
is
DEFINITE INTEGRALS
190
[IV,
95
It is easy to show that all the are less than any preassigried number rj provided that the angles Aw are taken sufficiently small, and that we may therefore neglect the term cAw in evaluating the limit.
approaches zero with Aw.
where
quantities analogous to
c
Hence the area sought
is
is
the limit of the
sum 2p 2 Aw/2,
that
is, it
equal to the definite integral
where w t and w 2 are the angles which the straight lines OA and OB make with the line Ox. An area bounded by a contour of any form is the algebraic sum If of a certain number of areas bounded by curves like the above. we wish to find the area of a closed contour surrounding the point 0, which is cut in at most two points by any line through 0, for to 2?r. The area of a con example, we need only let w vary from vex closed contour not surrounding O (Fig. 17)
is
equal to the dif
0AM BO
and OANBO, each of which may be calculated by the preceding method. In any case the area is represented by the line integral ference of the two sectors
C in the positive sense. This formula does For if we pass from not differ essentially from the previous one. we have to coordinates rectangular polar taken over the curve
x
dx
= cos w dp
=
p cos w,
p sin
y
w c?w,
x dy
dy
y dx
p
=
=p
2
&m w sin
>
w dp
+ p cos w c?w,
dta.
AMB
whose equation in oblique Finally, let us consider an arc coordinates is y =f(x~). In order to find the area bounded by this BB which are parallel arc AMB, the x axis, and the two lines AA ,
,
imagine a polygon inscribed in the arc AMB, and let us break up the area of this polygon into small trapezoids by lines parallel to the y axis. The area of one of these trapezoids is to the
y
axis, let us
IV,
IMPROPER AND LINE INTEGRALS
96]
lies in
~
form
in the
which may be written
(z,--i
#,)/()
Hence the area
the interval (x _ l; x^. t
191 sin
0,
where
|
in question is equal
to the definite integral
sin $
and
where x
of the points It
may
f(ix) dx,
A"
denote the abscissae
A and
B, respectively.
be shown as in the similar
case above that the area
any closed contour C whatever by the formula
is
given
x dy
B
A
bounded by
FIG. 18
y dx.
(O
Given a closed curve C (Fig. 15), let us draw at any point the portion of the normal which extends toward the exterior, and let a, ft be the angles which this direction makes with the axes Note.
M
of x
and
y,
the angle
ft
respectively, counted is
obtuse and dx
y dx
I
from
=
ds cos
=
\
J
\J (Am^B)
Along
Bm A z
the angle
in the line integral.
we
shall still
curve
may
have dx
If
ft is
ds cos
Along the arc Am^B Hence we may write
TT.
ft.
y cos
acute, but
we agree
=
to
dx
ft
is
ds.
negative along
to consider ds
Hence the area
ft.
Bm A 2
always as positive, of the closed
be represented by the integral
y cos
ft
ds,
where the angle ft is defined as above, and where ds is essentially This formula is applicable, as in the previous case, to a positive. contour of any form whatever, and it is also obvious that the same area is given by the formula x cos a
ds.
These statements are absolutely independent of the choice of axes. 96. Value of the integral
/xdy ydx. It is natural to inquire what will fxdy ydx, taken over any curve whatever,
be represented by the integral closed or unclosed.
DEFINITE INTEGRALS
192
[TV,
97
example, the two closed curves OAOBO and which have one and three double points, respec It is clear that we may replace either of these curves by a combination tively. Thus the closed contour OA OBO of two closed curves without double points. is equivalent to a combination of the two contours 0-40 and OBO. The integral taken over the whole contour
Let
us
for
consider,
ApBqCrAsBtCuA
(Fig. 19)
is
equal to the area of the portion less the area of the portion Likewise, the other contour
0.40 OBO.
may
be replaced by the two closed
curves tour
is
equal to the
sum
ApBqCrA and AsBtCuA, and
the integral taken over the whole con of the areas of ApBsA, BtCqB, and ArCuA, plus twice
the area of the portion AsBqCuA. This reasoning is, moreover, general. Any closed contour with any number of double points determines a certain number of partial areas 2 p of each of which it forms all the boundaries.
<TI,
The
,
integral taken over the
m
,
,
whole contour
is
equal to a
sum
of the
form
m
2 p are positive or negative integers which may be found by Given two adjacent areas the following rule separated by an arc ab of the contour C, imagine an observer walking on the plane along the contour in the sense
where mi,
,
,
:
,
determined by the arrows ; then the coefficient of the area at his left is one greater than that of the area at his right. Giving the area outside the contour the coeffi cient zero, the coefficients of all the other portions may be determined successively. If the
joining
AB
is not closed, we may transform it into a closed curve by given arc extremities to the origin, and the preceding formula is applicable to ydx taken over the radii vectores OA region, for the integral fxdy
its
this
new
and
OB V.
evidently vanishes.
FUNCTIONS DEFINED BY DEFINITE INTEGRALS
We
97. Differentiation under the integral sign. frequently have to deal with integrals in which the function tcr43e integrated depends not only upon the variable of integration but also upon one or more
we consider as parameters. Let f(x, a) be a continuous function of the two variables x and a when x varies from other variables which
x
to
X
and a varies between certain limits and a^ We proceed a which is defined by the
to study the function of the variable definite integral >,r
=
Cf(x,a)dx, Jxn
where a is supposed to have a definite value between a and a lf and where the limits x and X are independent of a.
IV,
FUNCTIONS DEFINED BY INTEGRALS
97]
We
have then
(33)
F(a
+
Aa)
- F(a) =
f
+ Aa) -f(x,
[/(*, a
193
a)] dx.
JjT
is continuous, this integrand may be made than any preassigned number c by taking Aa sufficiently small. x in absolute Hence the increment AF(a) will be less than e\X value, which shows that the function F(a) is continuous. If the function f(x, a) has a derivative with respect to a, let us
Since the function f(x, a)
less
write f(x,
where Aa, we
and
if
+
a
Aa)
- f(x,
a)
approaches zero with Aa.
e
= Aa [/. (x,
)
+
e]
,
Dividing both sides of (33) by
find
-q
be the upper limit of the absolute values of
value of the last integral will be less than ri\X the limit, we obtain the formula
x
c,
the absolute
\.
Passing to
^
(34)
da
In order to render the above reasoning perfectly rigorous we must it is possible to choose Aa so small that the quantity c will be less than any preassigned number rj for all values of x between
show that
the given limits x and X. This condition will certainly be satisfied For we have from if the derivative a (x, a) itself is continuous. the law of the mean
f
f(x,
a
+
Aa) -f(x, a) = Aa/, (x, a
+
0Aa),
<
$
<
1,
and hence
If the function for
fa
is
continuous, this difference e will be less than 77 a, provided that Aa is less than a properly
any values of x and
|
j
chosen positive number h (see Chapter VI, 120). Let us now suppose that the limits X and x are themselves func ti
denote the increments which correspond to an increment Aa, we shall have tions of a.
If A.Y
and
Aa:
DEFINITE INTEGRALS
194
F(a
+ Aa)- F(a) = f Jx
+
\_f(x,
+
a
Jx rx
o
+
/(a,
a + Aa) dx
f(x,
a+
first law of the mean two integrals and dividing by Aa,
applying the
F(a
4-
A F() _ C /(x, a
Aa)
Aa
-f(x, a)
^
Jx
last
for integrals to each of the
4- Aa-)
f(x, a)
,
Aa
J,
AT
-^/(^o + ^A^, As
97
f*X+&X /
I
or,
A,r)
[IV,
nr
+
Aa).
Aa- approaches zero the first of these integrals approaches the we find the formula
limit found above, and passing to the limit
^=
(35)
which
is
the general formula for differentiation under the integral
sign.
be reduced to a sum of ordinary evident that the preceding formula may be extended to line integrals. Let us consider, for instance, the line Since a line integral
definite integrals,
may always
it is
integral
F(a) taken over a curve
we
shall
=
f JAB I
AB
P(x,
which
y,
is
a} dx
+
Q(x, y, a) dy
independent of
a.
It is
evident that
have
F\a) where the integral
=
I
JAB is to
P a (x,
y,
a)dx
+
Q a (x,
y,
be extended over the same curve.
On
the
other hand, the reasoning presupposes that the limits are finite and that the function to be integrated does not become infinite between
We shall take up later (Chapter VIII, the limits of integration. in the which these conditions are not satisfied. cases 175)
IV,
FUNCTIONS DEFINED BY INTEGRALS
98]
195
to evaluate certain definite (35) is frequently used are more easily calcu which others them to reducing integrals by
The formula
lated.
Thus,
if
a
we have
is positive,
*
1
/
=
va
Jo
arc tan
whence, applying the formula (34) n
<-i
all
p. vo
1 times,
we
find
r -.i:t...<.-i>r Jo 7S (^
98. Examples of discontinuity. If the conditions imposed are not satisfied for values between the limits of integration, it may happen that the definite inte
Let us consider, for
gral defines a discontinuous function of the parameter. example, the definite integral
f J_ This integral always has a
imaginary except when
-
Hence the
making
l-2xcosa +
l
x2
the roots of the denominator are
which case
it is
evident that F(a) = 0. Sup = cos a + t sin a, the indefi
the substitution x
a dx - = f dt = a + x 2 J 1 + t2 I
.
arc tan
t.
2x cos
definite integral
F(a) has the value cos
a\
a
I
(1 sin
where the angles are
*<***
kit, in
/sin 1
1
finite value, for
a=
and posing that sin a ^ nite integral becomes
+
to be taken
cos
1
sin
a
I
arc tan
]
-
sin
\
a\
a
n/2 and x/2.
between
a x
cos
1
(
cos
1
sin
a
)
/
But
-a = -1,
In order to determine the difference of these angles is n/2. notice that the sign of the integral is the same as Hence F(a) = n/2 according as sin a is positive or negative. It follows that the function F(a) is discontinuous for all values of a of the form
and hence the sign uniquely that of sin a.
kit.
we need only
This result does not contradict the above reasoning in the
least,
however.
e to + e, for example, 1 to + 1 and a varies from For when x varies from the function under the integral sign assumes an indeterminate form for the sets of values a = 0, x = - 1 and a = 0, x = + 1 which belong to the region in ques
any value of e. would be easy to give numerous examples
tion for It
the integral
n
-4-
a:
sin
Ux f
mx
of this nature.
,
dx.
Again, consider
DEFINITE INTEGRALS
196 Making the
mx =
substitution
to be
99
find
mx
sin
X where the sign
we
y,
[IV,
y
taken
the sign of m, since the limits of the transformed is positive, but should integral are the same as those of the given integral if be interchanged if is negative. have seen that the integral in the second is
m
m
We
N
member is a positive number ( 91). Hence the given integral is equal to is positive or negative. If m = 0, the value of the according as integral zero. It is evident that the integral is discontinuous for = 0.
m
N is
m
VI.
APPROXIMATE EVALUATION OF DEFINITE INTEGRALS When
99. Introduction.
resort to certain
no primitive of f(x)
is
known we may
methods
for finding an approximate value of the definite integral The theorem of the mean for integrals f*f(x) dx. furnishes two limits between which the value of the integral must
and by a similar process we may obtain an infinite number of Let us suppose that $(x)
others.
<
<
s*b
s*t>
I <j>(x}dx< <J
a
If the functions <(#)
and
Ja
Let us consider, for example, the integral
lie.
C
dx
Vl-a; 4
Now Vl-z = Vl - x Vl + x\ between 1 and V2 for all values 4
2
Jo is,
and the factor
between Tr/2
}
r dx Vl-cc V2Jo Vl x and 7r/(2V2). Two even closer i
2
(1
a
+
z 2 )-
which results from the expansion of
1
/2
is
(1 -f 7/)-
greater 1/2
with a remainder carried to two terms. greater than the expression
dx
lies
two integrals
series is
z2
and unity.
l
dx
be found by noticing that
I
Vl +
of x between zero
integral lies between the
r that
c/a
the derivatives of two known two limits between which the value of
"Jo
Hence the given
I
^(a;) are
functions, this formula gives
the integral must
/-i
I f(x)dx<
1
r
}
x 2 dx
limits
than
1
may x 2/2,
by means of Taylor
Hence the
s
integral
APPROXIMATE EVALUATION
y]
IV,
The second lies
of these integrals has the value Tr/4
197 (
105)
hence /
;
between Tr/2 and 3 7r/8.
It is evident that the preceding methods merely lead to a rough In order to obtain closer idea of the exact value of the integral.
approximations we may break up the interval (a, 6) into smaller subintervals, to each of which the theorem of the mean for inte grals
may be
applied.
For definiteness
let
us suppose that the
Let function /(a:) constantly increases as x increases from a to b. a us divide the interval (a, b~) into n equal parts (b nh). Then, lies between the by the very definition of an integral,
^f(x}dx
two sums
= h\f(a) S = h\f(a + s
h)
+f(a +
+/(a +
-f
2A)
nh)\.
If we take (S + s)/2 as an approximate value of the integral, the The error cannot exceed S-s\/2= [(i - ) /2 n] [/(i) -/(a)] form in the written be value of (S + s)/2 may .
(/(a)
+ /O
|
,
2
/[a
+
(ro
- 1) A] +f(a + nh) 2
+
Observing that \f(a ih) the trapezoid whose height
+/[a +
(i
)
)
+1) h~\\h/2
is
the area of
h and whose bases are /(a + ih) and the whole method amounts to that we ih + may say f(a + 7t), curve under the the area y = f(x) between two neighbor replacing is
ing ordinates by the area of the trapezoid whose bases are the two This method is quite practical when a high degree of
ordinates.
approximation is not necessary. Let us consider, for example, the integral
dx
/ C/O
Taking n
= 4, we
and the error mate value of *
Found from
value being
rr/4.
find as the
is less TT
than 1/16
which
is
the formula \S
TRANS.
approximate value of the integral
=
.0625.*
This gives an approxi 3.1311
correct to one decimal place,
s\/2.
In fact, the error
is
about
.00260, the exact
DEFINITE INTEGRALS
198
[IV,
100
If the function f(x) does not increase (or decrease) constantly as x increases from a to b, we may break up the interval into sub-
intervals for each of
which that condition
is satisfied.
Another method of obtaining an approximate is the following. Let us determine
100. Interpolation.
value of the integral f f(x)dx a parabolic curve of order n,
y
=
=
+
a
<(z)
a-iX
+
1-
a n xn
,
which passes through (n -f- 1) points B B l} Bn of the curve y =f(x) between the two points whose abscissae are a and b. These points having been chosen in any manner, an approximate ,
value of the given integral
which is easily calculated. Let (x T/O), (#!, 7/i),
,
,
points
BQ
B!,
,
=
Z/o
<(*)
where the
(x n
f <j>(x}dx,
be the coordinates of the (n
?/)
,
The polynomial
.
b
furnished by the integral
is
A
-f
coefficient of
y
x _ (^ - XEQ) (Xo)
ijt
A*!
is
t
h
-\
i/i
Xf
\-y n
-\
Xn
,
a polynomial of degree n,
(x-x
i
_
~ Xi(*i
l
}(x- x
l)
(X
i
+
}
(x
}
-*<+!)
- ay) - Xn)
^
(*<
=x
which vanishes for the given values x x, x n except for x x and which is equal to unity when x Hence we have ,
,
{
U / a
The numbers x x
=
a
-f
:
Q
a),
a?!
=
a
-f-
$i(&
)
,
#
=
If
a
+
#(&
<
<
<
(b
(36)
where
K
is
KI
=
i
we
f,
are of the form
i
(b
,
.
6t 6H 5 1. Setting x = a + (b proximate value of the given integral takes the form
where
+1)
determined by
is
interpolation formula in the form
s
Lagrange
Bn
,
,
<
a)
(A"
y
+ K\y\
H
h
a)
t,
)
the ap
A .y,,),
given by the formula dt
"^^
77
I
^
.
divide the main interval
ratios are the
the numbers
(a, b) into subintervals whose same constants for any given function /(x) whatever, dn and hence also the numbers K are inde U ,
pendent of f(x).
,
,
Having calculated these
{ ,
coefficients once for all,
IV,
it
APPROXIMATE EVALUATION
101]
only remains to replace y
in the
yi,
yn by their respective values
,
formula
(36). If the curve f(x)
it is
whose area
is
to be evaluated is given graph (a, 6) into equal parts,
convenient to divide the interval
ically, it is
and
,
199
only necessary to measure certain equidistant ordinates of = 0, Thus, dividing it into halves, we should take
this curve.
= 1, which gives the following formula 61 = 1/2, 2 mate value of the integral
for the approxi
:
Likewise, for n 1
a
b
I =
Q-
=3
we
=
and for n
=
4yi
?
+
y2 )
formula
find the
C /0
o
+
u/o
+ 3#! +
3// 2
+
7/s)
,
4
~
=
7
(7z/o
~9o
Ir
The preceding method due
to
is
due
to Cotes.
The following method,
Let the interval
slightly different.
is
Simpson,
(a,
b~)
be
divided into 2n equal parts, and let y y 1} ?/ 2 y2n be the ordi nates of the corresponding points of division. Applying Cotes formula to the area which lies between two ordinates whose indices ,
,
,
are consecutive even numbers, such as y and y 2
whence, upon simplification, 1
101.
= -- [2/0 +
we
find
Simpson
+ 2(y + 2
2/2
s
and y t form
?/ 2
,
find an approximate value of the given area, in the
formula
---- -f y, H
?/2n
,
etc.,
we
:
_ 2)
In Gauss method other values are assigned The argument is as follows: Suppose that we
Gauss method.
the quantities
#,.
can find polynomials of increasing degree which differ less and less
from the given integrand for instance, that
/(*)
=
a
we can
+
a,x
where the remainder
+
in the interval (a, &).
f(x~)
Suppose,
write
a 2 x*
R 2H (x~)
is
+ less
+
a^ X
2n
~l
+ 7?
2n
(x)
than a fixed number
,
^
for all
DEFINITE INTEGRALS
200
b* The unknown, but they do not occur
values of x between a and
coefficients a, will be in
eral
in the calculation, as
Let x
see.
x 1}
,
x n _ be values of x between a and
-,
101
[IV,
we
and
b,
1
gen shall let
1 which assumes the same be a polynomial of degree n Then Lagrange s inter values as does /(ce) for these values of x. <(ce)
polation formula shows that this polynomial
may
be written in the
form
where
and
m
are at
ty k
most polynomials of degree n
I.
It is
clear that the polynomial m (x) depends only upon the choice of x n-i- O n th e other hand, this polynomial <,(#) must x x
o>
i>
">
x = xa _ assume the same values as does x m for x = x x = x l} For, supposing that all the a s except a m and also R^(x) vanish, m reduces to a m $ in (x). Hence the f(x) reduces to a m x and m difference x m (x) must be divisible by the product ,
,
l
.
(a:)
Pn (x) = It follows that a
1
"
(x
The
made
error
m
m (x)=
n, if ra
in
- a^).
(x
a-i)
Pn Q m _ n (x~), where Q m _ n (x) is a poly n and that x m if m n 1. m (x) = dx dx is by Ja (x~) replacing f f(x~) evidently
nomial of degree
- XQ) (x -
<
>
;
<J>
given by the formula
^Mr *" j"
_
VH
7?
4 i=0
2,i
( (
The terms which depend upon
xY i)\
1 1
^o
x *i (x \
\b
the coefficients
a
,
a1
,
-,
a n _ vanish l
identically, and hence the error depends only, upon the coefficients an + a -2n-\ an(l the remainder R 2n (x ). But this remain der is very small, in general, with respect to the coefficients a n o-n + a.2n _ Hence the chances are good for obtaining a <*>
">
i>
1 ,
,
,
l
.
high degree of approximation if we can dispose of the quantities x x lt x n _ l in such a way that the terms which depend upon , ,
a >
an +
i>
>
a 2n-i a ^ so vanish
necessary and s*b /
Ja
identically.
sufficient that the
I
Jo.
this
purpose
it is
nb
f*\>
P n Q u dx,
For
n integrals
P^.dx,
-,
I
fc/a
* This is a property of any function which is continuous in the interval according to a theorem due to Weierstrass (see Chapter IX, 199).
(a,
IV,
APPROXIMATE EVALUATION
102]
should vanish, where Q already seen the form
{
a polynomial of degree
is
88) that this condition
(
[<*-)
**.-;
We may
Pn =
0,
and these
assume that a
=
satisfied if
is
i.
we
We take
have
Pn
of
<*-*)]-
take for x
It is therefore sufficient to
the equation
201
,
x i}
,
x n _ the n roots of and b. l
roots all lie between a
and
1
+
b
1,
since all other cases
be reduced to this by the substitution x = (b + ) /2 + 2 (b a) /2. In the special case the values of x x lf _! are the roots of -,
may
,
The values of these roots and the polynomial Xn values of K for the formula (36), up to n = 5, are to be found to seven and eight places of decimals in Bertrand s Traite de Calcul Legendre
s
.
i
integral (p. 342). Thus the error in Gauss
method
C R*(x)dx
is
-"j^R^x,)
Jo,
t
=
f
*,
*^ a
where the functions ^ (x) are independent of the given integrand. In order to obtain a limit of error it is sufficient to find a limit of R^(x), that is, to know the degree of approximation with which {
the function f(x) can be represented as a polynomial of degree 2n 1 in the interval (a, &). But it is not necessary to know this
polynomial
itself.
Another process for obtaining an approximate numerical value of a given definite integral is to develop the function f(x) in series and integrate the series term by term. We shall see later (Chapter VIII) under what conditions this process is justifiable and the degree of approximation which it gives. 102. Amsler s planimeter. A great many machines have been invented to measure mechanically the area bounded by a closed plane curve.* One of the most ingenious of these is Amsler s planimeter, whose theory affords an interest
ing application of line integrals. Let us consider the areas AI and A* bounded by the curves described by two points AI and A 2 of a rigid straight line which moves in a plane in any manner arid finally returns to its original position.
A,
dinates of the points AI and lar axes. Let I be the distance
*
A
Let
(xi, 2/1)
and
(x 2 ,
y 2 ) be the coor
respectively, with respect to a set of rectangu
AiA 2
,
and
the angle which
A\A Z
makes with
description of these instruments is to be found in a work by Abdankla courbe integrate et ses applications (Gauthier-
Abakanowicx: Les integraphes, Villars, 1886).
DEFINITE INTEGRALS
2U2
In order to define the motion of the line analytically,
the positive x axis.
102
[IV,
i, j/i,
and 6 must be supposed to be periodic functions of a certain variable parameter t which resume the same values when t is increased by T. We have x 2 = %i + I cos 6, = V\ + I si n an(l hence 2/2
+ PdO
yidx\
+
sintfdxi
I(cos0dyi
+
areas AI and A 2 of the curves described by the points A\ and general conventions made above ( 96), have the following values
Az
The
AI
=
r
l
-
-
xidyi
J
A2
yidxi,
=
1 - r x 2 dy 2
we
Hence, integrating each side of the equation just found, A2
=
AI -f
- CdO + -
fcos6dyi
j
-
y z dx 2
J
under the
,
:
.
obtain the equation
- s m0dx + f (xicosfl + l
yis\n0)d0\
,
of the integrals correspond to the values t and t Q + T is an integer which evident that fd8 = 2Kn, where On the other hand, in which the straight line moves.
where the limits of each of the variable
t.
the
It is
way
K
depends upon integration by parts leads to the formulae /
Xi cos 6 d0
f yi But
Xi sin
=
01
y\ cos
+
/
sin
dx\
cos 9 dyi
have the same values for t be written in the form
and y\ cos
tQ
,
.
and
t
=
to
+
T.
Heuce
may
the preceding equation
A2
d0
sin
Xi sin
=
AI
+ Knl2 +
CcosOdyi
I
si
Now let s be the length of the arc described by A i counted positive In a certain sense from any fixed point as origin, and let a be the angle which the positive Then we shall have direction of the tangent makes with the positive x axis. cos
where
sin 6 dxi
dyi
V is the angle
positive direction
=
(sin
a
cos
sin
which the positive direction
A^A Z
cos a) ds
=
sin
V da
,
makes with the The
of the tangent
of the straight line taken as in Trigonometry.
preceding equation, therefore, takes the form A2
(38)
=
AI
+ Kxt2 + ifsinVds.
Similarly, the area of the curve described by line is given by the formula
A3
(39)
=
A!
+
Kl
*
any third point
A
3
of the straight
+ fsiuVds, I
where I is the distance AiA & Eliminating the unknown quantity fsinVds between these two equations, we find the formula .
1
Aj
-
IA
=
(*
-
1)
AI
+ Kxll\l - Hi
IV,
APPROXIMATE EVALUATION
102]
which may be written A! (23)
(40)
where
(ik)
in the
+
203
form
Ao (31 )
+
+ K* (12) (23) (31) =
A 3 (12)
denotes the distance between the points Ai and
taken with
its
A\A
Ak
(i,
k
=
+
of length (a
application of this formula, let us consider 2 describe the 6), whose extremities A\ and
A
same closed convex curve C. The point A 3 which divides the line ments of length a and 6, describes a closed curve which lies wholly
into seg
,
C"
=
inside C.
we have
In this case A2
1, 2, 8)
As an
proper sign.
a straight line
,
Ai,
=
(12)
whence, dividing by a
+
a
+
=-
(23)
b,
6,
=
A3
AI
(31)
6,
=-
a,
K=
I
rtab.
But AI Hence this area is AS is the area between the two curves C and C independent of the form of the curve C. This theorem is due to Holditch. If, Instead of eliminating JsinFds between the equations (38) and (39), we eliminate AI, we find the formula .
A3
(41)
Amsler
s
=
+ Kx(V* -
A2
I
2 )
+
planimeter affords an application
A
(I
-
l)CsinVds.
of this formula.
Let
AiA 2 A s
be a
The point being fixed, the point 2 with another rod (L4 2 rigid rod joined at 3 to which is attached a sharp pointer, is made to describe the curve whose area
A
.
,
The point A% then sought. describes an arc of a circle or is
an entire circumference, accord ing to the nature of the motion. In any case the quantities A 2 ,
I,
I
are
all
-K",
known, and the area
AS can be calculated
if
which
the in
to be taken over the curve C\ described by the point A\, can be evaluated. This end A\ carries a graduated tegral Jsin Vds,
is
f\ **
V
FIG. 20
,
circular cylinder whose axis coin cides with the axis of the rod
AiA 3 and which can turn about this axis. Let us consider a small displacement of the rod which carries AiA 2 A 3 into the position AiA ^Az. Let Q be the intersection of these straight lines. About Q
along
its
draw the
Ai a and drop the perpendicular A{P from imagine the motion of the rod to consist of a sliding own direction until AI comes to or, followed by a rotation about Q which
as center
AI upon
,
AiA 2
.
circular arc
We may
In the first part of this process the cylinder would slide, with brings a to A{. out turning, along one of its generators. In the second part the rotation of the cylinder is measured by the arc aA{. The two ratios aA\/A{P and
A{P/axcAiA{ approach zero.
Hence ccA{
=
1
and sinF, respectively, as the arc A[A\ approaches + e), where e approaches zero with As. It follows
As (sinF
that the total rotation of the cylinder SAs(sinF + e), that is, to the integral this rotation is sufficient for the
is
proportional to the limit of the sum Hence the measurement of
JsinFds.
determination of the given area.
DEFINITE INTEGRALS
204
[IV, Exs.
EXERCISES 1. Show that the sum 1/n + l/(n + !) + + l/2n approaches log 2 as n increases indefinitely. l [Show that this sum approaches the definite integral fQ [1/(1 + x)]dx as its
limit.] 2.
As
in the
preceding exercise, find the limits of each of the sums
n
n2 +
-5-
+
n2 +
1
1
..-
+
+
n2
(n
-
I)
2
1
1
Vn2 - 1
Vn2 -
by connecting them with certain the
+
22
Vn* -
22
-
(n
I)
2
In general, the limit of
definite integrals.
sum
as n becomes infinite, is equal to a certain definite integral whenever 1 in i and n. a homogeneous function of degree
Show
3.
"
that
the
value
of
the
definite
integral
f
/2
(i,
n)
log sin x dx
-(jr/2)log2. [This
may
be proved by starting with the .
sm
it
2*
.
or else
(n
.
sm
sin
trigonometric formula
n
\)it ^
-
,
n
n
n
known
2"-
by use of the following almost self-evident
equalities
:
JT
/I,
log sin x dx
I
=
By
,
log cos x dx
I
=
O
/o
4.
fy.
C
1
2
2
/sin2x\
.
log /
I
(
\
u
2
_.
dx. ]
/
the aid of the preceding example evaluate the definite integral
1
tan x dx
.
2/ 5.
Show
that the value of the definite integral ,1
/ Jo
is
1
+
x2
(jr/8)log2.
[Set x 6*.
=
tan
and break up the transformed
integral into three parts.]
Evaluate the definite integral /ITT I Jo
log (1
2a
cos x
+ a2
)
dx
.
[POISSOK.]
is
is
EXERCISES
IV, Exs.]
205
[Dividing the interval from to it into n equal parts and applying a well-known formula of trigonometry, we are led to seek the limit of the expression
ra
?r
-log n La
1
2n_!
-
n becomes
as
a2
>
1, it is 7t
Show
1.
If
infinite.
log
a2
a
1
between
lies
Compare
.
+
+
and
1
1,
this limit is zero.
If
140.]
that the value of the definite integral
sinxdx
/ Vl
2or cos
J/o o where a
is
positive, is 2
Show
15*.
if
a
<
1,
and
is
2/a
x
if
+ a2 a
>
1.
that a necessary and sufficient condition that /(x) should be inte-
grable in an interval (a, b) is that, corresponding to any preassigned a subdivision of the interval can be found such that the difference S
corresponding sums Let/(x) and
S and
than
s is less
number
e,
the
s of
e.
be two functions which are continuous in the interval (a, 6), be a method of subdivision of that interval. If ,, 77. are any two values of x in the interval (x,-_i, x,-), the sum 2/(,-) (?;,) (x, x,-_i) dx as its limit. approaches the definite integral f^f(x) 9.
and
<j>(x)
let (a, Xi, Xj,
,
b)
(x)
10. Let/(x) be a function which is continuous and positive in the interval (a, that the product of the two definite integrals
b).
Show
/>* is
a
minimum when
Xi.
where c
=
is
a constant.
Let the symbol I* denote the index of a function Show that the following formula holds: 1
11.
and
the function
f/l>
e
=+
1
if
and
if
/(x
/(x ) [Apply the last
tions /(x)
and
)
>
and
/(Xi)
<
0, e
=-1
if
f(x
/(Xi) have the same sign. formula in the second paragraph of
)
<
(
77)
and
between
f(xi)
>
0,
and
77 to each of the func
l//(x).]
12*. Let U and V be two polynomials of degree n and n 1, respectively, which are prime to each other. Show that the index of the rational fraction V/U between the limits oo and -f oo is equal to the difference between the number of imaginary roots of the equation U + iV = in which the coefficient of i is positive and the number in which the coefficient of i is negative.
[HERMITE, Bulletin de
la Socidte
13*. Derive the second theorem of the parts.
matMmatique, Vol. VII,
mean
for integrals
p. 128.]
by integration by
DEFINITE INTEGRALS
206 [Let/(x) and b) and the
be two functions each of which
(x)
[IV, Exs.
is
continuous in the inter
which, /(x), constantly increases (or decreases) and has a continuous derivative. Introducing the auxiliary function
val (a,
first of
*(z)
and integrating by
parts,
f /(x)
va Since /
we
= Ja
f*(x)dx
find the equation
= f(b) *(&) - f / va
dx (x)
always has the same sign,
(x)
of the
mean
14.
Show
for integrals to the
new
it
(x)
*(x) dx
.
only remains to apply the
first
theorem
integral.]
directly that the definite integral
ydx extended over a
fxdy
closed contour goes over into an integral of the same form when the axes are replaced by any other set of rectangular axes which have the same aspect. 15.
Given the formula /**
cos Xx dx
I
Ja
=
1
X
(sin
\b
sin Xa) ,
evaluate the integrals /^fc
/ift
x 2P + 1 sinXxdx,
i <J
x 2^cosXxdx.
I
Jn
a
Let us associate the points (x, y) and (x y ) upon any two given curves The point whose respectively, at which the tangents are parallel. coordinates are x\ = px + qx yi = py + qy where p and q are given constants, describes a new curve C\. Show that the following relation holds between the 16.
C
and
,
C",
,
,
corresponding arcs of the three curves Si
17.
Show
c
(
=
:
ps
qs
that corresponding arcs of the
=
x
tf(t)
- f(t)
+t
(t),
C
.
two curves x
=V
(t)
have the same length whatever be the functions /(f) and
M
From
a point of a plane let us n given curves Ci, C*2, Cn which lie distance The locus of the points = holds between the n In) F(li, k) 18.
,
MP
(
.
)
- f(t)
-*
()
,
\
(t).
draw the normals -MPi, in the same plane, and
,
MPn
to
k be the M, for which a relation of the form distances is a curve T. If lengths let
Z,-,
proportional to cF/dli be laid off upon the lines MP^ respectively, according to a definite convention as to sign, show that the resultant of these n vectors gives the direction of the normal to F at the point M. Generalize the theorem for surfaces in space.
that the
C be any closed curve, and let us select two points p and p upon the C at a point TO, on either side of TO, making mp = mp Supposing distance mp varies according to any arbitrary law as TO describes the
curve
show that
19.
Let
tangent to
(7,
.
the points p and p describe curves of equal area. mp is constant.
the special case where
Discuss
EXERCISES
IV, Exs.]
207
any closed convex curve, let us draw a parallel curve by laying oft I upon the normals to the Show that the area given curve. between the two curves is equal to it I2 + s, where s is the length of the given 20. Given
a constant length curve.
21. Let C be any closed curve. Show that the locus of the points A, for which the corresponding pedal has a constant area, is a circle whose center is fixed.
C
[Take the equation of the curve
x cos
t
+
form
in the tangential
= /().]
y sin t
22. Let C be any closed curve, C\ its pedal with respect to a point A, and C 2 the locus of the foot of a perpendicular let fall from upon a normal to C. Show that the areas of these three curves satisfy the relation A = AI A2
A
.
and u are the polar coordinates of a point on d, the coordinates of the corresponding point of C2 are p and u + n/2, and 2 those of the corresponding point of C are r = Vp 2 + and = w + arc tan p /p.] [By a property of the pedal
(
36),
if
p
p"
23. If a curve is
rigidly
Show
C
without slipping on a straight line, every point A which C describes a curve which is called a roulette.
rolls
connected to the curve
that the area between an arc of the roulette and
its
base
is
twice the area
A
of the corresponding portion of the pedal of the point with respect to C. Also show that the length of an arc of the roulette is equal to the length of the corre
sponding arc of the pedal.
ro
[SXEINEK.]
X
Y
and be the coordi [In order to prove these theorems analytically, let nates of the point with respect to a moving system of axes formed of the
A
M
OM
on C. Let s be the length of the arc tangent and normal at a point counted from a fixed point on C, and let w be the angle between the tangents at and M. First establish the formulae
and then deduce the theorems from them.] 24*.
The error made
in
Gauss method of quadrature may be expressed in
the form
/(2n 1
where
lies
.
between
>()
2n
2 1
r
2
x
2n +
and +1.
1
Ll
1.2..3... .
r
2 ,,
(2n
71
.-,
-ft
1)J 100
.,
,
[MANSION, Comptes renews, 1886.]
CHAPTER V INDEFINITE INTEGRALS
We
shall review in this chapter the general classes of elemen functions whose integrals can be expressed in terms of ele tary functions. Under the term elementary functions we shall mentary include the rational and irrational algebraic functions, the exponen tial
function and the logarithm, the trigonometric functions and and all those functions which can be formed by a
their inverses, finite
number
When
of combinations of those already named.
the
indefinite integral of a function f(x) cannot be expressed in terms of these functions, it constitutes a new transcendental function.
The study is
of these transcendental functions
and
their classification
one of the most important problems of the Integral Calculus.
I.
INTEGRATION OF RATIONAL FUNCTIONS
103. General method.
Every rational function
/(a:) is
an integral function E(x) and a rational fraction P(x) is prime to and of less degree than Q(#).
imaginary roots of the equation be decomposed into a other of the two types tion
may
A 1
(cc-a)"
The
fractions of the
Q(x~)
sum
sum
the
of
where P(x~)/Q(x"), If the real and
be known, the rational frac
of simple fractions of one or the
MX + N - a) + [(x 2
2
/3
]
type correspond to the real roots, those of the second type to pairs of imaginary roots. The integral of the integral function E(x) can be written down at once. The inte first
grals of the fractions of the first type are given
Adx (x-a) A dx x
a
by the formulae
A >
(m-l)(a;-a)
=A
log (x
a),
if
m=
1
.
For the sake of simplicity we have omitted the arbitrary constant C, which belongs on the right-hand side. It merely remains to examine 208
V,
RATIONAL FLECTIONS
103]
209
the simple fractions which arise from pairs of imaginary roots. In order to simplify the corresponding integrals, let us make the substitution
x
The
=
a
-\-
dx
fit,
ftdt.
integral in question then becomes
MX +
N
-
+
r
J
[(x
a)*
L-
W
CMa + N+MQt
1
-
d
0*- J
and there remain two kinds of integrals
c
J Since tdt grals
is
r
tdt
is
given,
(i
n
f
J
n
>
1,
2
)
dt (i
+ +
a )"
2
_ __ t
,
the
first
of these inte
by the formula i
1,
2
/?
by the formula tdt
The only
2
half the differential of 1 if
J or, if
t
*
:
r
tat
+
(1+
1
integrals
which remain are those
r
of the type
dt
J If
n
= 1,
the value of this integral dt 1
If
n
is
+
=
arc tan
is
t
=
x arc tan
-
a
t*
greater than unity, the calculation of the integral
may
be
reduced to the calculation of an integral of the same form, in which the exponent of (1 -f- 2 ) is decreased by unity. Denoting the inte gral in question
by /, we may write 1
-j-
~7T
/i (1 From
2
+
t
z
5T~ dt t )"
= r I
J
the last of these integrals, taking
tdt
<
INDEFINITE INTEGRALS
210
__ __
and integrating by
C
t*dt
J
we
parts,
103
formula
find the
C
1
t
dt
2(-l)J
1
2(n-l)(l+
(i+O"~
[v,
Substituting this value in the equation for 7n that equation becomes ,
_
n
-
2n o 2n
3 O *n-l 2
t I
-!)(! +
2 (n
*
2 )"~
Repeated applications of this formula finally lead to the integral arc tan t. Retracing our steps, we find the formula
=
(2n
where
72
()
-
3) (2n
5)
3
-
1
.
which
a rational function of
is
We will merely the numerator
-
observe that the denominator
easily calculated. 2 1 and that (1
is
+
is
)"~
,
of degree less than 2n 2 (see 97, p. 192). It follows that the integral of a rational function consists of is
terms which are themselves rational, and transcendental terms of one of the following forms :
log (SB
-
log [(x
a),
+
2
a)
s
arc tan
/8 ],
y _ -
f-f
P
Let us consider, for example, the integral /[l/(# 4 The 1)] dx. denominator has two real roots -f 1 and and two 1, imaginary roots
+i
and
We may
i.
1 x*
-
~ 1
therefore write
B
A
x-l
Z
+
+
Cx
7)
l
In order to determine A, multiply both sides by x 1 and then set x = 1. This gives A = 1/4, and similarly B = The iden 1/4. assumed written in form therefore be the tity may r.r
or,
+D
simplifying the left-hand side,
D ~_ Cx + 2
-1 It follows that
C=
2(1 + x and 7) =
1 T"
*^
3
x
1/2, and
we
\ -Ly "I
!/
.^ / V Trl*^
1
^^ 11/
*/
1-.-
1
,
(x
-
1\
( nr* \
which gives rfa;
have, finally,
1
1 A / ~t V t
-L
+
1
)
1
\ 1^
1
^ /
V,
RATIONAL FUNCTIONS
104]
211
The preceding method, though absolutely general, is not The work may often be shortened by using the simplest. always device. Let us consider, for example, the integral a suitable Note.
dx
-
2
1)"
n
If
>
1,
we may
either break
tions byliieans of the roots formula similar to that for
make
=
the substitution x
up the integrand into partial frac and 1, or we may use a reduction But the most elegant method is to which gives )/(! 2),
+1 /.
(1 -f
4*
dx
(I-*)
(!-*)
/7 (*
-
= 1)-
2dz
=
O
/"*
/1
~\2
2
rf*
"
4"-
J
*
~
2n 2 by the binomial theorem, it only remains Developing (1 z) to integrate terms of the form Az* where \L may be positive or t
negative. 104. Hermite s method.
We
have heretofore supposed that the
fraction to be integrated was broken up into partial fractions, which presumes a knowledge of the roots of the denominator. The fol
lowing method, due to Hermite, enables us to find the algebraic part of the integral without knowing these roots, and it involves only elementary operations, that
is
to say, additions, multiplications,
and divisions of polynomials. Let f(x)/F(x) be the rational fraction which is to be integrated. We may assume that f(x) and F(x) are prime to each other, and
we may
suppose, according to the theory of equal roots, that the polynomial F(x) is written in the form
X l} A 2 A ^ are polynomials none of which have multiple and no two of which have any common factor. We may now break up the given fraction into partial fractions whose denomina X pp tors are X lt X\, where
,
,
roots
,
:
X*
X\ where est
^4, is
common
a polynomial prime to X For, by the theory of high divisor, if X and Y are any two polynomials which are t
.
INDEFINITE INTEGRALS
212 prime
Z
and
to each other,
nomials
that
BX + AY= Z. X = X Y = X\ Xpp and Z =/(*). lt
Then
,
this identity
+ AX\--.Xl=f(x),
BXi or,
104
any third polynomial, two other poly
A and B may always be found such
Let us set becomes
[v,
dividing by F(x),
from the preceding identity that if f(x) is prime to X*. prime to X 1 and B is prime to X\ F(x~), Kepeating the the fraction process upon It also follows
A
is
B and so on, we
finally reach the
It is therefore sufficient to
form given above.
show how
to obtain the rational part
of an integral of the form
/A
dx
~"
where
is <(.x)
a polynomial which
prime to
is
by the theorem mentioned above, and C such that
we can
derivative. Then, two polynomials B
its
find
and hence the preceding integral may be written
f A_dx_ C J ~J
B<}>+
4>*
If
n
is
in the
form
C $dx fBdx r 1+ ~~J J
T(V
V
f
#>"-
greater than unity, taking
u=C, and integrating by
parts,
C c 4Sdx = J 4?
we
v
=
get
C
1
(n-\}r~
l
C C
n-lj
fi-"
whence, substituting in the preceding equation, we find the formula
C A dx
J
p
C
~( W
-1)^"-
C A^dx 1+ J -&=*
V,
RATIONAL FUNCTIONS
104]
where A!
is
2, we may apply the same the process may always be in the denominator is equal to
a new polynomial.
process to the
new
If n and so on
integral,
213
>
:
continued until the exponent of and we shall then have an expression of the form <
one,
A dx
+
*(*>
where R(x)
C \Ldx
V
J
a rational function of x, and ^ is a polynomial whose but which may always suppose to be less than that of is
degree we is not necessarily prime to To integrate the latter form we must know the roots of but the evaluation of this integral will intro ,
<.
<,
duce no new rational terms, for the decomposition of the fraction leads only to terms of the two types \[//
Mx + N
A
x-a
2 (z-a) +
2 /3
each of which has an integral which is a transcendental function. This method enables us, in particular, to determine whether the integral of a given rational function
The necessary and
is itself
a rational function.
should be true
sufficient condition that this
that each of the polynomials like ^ should vanish has been carried out as far as possible. It will
for
/
is
be noticed that the method used
when
formula Let us now
in obtaining the reduction
essentially only a special case of the preceding
method.
consider the more general integral
+ 2Bx +
(Ax*
From
C)
the identity
A(Ax* + 2Bx + C)- (Ax + it is
evident that
C J (Ax* +
we may
= C)
A AC -
Integrating the last integral by parts, -f
B)
/
A
O
(Ax*
= AC-
C
dx
B* J (Ax*
2Bx +
AC-B*J -SJ (Ax \
2
B*
write
**
2Bx +
B)
Ax + B Tt + 2Bx + i
C\
.
y^v
C)
we
n ..
dx
+ B) (Ax ^
1 C)"-
+ B)dx + 2Bx + C)
(Ax (Ax*
find
Ax + B
= 2(n-
2n-2J
(Ax*
is
the process
+ 2Bx +
C)
n
~
214
INDEFINITE INTEGRALS
whence the preceding
relation
[V,
104
becomes
Ax + B (Ax*
+ 2Bx +
2(n- \)(AC -
C)
2n-3 -
2n
^
/*
AC -
2
+ 2Bx
B*)(Ax* 2
Ax (4z*
J
+
i
(
X
2
+
C]
Continuing the same process, we are led eventually to the integral
dx ^Ix 2
which
is
a logarithm
B2 -
if
+ 2Bx + C and an arctangent
AC>0,
As another example, consider the
C 5x + 3x J (x + 3x+ 3
From
B2 -
+
1)
AC<0.
1
.
dx. ,
the identity
+
5x 3 it is
if
integral
we may
evident that
+ 3x 3x + + (x /5x 8
Integrating the
first
3
1
=
6x(x
2
+
1)
+
C
1
dx l)3
+
6x(x 2
+
(x
+
I)
whence the value of the given 5x 8
l)
=J ^^3x +
i)tfg
3X
-
3
+
3x
write
integral on the right by
6 ( g8
Cx J (X
-
3x
+
(X
3
+
integral
3x
-
1
3X is
.
-dx
+
-J we
parts,
-x
= ~ 3
C
dX
l)*
+ 1)2
dx
(xM+
find
r
J
+
OX
dx (x (
8
+
3x
+
1)2
seen to be
-
=
z
Note. In applying Hermite s method it becomes necessary to solve the fol lowing problem given three polynomials A, B, C, of degrees m, n, p, respectively, two of which, A and B, are prime to each other, find two other polynomials u and v such that the relation Au + Bv = C is :
identically satisfied.
In order to determine two polynomials u and v of the least possible degree which solve the problem, let us first suppose that p is at most equal to m + 1. Then we may take for u and v two polynomials of degrees n - I and m - 1, The respectively. + n unknown coefficients are then given by the system of
n-
m
m+
n
linear
non-homogeneous equations found by equating the
For the determinant of these equations cannot vanish, since, find two polynomials u and v of degrees n - 1 and m - 1 or the identity Au + Bv = 0, and this can be true only when
common
If the
if it
coefficients.
did,
we could
which satisfy and B have a
less
A
factor.
degree of
C
is
and obtain a remainder
m + n, we may divide C by AB m + n. Then C = A BQ + C
equal to or greater than whose degree is less than
C"
and, making the substitution u - BQ = MI, the relation Au C Aui + Bv This is a problem under the first case. .
,
+ Bv = C reduces
to
V,
RATIONAL FUNCTIONS
105]
105. Integrals of the type
+
/R(x, ^Ax?
integrals of rational functions grals of irrational functions.
is
it
We
2Bx
215
+
c) dx.
After the
natural to consider the inte
commence with the
shall
case in
a rational function of x and the square root of a polynomial of the second degree. In this case a simple substitu tion eliminates the radical and reduces the integral to the preceding
which the integrand
is
This substitution
case.
the radical
is
self-evident in case the expression under b. If we set ax ax b t*,
+ =
+
of the first degree, say
is
the integral becomes
^ax^b]dx =
Cll(x,
J
J/V\
,
a
t]
>
a
J
and the integrand of the transformed integral is a rational function. If the expression under the radical is of the second degree and has two real roots a and b, we may write
A(x-a)(x-b) = (x-b)
-
and the substitution x
or
= Aa - bt A t*
2 >
actually removes the radical. If the expression under the radical sign has imaginary roots, the above process would introduce imaginaries. In order to get to the
bottom of the matter,
Then x and y equation
y denote the radical
let
^Ax
2
-f
2Bx
are the coordinates of a point of the curve
+
C.
whose
is 2
y
(1)
= Ax + 2
2Bx
+
C,
and
it is evident that the whole problem amounts to expressing the coordinates of a point upon a conic by means of rational functions of a parameter. It can be seen geometrically that this is possible.
For,
if
a secant
y
-
J3
=
t(x
-
a)
be drawn through any point (a, /3) on the conic, the coordinates of the second point of intersection of the secant with the conic are given by equations of the functions of t. If the trinomial
cient
A must be
Ax 2
first
degree,
+ 2Bx + C
and are therefore rational
has imaginary roots, the coeffi not, the trinomial will be
positive, for if it is
negative for all real values of x.
In this case the conic (1)
is
an
INDEFINITE INTEGRALS
216
A
hyperbola.
[V,
105
straight line parallel to one of the asymptotes of
this hyperbola,
y
= x Vyi + t,
cuts the hyperbola in a point whose coordinates are
-
C
2B
A
C
r~
t*
2t
A
t*
-
2B
an ellipse, and the trinomial A x 2 + 2Bx + C must have two real roots a and b, or else the trinomial is negative If
<
0,
the conic
is
The change
for all real values of x.
cisely that
moving secant
=
y
As an example
let
bola in
Making
.
2
= x + k is an hyperbola, and the straight line which is t, parallel to one of the asymptotes, cuts the hyper a point whose coordinates are 2
auxiliary conic if
+y=
a)
+ k) Vz + k
2
The
t(x
us take the integral
(x
x
of variable given above is pre this conic by the
which we should obtain by cutting
the substitution indicated by these equations,
_=
dt ft* ~
+
C dx _
k\
4tdt
find
2
7~
J
*
we
or, returning to the variable x,
dx
_
where the right-hand side In general,
if
B
AC
2
is
+
Vcc 2
x
is
(Ax
+
2Bx
_
x
1
determined save for a constant term
not zero, we have the formula
Ax + B
1
Ix 2
k
+ Cy
B VAx 2 + 2Bx + C
AC
2
In some cases it is easier to evaluate the integral directly without removing the radical. Consider, for example, the integral
dx
+
2Bx
+C
V,
RATIONAL FUNCTIONS
105]
ff the coefficient
A
is
positive, the integral
_ C J ^A*x + 2ABx + AC J = setting Ax + B r
^/Adx
x
2
or
217 be written
may
VJdx + ) + AC 2
B*
t,
i
dt
r
^AJ
-
-VA
B*
t-AC
Returning to the variable
dx
x,
we have
the formula
1
f 2Bx
B
+C
2 If the coefficient of x
+A
Ax*
negative, the integral
is
may
+ 2Bx be written in
the form
/7
/*
doc
I
B 2 - (Ax - BY
V- Ax + 2Bx + C J VJ 2
The quantity A C
+B
2
is
Hence, making the
necessarily positive.
substitution
Ax - B =
t
+B
^/A C
2 ,
the given integral becomes
_
r dt V.4 I J Vl Hence the formula
I t
2
V.4
in this case is
dx
V- Ax +
+
Ax
1
= 2Bx
2
-^=
C
1 show that the argument of the arcsine varies from x the two of varies between roots the trinomial. + In the intermediate case when ^4=0 and B 0, the integral is
It is easy to
to
B
arc sin
1 as
=
algebraic
:
f J Integrals of the type
dx
- a) V^x + (x 2
2Bx
+C
INDEFINITE INTEGRALS
218
[V,
reduce to the preceding type by means of the substitution x We find, in fact, the formula
r
J
r
dx a) -^/Ax*
(x
+
2Bx
+
100
= a -f 1/y.
dy
J
C
where
A!
= Aa + 2Ba +
B = Aa +
2
C,
l
B,
It should be noticed that this integral is algebraic if and only if the quantity a is a root of the trinomial under the radical. Let us now consider the integrals of the type f Va: 2 A dx. Inte
+
grating by parts,
we
rVa;"
On
find
+ A dx = x Va: + A 2
hand we have
the other
2
fx
I. = C Va;
dx
2
I
.
Va: 2
+
J
A
= From
/
these two relations
Va: 2
it is
-f
A dx
-\-Adx-A
C J/
+ A7
Adx -
Va; 2
+
log (x
J
+
Va: 2
+ A)
.
easy to obtain the formulae
c
(2)
r I
J
2
+ A + - log
(or
+
Va: 2
+
^),
2 *
The following formulae may be derived
in like
manner:
x*dx (5)
arc sin
z
a
106. Area of the hyperbola. The preceding integrals occur in the evaluation of the area of a sector of an ellipse or an hyperbola. Let us consider, for
example, the hyperbola
V,
106]
and
let
x
axis,
RATIONAL FUNCTIONS
AMP
us try to find the area of a segment bounded by the arc AM, the This area is equal to the definite integral
and the ordinate MP.
6
a that
is,
by the formula
MP = y =
(6/a)
Vx 2
a2 dx
,
(2),
a2
But
219
Vx 2
-
a2 log (?-
a 2 and the term (b/2a) x ,
area of the triangle OMP. Hence the area the arc and the radii vectores OA
S
Vx 2
a2
of the sector
AM
and OM,
S=
is
precisely the
0AM, bounded
by
is
1 e*
/x loj
I
+ Vx 2 -
a2\ I
a
V
2
)
= -1 oft log ( -x .
,
2
\a
This formula enables us to express the coordinates x and y of a, point of the hyperbola in terms of the area S.
M
In fact, from the above and from the
equation of the hyperbola, show that
it is
easy to FIG. 21
y
=
-
(e"
The functions which occur on the right-hand and sine :
h b
e
side are called the hyperbolic
cosine
cosh x
=
e*
+
e~ x sinh x
=
2
The above equations may x
2
therefore be written in the form
=
a cosh
2S ab
y
=
.
.
, b sinh
ab
These hyperbolic functions possess properties analogous to those of the trigo nometric functions.* It is easy to deduce, for instance, the following formulae x = = cosh x cosh y + = sinh x cosh y +
cosh 2 x cosh
(x
sinh (x * is to
+ +
y) y)
sinh 2
:
1,
sinh x sinh y,
sinh y cosh
A
x.
table of the logarithms of these functions for positive values of the argument be found in HoueTs Recueil desformules
numeriques.
INDEFINITE INTEGRALS
220 may be shown may be expressed It
[V,
107
manner that the coordinates of a point on an ellipse terms of the area of the corresponding sector, as follows
in like in
:
x
=
a cos
2S y
,
ab
=
,
2S
.
b sin
ao
In the case of a circle of unit radius, and in the case of an equilateral hyperbola whose semiaxis is one, these formulae become, respectively,
x z
= =
cos2S, cosh2S,
= sin2-S; = sinh2/S.
7/
y
evident that the hyperbolic functions bear the same relations to the equi lateral hyperbola as do the trigonometric functions to the circle. It is
107. Rectification of the parabola. Let us try to find the length of the arc of = x 2 between the vertex and any point M. The general
a parabola 2py formula gives
.
W
d pj7w^y *
Jo or,
applying the formula
J
2
-
p
\
The algebraic term in this result is precisely for we know that OT = x/2, and hence
we draw
P
(2),
2p
If
r^sv
=
z
x2
x*
x2
4
4p*
4
the straight line connecting
T to
the length
x 2 (x 2
-4-
4p
MT
of the tangent,
v^)
2
the focus F, the angle
MTF will
be a right angle. have
Hence we
FT
/
V
n2
a-2
H.J
1
=I
whence we may deduce a
curi
ous property of the parabola. Suppose that the parabola rolls without slipping on the x "
i
T
T
,
FIG. 22
7""
axis, and let us try to find the locus of the focus, which is sup posed rigidly connected to the
parabola. When the parabola = arc OM. The point T has come into a to the x axis, tangent at T = 3/T, and the focus F is at a point F which is position T such that
M
is
found by laying nates
X and
OM
M
off
Y of
T F = TF
the point
F
on a
line parallel to the
are then
y
axis.
The coordi
RATIONAL FUNCTIONS
V,108] and the equation tions.
From
the
221
by eliminating x between these two equa
of the locus is given first we find
!
x to
+ Vx 2 + p 2 = pe P
_
,
which we may add the equation
Vx2 +
x since the product of the two equations, we find
two left-hand
and the desired equation of the locus
2
_
=
sides
P
pe
j>
is
equal to
p*.
Subtracting these
is
2
p
This curve, which is
somewhat
is called the catenary, is quite easy to construct. similar to that of the parabola.
108. Unicursal curves.
Let us now consider, Let
Its
form
in general, the inte
grals of algebraic functions.
F(x, y)
(6)
=
be the equation of an algebraic curve, and let R(x, y) be a rational function of x and y. If we suppose y replaced by one of the roots of the equation (6) in R(x, y), the result is a function of the single variable x, and the integral
called an Abelian integral with respect to the curve When (6). the given curve and the function R(x, y) are arbitrary these inte grals are transcendental functions. But in the particular case where is
is unicursal, i.e. when the coordinates of a point on the curve can be expressed as rational functions of a variable param eter t, the Abelian integrals attached to the curve can be reduced at
the curve
once to integrals of rational functions.
For, let
be the equations of the curve in terms of the parameter t as the new independent variable, the integral becomes
j
R(x, y}dx
and the new integrand
is
=
evidently rational.
t.
Taking
INDEFINITE INTEGRALS
222
shown
It is
in treatises on Analytic
[v,
108
Geometry* that every
uni-
cursal curve of degree n has (n l)(n conversely, that every curve of degree n
2)/2 double points, and, which has this number of
double points is unicursal. I shall merely recall the process for obtaining the expressions for the coordinates in terms of the param
Given a curve C B of degree n, which has 8 = (n !)(. 2)/2 double points, let us pass a one-parameter family of curves of degree 3 ordinary points 2 through these 8 double points and through n n eter.
on Ca
These points actually determine such a family, for
.
-
1,
whereas (n 2)(n -(-l)/2 points are necessary to determine uniquely 2. Let P(x, y) + tQ(x, ?/) = be the equation a curve of order n of this family, where t is an arbitrary parameter. Each curve of the family meets the curve
C n in
2) points, of
n(n
which a certain
num
3 ordinary points chosen ber are independent of t, namely the n above and the 8 double points, each of which counts as two points of
But we have
intersection.
- 3 + 28 = n - 3 + (ft -l)(n - 2) = n(n - 2) -1, and there remains just one point of intersection which varies with t. The coordinates of this point are the solutions of certain linear equa tions whose coefficients are integral polynomials in t, and hence they are themselves rational functions of
t.
Instead of the preceding
we
might have employed a family of curves of degree n 1 through the 3 ordinary points chosen at (n l)(w 2)/2 double points and 2n
Cn
pleasure on If
=
n
is
degree
.
=
2)/2 (n l)(w therefore unicursal, as
2,
0,
every curve of the second seen above. If n = 3,
we have
=
the unicursal curves of the third degree 1, l)(w (n 2)/2 are those which have one double point. Taking the double point as origin, the equation of the cubic is of the 4>s
where
and
(x, y)
+
=
fa (x, y)
form ,
2 are homogeneous polynomials of the degree of their tx through the double point meets the cubic secant y in a single variable point whose coordinates are 3
<
indices.
A
<
=
(!,
Q
<MM)
*See,
e.g.,
Q *i(M)
<&(!, "
Niewenglowski, Cours de Geometric analytique, Vol.
II,
pp. 99-114.
V,
.RATIONAL FUNCTIONS
108]
223
A unicursal curve of the fourth degree has three double points. In order to find the coordinates of a point on it, we should pass a family of conies through the three double points and through another point chosen at pleasure on the curve. Every conic of this family would meet the quartic
in just one point which varies with the gives the abscissae of the points of
The equation which
parameter.
would reduce to an equation of the first the factors degree corresponding to the double points had been removed, and would give x as a rational function of the intersection, for instance,
when
We should proceed to find y in a similar manner. parameter. As an example let us consider the lemniscate which has a double point at the origin and two others at the imagi nary circular points. A circle through the origin tangent to one of the branches of the lemniscate, x*
+
y
2
=
- y}
t(x
,
meets the curve in a single variable point. equations,
or,
we
Combining these two
find
dividing by x
y,
This last equation represents a straight line through the origin which cuts the circle in a point not the origin, a _o
+ a + a*
3 *(<
t*
2 )
whose coordinates are
+
2 2 _ a t(t
t*
a2)
a<
These results may be obtained more easily by the following process, which is at once applicable to any unicursal curve of the fourth degree one of whose double points is known. The secant = \x cuts the in lemniscate two whose coordinates are y points
The expression under the by cal.
radical is of the second degree. Hence, 2 105, the substitution (1 A) X)/(l (a/t) removes the radi It is easy to show that this substitution leads to the expressions
just found.
-
+
=
INDEFINITE INTEGRALS
224
[V,
109
Note. When a plane curve has singular points of higher order, it can be shown that each of them is equivalent to a certain number of
In order that a curve be unicursal,
isolated double points.
it is suffi
cient that its singular points should be equivalent to (n l)(n 2)/2 For example, a curve of order n which has isolated double points. 1 is unicursal, for a secant through a multiple point of order n the multiple point meets the curve in only one variable point.
109. Integrals of binomial differentials. in
ing types
/
:
R\_x, (ax
+
1
b)
\dx
where a, a ,
R
a ,
xa \ xa
b, ~V
-f-
+ d)dx,
ex
-)dx,
",
and where the exponents For the first type it is
denotes a rational function
are commensurable numbers.
a",
sufficient to set
+b=
R(X, ~vax
I
,
R(x
ax
the other integrals
Among
which the radicals can be removed may be mentioned the follow
ax
-f b
=
t
q
In the second type the substitution
.
leaves merely a square root of an expression of the second degree, which can then be removed by a second substitution. t*
=
Finally, in the third type we may set x denominator of the fractions n-, a ,
three exponents m, n,
,
,
we may
common
Let us suppose that the If p is an integer, the
differentials.
p made
rational
just seen.
by means of the substitution
In order to discover further cases
\
x (ax n
The transformed
a
are commensurable.
dx
J
is
consider a class of
of integrability, let us try the substitution ax n
\
D
form
which are called binomial expression may be x = t D as we have
D where
a",
In connection with the third type differentials of the
t
a
+
integral
is
of the
same form
+
t.
This gives
/
+ by dx = -H |W^ naj \ a /
the exponent which takes the place of p is (m the integration can be performed if (m l)/w
b
"
dt.
as the original,
+ is
1.
1) /n an integer.
and
Hence
RATIONAL FUNCTIONS
V,109]
On
the other hand, the integral
whence
it is
may
225
be written in the form
clear that another case of integrability
l)/w = (m + l)/w + p
is
that in which
an integer. To sum up, the np (m one of the three numbers whenever can be performed integration In no other case can the is an integer. (in +p 2^ +l)/n, (m+V)/n
+
4-
by means of a finite number of elementary n, and p are rational.
integral be expressed
functional symbols In these cases it
is
when m, is
convenient to reduce the integral to a simpler n Setting ax
form in which only two exponents occur.
x
/As = (}
a
l
n
t
dx
,
we
find
n = -1/Asl-, dt,
n a
r
x m (ax n
= bt,
IP /*\!ii lt + bydx = n J "
t
)
"
1
+
1
n
i
\a/
Neglecting the constant factor and setting q
=
(m
+
l)/n
1,
we
are led to the integral /
tydt.
The
cases of integrability are those in which one of the three num p q is an integer. If p is an integer and q r/s, we
bers p, q,
+
should set
t
l+t =
u*.
=
us
.
If q
Finally,
if
should set an integer and p r/s, we + q is an integer, the integral may be
p
is
written in the form
and the substitution 1 + t = tus where p As an example consider the integral
r/s,
,
x
Here is
m = 1,
n
an integrable
= 3, p = 1/3, case.
Vl +
x 3 dx
and (m
3 Setting x
=
t,
removes the
.
+ l)/w + p = 1.
Hence
the integral becomes
dt,
and a second substitution
1
+ = t
radical.
tu 8 removes the radical.
this
INDEFINITE INTEGRALS
226
II.
110.
ELLIPTIC
Reduction
of degree/?
AND HYPERELLIPTIC INTEGRALS
of integrals.
which
is
110
[V,
prime
Let P(x) be an integral polynomial
The
to its derivative.
integral
where R denotes a rational function of x and the radical y
Vp(ce),
cannot be expressed in terms of elementary functions, in general, when p is greater than 2. Such integrals, which are particular cases of general Abelian integrals, can be split up into portions which and logarithmic functions and a certain number
result in algebraic
which give rise to new transcendental functions which cannot be expressed by means of a finite number of elemen of other integrals
We
tary functional symbols. proceed to consider this reduction. The rational function R(x, y) is the quotient of two integral polynomials in x and y. Replacing any even power of y, such as 2q 9 g+ l q y by [P(V)] and any odd power, such as y* by y [_P(x)~] we may evidently suppose the numerator and denominator of this frac >
,
>
,
tion to be of the first degree in y,
A in R(x, J) v
C
+ By + Dy
>
where A, B, C, D are integral polynomials numerator and the denominator each by C by P(x), we may write this in the form R(x, y}
Multiplying the 2 Dy, and replacing y
in x.
--
K
K
are polynomials. The integral is now broken of which the first up parts, JF/K dx is the integral of a rational function. For this reason we shall consider only the second
where
F, G,
into
integral
and
two
fOy/K dx, which may
also be written in the
form
fpft where
M and N are
integral polynomials in
M/N
sum
of partial fractions
may
x.
The
rational frac
be decomposed into an integral part E(x) and a
tion
V,lio]
AND HYPERELLIPTIC INTEGRALS
ELLIPTIC
227
where each of the polynomials A is prime to its derivative. shall therefore have to consider two types of integrals, ,-
l
/xV If the degree of P(x) in terms of the first p
Adx
C
dx
Ym may be expressed Yp _ 2 and certain
p, all the integrals 1 of them, F , Yi}
is
We
-,
,
algebraic expressions.
For, let us write
P(X)
=
a xp
+
+
a^x*
It follows that
~JZ(
A
2 mP(x) _ 2mx 1
+
VP(Z)
x m P (x)
2
The numerator highest term
is
of this expression
(2m
+ p~)a
~ x m+p l
of degree
is
m +p
1,
and
its
Integrating both sides of the
.
above equation, we find
2x
Vp(xj=(2m+p}a Ym+p _ + l
.--,
where the terms not written down contain integrals of the type Y whose indices are less than m + p 1. Setting m = 0, 1, 2, ,
succes successively, we can calculate the integrals Yp _ l} Yp 1 integrals sively in terms of algebraic expressions and the p ,
YO>
YI,
,
Yp _ 2
.
With
respect to the integrals of the second type we shall distin the two cases where is or is not prime to P(x)> guish
X
1) If X is prime to P(x), the integral Z n reduces an algebraic term, a number of integrals of the type
the
to
sum of
Yk and a new ,
integral
B dx
f-X V P(x) where
B
is
a polynomial whose degree
is less
than that of X.
and also to P(ar), X n is prime to PA". Hence two polynomials A and fi can be found such that + p.X P = A, and the integral in question breaks up into two parts: Since
X is prime
to its derivative
X
XX"
f ii== f *^_ J X n ^P(x) J VP(X)
i
c J
Xn
INDEFINITE INTEGRALS
228
The
first
integral,
a
is
part
when n
sum
>
In the second
of integrals of the type Y.
1, let
110
[V,
us integrate by parts, taking 1
f I
~\./ T-)
tt
*3/
TT~"
**
(n
-I)*-
1
which gives
r^px dx = X
J
n
(n-l)X
n~
rwp + pp*
i
-pV7>
n-lJ
l
l
^
1
2J"-
VP(a:)
The new integral obtained is of the same form as the first, except is diminished by one. that the exponent of Repeating this is i.e. as long as the exponent of as often as possible, process of the form a result obtain we than finally unity, greater
X
X
VP
J
J .YVP
P(x)
Xn ~
where B, C, D are all polynomials, and where the degree of always be supposed to be less than that of X.
B may
have a common divisor D, we shall have X = YD, SD, where the polynomials D, S, and Y are all prime to each Hence two polynomials X and other. may be found such that n in the form written be = the and A XD + integral may 2) If
X
and
P
P=
/u.
p.Y",
/Adx jpVp The
first
of the
new
C \dx
C
J r n Vp
J Dn
integrals
is
fidx
of the type just considered.
The
second integral,
D is a factor of P, reduces to the sum of an algebraic term a number of integrals of the type Y. and
where
For, since
Dn
nomials A! and
prime to the product D S, we can find two poly such that X^" -f piD S = /x. Hence we may write
is
//,!
C J Replacing form
P
by DS,
dx_ n >
let
= C\dx + Cr^
VP J VP
J
D"
VP
us write the second of these integrals in the
D
AND HYPERELLIPTIC INTEGRALS
ELLIPTIC
V,110]
and then integrate
it
by
parts, taking
229
--11 -
V
which gives
r,
Cjidx^
J
|
J
D"VP
This
again a reduction formula but in this case, since the expo even 1/2 is fractional, the reduction may be performed
is
;
nent n
D
occurs only to the first power in the denominator, and an expression of the form obtain finally
when
C
_
p.dx
J D VP n
where
H and K are
To sum up our
KVP
"fP
we
C Hdx
V
polynomials.
we
results,
see that the integral
M dx can always be reduced to a sum of algebraic terms and a number of integrals of the
two types JO
/
///)
\JiJCi
VP
J
/rpftl
where
ra is less
derivative
X
and
"V
-**-\
/7 Tf UJU
xVp
p 2, where X and where the degree of
than or equal to also to P,
is
^
prime is less
to its
than
This reduction involves only the operations of addition,
that of X.
multiplication, and division of polynomials. are known, each of the rational If the roots of the equation into a sum of partial fractions of can be broken fractions
X=
X /X
up
l
the two forms
A
Bx + C
x-a
(x-af + p*
where A, B, and C are constants.
r J (xx which reduce to allow a to
dx a a)
VP(V>
to a single type,
have
This leads to the two new types
C (Bx J [_(xx - a) 2 +
we agree sort are this of Integrals
namely the
imaginary values.
2 /3
first
of these, if
INDEFINITE INTEGRALS
230
[V,
called integrals of the third kind. Integrals of the type called integrals of the first kind when ra is less than p/2
110
Ym
are
1,
and
m
is equal to or greater are called integrals of the second kind when 1. than p/2, Integrals of the first kind have a characteristic
they remain
property, indefinitely,
when
finite
the
and also when the upper
89, 90); but the essential distinction
(
The
pointed out
Up
Note.
real distinction
later.
we have made no assumption about the polynomial P(x). If p is an odd number, it may
to the present
of the
p
degree
is a root of P(x) between the integrals of
must be accepted provisionally at this between them will be
the second and third kinds
time without proof.
upper limit increases
limit
always be increased by unity. 1 nomial of degree 2q
For, suppose that P(x)
is
a poly
:
Then
let
P(x)
=
A x*-
us set x
=
a
-f-
+
1
1/y,
+
A.x^-*
where a
.-.
+ A.
2q
_v
.
This
not a root of P(z).
is
gives
P(x)=P(a) where P^
(y) is a
I i y
+ P (a)
+
\
+ I__Lii p(2<,-l)( fl
...
1
=
P,((/\ fJ-134, 2 "
<
1)! y-
(2q
polynomial of degree
1
-i-j
i/
Hence we have
2q.
and any integral of a rational function of x and Vp(a-) is trans formed into an integral of a rational function of y and \/P l (y). Conversely, if the degree of the polynomial P(x) under the radi cal is an even number 2q, it may be reduced by unity provided a root of
x
=
a
-f-
P(x)
is
known.
For,
if
1/y.
y
where Pi(y)
a
is
a root of P(x),
let
us set
This gives
is
of degree 2y
(2?)! 1,
and we
y
y
shall have
Hence the integrand of the transformed integral will contain no other radical than
V,
lllj
ELLIPTIC
AND IIYPERELLIPTIC INTEGRALS
111. Case of integration of the
in algebraic
terms.
We have
just seen that
231
an integral
form
C R[X, VP(x)]dx can always be reduced by means of elementary operations to the sum of an inte gral of a rational fraction, an algebraic expression of the form G VP(x)/L, and Since we can also a number of integrals of the first, second, and third kinds. find
by elementary operations the rational part of the integral of a rational it is evident that the given integral can always be reduced to the form
fraction,
]dx = F[x, VP(xj] + T, where
F is
a rational function of x and VP(x), and where
T
is
sum
a
of inte
X
being prime to its deriva grals of the three kinds and an integral fXi /Xdx, Liouville showed that if the given integral tive and of higher degree than X\ .
integrable in algebraic terms, fore have, identically,
is
it is
R[x, VP(x)~\
and hence
T=
=
should there
~
0.
Hence we can
discover by
whether a given integral the
We
equal to F[x, VP(x)].
same process
is
means of multiplications and
divisions of polynomials
integrable in algebraic terms or not,
and
in case
it is,
gives the value of the integral.
112. Elliptic integrals.
If the
polynomial P(x)
is
of the second
degree, the integration of a rational function of x and P(x) can be reduced, by the general process just studied, to the calculation of the integrals
/dx VP(z) which we know how
The next simplest is
C
J
dx (x
-
to evaluate directly
a)VP(z) (
105).
that of elliptic integrals, for which P(x) of the third or fourth degree. Either of these cases can be case
is
reduced to the other, as we have seen just above.
Let P(x) be a
polynomial of the fourth degree whose coefficients are all real and whose linear factors are all distinct. proceed to show that
We
a real substitution can always be found which carries P(x~) into a polynomial each of whose terms is of even degree. Let a, b, c, d be the four roots of P(x). Then there exists an
involutory relation of the form (7)
Lx
x"
+ M(x +
x")
+N=
0,
INDEFINITE INTEGRALS
232 which
is satisfied
coefficients L,
by x
=
a,
which are evidently
=
a
b
-+-
c
Let a and
ft
112
For the
x"
0,
0,
we take
satisfied if
M = cd
d,
= d. and by x = c, the two relations satisfy b,
+ M(a + b) -f N + M(c + d) + N =
Lab Led
L
x"
M, N need merely
[V,
N = ab (c + d)
ab ,
cd (a
be the two double points of this involution,
+ b).
i.e.
the
roots of the equation Zt*
These roots will both be (cd
that
-
ab)*-(a
+ 2ATw + N = 0.
real if
+b-c-
d) [ai
(c
+
d)
-
cd(a
+
J)]
>
0,
is, if
(a-c)(a-d)(b-c)(b-d)>Q.
(8)
The
2
roots of P(x) can always be arranged in such a way that this If all four roots are real, we need merely is satisfied.
condition
Then each factor in (8) is positive. we should choose a and b as the real two conjugate imaginary roots. Then the
choose a and b as the two largest. If only two of the roots are real,
roots, and c and d as the c and a d are conjugate imaginary, and so are the two factors a c and b d. other two, b Finally, if all four roots are imaginary, we may take a and b as one pair and c and d as the other pair of
conjugate imaginary roots.
In this case also the factors in (8) are
It should also be noticed that these conjugate imaginary by pairs. real. methods of selection make the corresponding values of L, M, The equation (7) may now be written in the form
N
~_ +
(9}
If
we
x
set (x
a)/(x
4-
>
ft
ft~)
= y,
x "~ a
X
"
or x
-
=
=
(3
(ft//
1),
<*)/(y
we
find
where P\(y) is a new polynomial of the fourth degree with coefficients whose roots are a
a
a-p It is
b
b-
a
c
ft
c-
a
d
a
ft
d-
ft
real
evident from (9) that these four roots satisfy the equation
V,
ELLIPTIC
112]
y
=
-fby pairs of odd degree. If the four roots y"
shall
AND HYPERELLIPTIC INTEGRALS hence the polynomial
;
/ i(y) contains
233 no term
= c + d, we b, c, d satisfy the equation a + b one of the double points of the involution lies Setting a N/2M, the equation (7) takes the form ,
= 0, and
have L
at infinity.
+ x = a +y a
x
a
x"
0,
in order to obtain a polynomial and we need merely set which contains no term of odd degree. We may therefore suppose P(x) reduced to the canonical form
any elliptic integral, neglecting an algebraic term and an integral of a rational function, may be reduced to the sum of integrals of the forms It follows that
C
dx
^AtX*+AiX*+Ai
C
xdx
J ^/A^+AiXt+At
x*dx
J ^A
9
x*+A
l
and integrals of the form dx
/;(x The
a
integral
dx
If we consider x, on the the elliptic integral of the first kind. inverse function of this other hand, as a u, function is called an of the above The second integrals reduces to an elliptic function. is
2 elementary integral by means of the substitution x
= u.
The
third
integral
x 2 dx
is
Legendre
s
integral of the second kind.
Finally,
we have the
identity
/dx (x
The
-
a)Vp(x)
C
xdx _ C ~J (x - a )VP(x) 2
V
2
dx 2
(x
integral
dx 2
(x is
Legendre
s
+
/i)
VJ
x4
+ /Ijx + A.
integral of the third kind.
2
4
-
a2 ) VP(^)
INDEFINITE INTEGRALS
234
These elliptic integrals were so named because they were met with in the problem of rectifying the ellipse. Let
x
=
a cos
y
,
=
2 setting a
or,
Hence the
= dx + dy* = (a b = e a ds = a Vl 2
Then we
shall have
+
<
t>*
cos 2
2 <)
d<j>
,
,
integral
stitution cos
sin 2
2
2
3
2
first
b sin
be the coordinates of a point of an ellipse. ds 2
113
[V,
<
=
t,
2
e cos
2 .
<
d
which gives an arc of the takes the form
ellipse, after the
sub
an ellipse is equal to the sum of an and an integral of the second kind.
inte
It follows that the arc of
gral of the first kind
Again, consider the lemniscate defined by the equations .
t*
An
y
a4
=
a"
t
4
+
a4
easy calculation gives the element of length in the form ds 2
Hence the the
+
first
=
dx 2
+
dif
arc of the lemniscate
a
=-
is
t
-
~p
dt*.
Ct
given by an elliptic integral of
kind.*
It sometimes happens that an integral of the where P(x) is a polynomial of the third or fourth degree, can be expressed in terms of algebraic functions and a sum of a finite number of logarithms of algebraic functions. Such integrals are called pseudo-
113. Pseudo-elliptic integrals.
form
f F[x, VP(x)]
elliptic.
dx,
This happens in the following general case.
(10)
Lx
x"
+ M(x +
x")
+
Let
N=0
an involutory
be the
relation which establishes a correspondence between two pairs of four roots of the quartic equation P(x) = 0. If the function f(x) be such that
the relation
(ID is identically satisfied, the integral
Lx /[/(x)/VP(x)] dx
is
pseudo-elliptic.
* This is a common property of a whole class of curves discovered by Serret (Cours de Calcul differentiel et integral, Vol. II, p. 264).
V,
ELLIPTIC
113]
Let
a and
AND HYPERELLIPTIC INTEGRALS
235
As we have
p be the double points of the involution. may be written in the form
already
seen, the equation (10)
(12)
Let us
-
x
now make
x"
/3
the substitution (z
-
a)/(x
(a-^, (l-y)
p
a
=
P)
P(2)
=
p)
dy
This gives
y.
^L, (1-y)*
and consequently dx
_
(
a polynomial of the fourth degree which contains no odd powers the other hand, the rational fraction f(x) goes over into a which satisfies the identity For if rational fraction + y) = 0.
where PI (y) of y ( 112).
is
On
4>(y),
<(
two values of x correspond by means of (12), they are transformed into two = 0. It is evident which satisfy the equation y + values of y, say y and 2 Hence that is of the form y^(y z ), where ^ is a rational function of y the integral under discussion takes the form y"
y" ,
.
4>(y)
and we need merely
set y 2
Thus the proposition
is
=
reduce it to an elementary integral. merely remains actually to carry out
z in order to
proved, and
it
the reduction.
The theorem remains provided that
we
true
when
think of one of
the polynomial P(x) its
is
of the third degree,
The demonstration
roots as infinite.
is
exactly similar to the preceding. If, for example, the equation P(x)
= is a reciprocal equation, one of the = 1. Hence, involutory relations which interchanges the roots by pairs is x if f(x) be a rational function which satisfies the relation /(x) + /(1/x) = 0, x"
the integral l)/(z
(x
+
/[/()/ VP(x)] dx 1)
=
y,
y
2
=
z,
is
pseudo-elliptic,
performed
in order,
and the two substitutions it into an elementary
transform
integral.
Again, suppose that P(x)
a polynomial of the third degree,
= b = 0, c = 1, d = I/A; 2 There which interchange these roots by pairs
Let us set a tions
is
.
o>,
exist three involutory rela
:
l-Jk 2 Hence,
if
/(x) be a rational function
which
satisfies
x"
one of the identities
INDEFINITE INTEGRALS
236
[V,
114
the integral
f(z)dx
Vx(lis
z
From
pseudo-elliptic.
=
this others
2 2 , the preceding integral
whence
it
may
For instance,
be derived.
if
we
set
becomes
new
follows that this
)(!
integral
is
also pseudo-elliptic
if
/(z
2 )
satisfies
one of the identities
The
first
III.
of these cases
was noticed by Euler.*
INTEGRATION OF TRANSCENDENTAL FUNCTIONS
114. Integration of rational functions of sin
known
x and
cos x.
It is well
and cos a? may be expressed rationally in terms Hence this change of variable reduces an integral
that since
=
of tan ce/2 of the form
t.
sinx, cosxjdx to the integral of a rational function of
x
= 2 arc tan
t
,
ax
=
-
2dt
-
-^
sin x
>
For we have
t.
2t
=
>
2
cos x
and the given integral becomes
where &(t)
is
For example,
a rational function.
-
C -dx = Cdt = sin J t
\
J
I
log
t
;
a;
hence
dx
/dx sin
* See
Hermite
3
a;
=
x
log tan--
2
lithographed Cours, 4th ed., pp. 25-28.
= l-t
2 >
TRANSCENDENTAL FUNCTIONS
237
The integral f [I/cos x~]dx reduces to the preceding substitution x y, which gives 7r/2
by means of the
V,
114]
=
dx-
/dx
x\ -
ITT
=-
log tan
cos x coscc
T
=
I
\4
ITT
x -
\4
2
-+
log tan
2/
The preceding method has the advantage of generality, but it is often possible to find a simpler substitution which is equally suc cessful. Thus, if the function /(sin x, cos x~) has the period IT, it is a rational function of tana;, F(tan x). The substitution t&nx t
=
therefore reduces the integral to the form
f F(tan x~) dx = As an example
let
us consider the integral
dx
A where A, B, C, period
IT
1
*
cos"
cos 2 x
D are
B
-f-
x
=
=
we
t,
+ C sin
x
sin
)
Hence the given
x
+D
find
1
x cos x
>
1
t
-L ~~T~
2
The integrand evidently has the
any constants.
and, setting tan x
;
sin x cos
*
sin 2 x
=
~T~
integral becomes
r
J The form
of the result will
of the denominator.
we
depend upon the nature of the roots Taking certain three of the coefficients zero,
-
find the formulae
/dx cos 2
x
r dx = -; J sin x cos x
= tan:r,
/
dx
/; When R (cos x) first
case
r-r-
=
log tan x,
COt X.
of the form
R(sm x) cos x, or of the form of variable is apparent. In the proper change should set sin x t in the second case, cos x t.
the integrand
is
since, the
we
=
=
;
It is sometimes advantageous to
make
a
to simplify the integral before proceeding
For example,
let
first
with the general method.
us consider the integral
dx a cos x
+
b sin
x
substitution in order
-f
INDEFINITE INTEGRALS
238
114
[V,
where a, b, c are any three constants. If p is a positive number and an angle determined by the equations <
= p cos
a
we
shall
p
= p sin
b ,
,
have
= V/ a
a
n
,
-f-
cos
)
o>
a 7== Va
=
2
+b
and the given integral may be written
/dx p cos (x
where x
tan y/2
= y.
2
in the
_ C J
c
-)+
b
p== Va +
=
d>
>
62
form
dy y
p cos
+
c
Let us now apply the general method, setting the integral becomes
Then
t.
sin
>
2
2dt
and the rest of the calculation presents no difficulty. Two different 2 2 2 2 c = a + b forms will be found for the result, according as p 2 c is
positive or negative.
The
integral
m cos x
"
n sin x + *p dx + o sin x + c For, let u = a cos x + b sin x + preceding.
f may and
be reduced to the let
.
;
:
us determine three constants
m cos x is
-f
a cos x
-\-
n
sin
x
+
X,
7>
p.,
=
and
MI
+
c,
such that the equation
v
du u.
\-
ax
v
The equations which determine these num
identically satisfied.
bers are
m = \a + p.b, the
first
n
= Xb
p=
p,a,
two of which determine X and
/A.
The
\c
+
v,
three constants hav
ing been selected in this way, the given integral
may
be written in
the form
+
/du Xt<
fi
+
v
.
u
Example. Let us try
dx
=
\x
-\-
u.
log u 4- v
/
J
a cos x
to evaluate the definite integral
dx 1
+
ecosx
where
|e|
+ b sin x +
c
y,
TRANSCENDENTAL FUNCTIONS
115]
it first
Considering
C
J
as
an indefinite
dx 1
_
C
2
+ ecosx
integral,
J
we
find successively
dt
+
l
vT^J
e)V
by means of the successive substitutions tanx/2 Hence the indefinite integral is equal to
As x
varies
from
arc tan
\l
I
\i+
\
to x,
the arctangent varies to ;r/V(l - e 2 ).
ll-e.
/
2 .
TIT^ V(l from to
e)/(l it/2.
C
2
~
+ (l-
e
239
=
t
f,
du 1
+ w2
= u V(l +
e)/(l
e).
x\
tan -
J
2/
c
+ e) tan x/2
increases from
Hence the given
to
+
definite integral
oo,
is
and
equal
There are also certain classes of integrals which reduction formulae exist. For instance, the formula for
115. Reduction formulae.
for
the derivative of tan n
~
l
1
j-
whence we
(tan"-
be written
may
a;)
= (?i
2
-l)tan"-
x(l+
tan 2 a;),
find ~ ka,n"xdx
=
l
n-l /tan"
x --
C tan"~
/
J
The exponent
of tan x in the integrand is diminished by two units. Repeated applications of this formula lead to one or the other of the two integrals I
=x
dx
The analogous formula
C L/UU J II
I
,
tan x dx
log cos x
for integrals of the type 1
c*r\i~ ** T*
=
fiw W-i-C
^^ -
-
^ ______ cot"-
n-l
r I I
J
r/-k r i
CUu
*
.
/cot"
*
X fit* (L3C
x dx
is
*
,
In general, consider the integral sin m x cos n xdx,
m
and n are any positive or negative integers. When one of these integers is odd it is best to use the change of variable given above. If, for instance, n 1, we should set sin x t, which 2p
where
=
=
+
2 p reduces the integral to the form />(! t ) dt. Let us, therefore, restrict ourselves to the case where
both even, that
is,
to integrals of the type
= f sin si J
2
"
a;
cos j "ccc?x,
m
and n are
INDEFINITE INTEGRALS
240
which may be written
T
*m,m
form
in the
=
sin 2m
I
~
xcos 2n xsmxdx.
l
J
2n Taking cos z since dx as the differential of an integration by parts gives
T
n
l/(2n
[
pQgZn-fl^ 2m 1 C sm = -sin 2m - xr sin r+2ft + 1 2n+lJ l
Lm
-
"*"
1
^ cos 2
2 (m
+1 a;,
a
ajcos*"aj(l-sin x)cte,
"
+1 a;
2m
1
j
~
-**
2n
I)]cos
form
in the
sin 2 7"
-f
8
/
which may be written
116
[V,
+ n)
2 (m
+
ri)
-*-!,
This formula enables us to diminish the exponent m without alter If m is negative, an analogous formula ing the second exponent. be obtained by solving the equation (A) with respect to /m _ lin
may
and replacing
m by
1
sin 1
j
m ~2
**..
:
? a;
cos 2n+1 a;
The following analogous
1-2WI
easily derived, enable
~1
2n
a;
T
1
^ m ,-i
:
2 (m
+ w)
2(m + 2
l2n
**,-
,-
^l-m,n
:
m+1 a;cos 1 ~ 8 *a; _ sin
^-
t
(
sin 2m+1 a;cos 2n
M
which are
formulae,
us to reduce the exponent of cos a;
r
m + 1)
2 (n
1-2OT
(m
w)
+1
l2n
n)
j -*,-+!*
Eepeated applications of these formulae reduce each of the num m and n to zero. The only case in which we should be unable to proceed is that in which we obtain an integral /TOiB where m + n = 0. But such an integral is of one of the types for which reduction for mulae were derived at the beginning of this article.
bers
,
m
116. Wallis formulae. There exist reduction formulae whether the exponents and n are even or odd. As an example let us try to evaluate the definite integral n
Im =
I
Jo
where
m
is
a positive integer.
7T
2 |
Jo
An
s
m m xdx,
integration
by parts gives E
7T
siu m
-1
xsinxtZx
=
[cosxsin-^]^o + (m
2
1)
|
Jo
sin m
-2
cos 2 xdx,
V,
TRANSCENDENTAL FUNCTIONS
117]
m ~ l x vanishes at both whence, noting that cosz s\n
limits,
241
we
find the
formula
7T
Im
=
(m
-
1)
2 f sin- 2 x(l -
Repeated applications of even, or to Ii
is
=1
m successively
replacing
-*a
or,
- -1 =m
Im
(13)
m
=
(m
1
--T05
this
m
if
by T
-M
-
l)(/m _
2
-
/m),
formula
ivhich leads to the recurrent
if
=
sin 2 x)<Jx
Jo
is
formula reduce the given integral to IQ = it/2 In the former case, taking m = 2p and odd.
2, 4, 6,
3 7
=
T
Im-2-
m
,
T -*
2
we
2j>,
find
JT2p
* >
-
P- 1
2
T
2n~
multiplying these equations together,
_ Similarly,
we
1
.
3
.
6
2
.
4
-
(2p .
6
.
6
1)
5 2
2p
find the formula
12.P
2
=
+1
1
.
3
.
4
.
5
2p (2j>
+
1)
A
curious result due to Wallis may be deduced from these formulae. It is m + 1 x is less than evident that the value of Im diminishes as increases, for sin
m
sin m z.
and
if
Hence
we replace I2 p + ^2 P new inequalities i>
IS P
,
-I by their values from the formulae above,
we
find the
where we have
set, for brevity,
2244 1335 It is
evident that the ratio
nitely.
-
2p 2p
2 1
2p
2p
-
1
the limit one as
n/2Hp approaches
H
factors increases indefinitely.
p
increases indefi
of product p as the number The law of formation of the successive factors is
It follows that 7T/2 is the limit of the
apparent.
117.
The
integral
/cos (ax
+ b) cos (a
x
+b
dx.
)
Let us consider
+
a product of any number of factors of the form cos (ax b), where a and b are constants, and where the same factor may occur several times.
The formula cos u cos
v
=
^+
cos (u
?
)
H
cos (u
^
v)
242
INDEFINITE INTEGRALS
[V,
117
enables us to replace the product of two factors of this sort by the of two cosines of linear functions of x hence also the product of n factors by the sum of two 1 factors each. products of n
sum
;
Kepeated applications of this formula sum of the form 2 H cos (Ax
gral to a
If
immediately integrable. /
A
sin
=
B}dx
+ B),
not zero,
is
,
cos (Ax -f
finally reduce the given inte
(Ax
each term of which
is
we have 4-
B}I
+
C,
J\.
when A = 0, /cos B dx = x cos B + C. This transformation applies in the special case of products of the form cos m a; sin n while, in the particular case
ic,
where
m
and n are both positive
For this product
integers.
may
be written
and, applying the preceding process, we are led to a sum of sines and cosines of multiples of the angle, each term of which is immediately integrable.
As an example
let
us try to calculate the area of the curve
which we may suppose given in the parametric form x = acos 0, to 2-rr for the whole curve. The y = b sin 0, where 6 varies from formula for the area of a closed curve,
A
=
-
xdy
/
ydx,
i/(C)
gives I
Jo
But we have the formula 2 (sin 6 cos 0)
Hence the area
= - sin
2
of the given curve
Sab
20
= - (1 -
cos
is 27r
[~
sin40~l
3-rrab
V,
TRANSCENDENTAL FUNCTIONS
117]
It is
now
easy to deduce the following formulae
243
.
C,
A
may be noticed in these formulae. The integrals = f* dx have the period 2?r xdx and F(x) f*sin. n is even, these integrals when other the n is odd. On when hand, general law n
cos"x
<J>(cc)
increase by a positive constant when x increases by 2?r. dent a priori that these statements hold in general. For r27r +
/-2Jr
F(x
+
2-Tr)
=
sin";rcfa;
I
+
Jo
It is evi
we have
a:
sin"
I
xdx,
Jz-rr
or /o2ir
F( x
+ 27r) =
I
I
since sin x has the period 2?r. 2v
f
sin
n
sin n a;rfa;
=
F(x~)
x dx
is
If
n
is
it is
even,
a positive quantity.
integral vanishes, since sin (x
+
I
sin n
co?a;,
i/O
i/O
i/O
integral
/^2ir
r>x
sinn a;c?ic+
+ TT) =
If
n
evident that the is
odd, the same
sin x.
Note. On account of the great variety of transformations appli cable to trigonometric functions it is often convenient to introduce them in the calculation of other integrals. Consider, for example, this integral the integral Setting x x*)*~\dx.
=
/[!/(!+
becomes f cos
= sin
d<j>
<j>
+
C.
dx
which
is
ts,n,
Hence, returning to the variable
the result already found in
105.
x,
INDEFINITE INTEGRALS
244 The
118
[V,
wx dx.
Let us now consider an integral R where /R(x)e" dx, (x) is a rational function of x. Let us suppose the function R(x) broken up, as we have done several times, into a sum of the form 118.
integral
/R(x)e x
of the form
where E(x), A l A 2 A p X 1} Xp are polynomials, and X is to its derivative. The prime given integral is then equal to the sum of the integral / E(x)e ax dx, which we learned to integrate in 85 by a suite of integrations by parts, and a number of integrals of the form ,
,
,
,
,
t
There exists a reduction formula for the case when n than unity.
is
greater
X is prime to its derivative, we can determine and which satisfy the identity A = \X + p.X
For, since
two polynomials Hence we have
A.
/*
.
and an integration by parts gives the formula
f
J
-1
;
dx.
X"
Uniting these two formulas, the integral under consideration reduced to an integral of the same type, where the exponent n reduced by unity.
Eepeated applications of
is is
this process lead to
the integral
dx,
X
where the polynomial B may always be supposed to be prime to and of less degree than X. The reduction formula cannot be applied to this integral, but if the roots of X be known, it can always be reduced to a single new type of transcendental function. For
defmiteness suppose that all the roots are real. Then the integral in question can be broken up into several integrals of the form
x
a
dx.
v,
TRANSCENDENTAL FUNCTIONS
119]
245
=a+ u= Neglecting a constant factor, the substitutions x enable us to write this integral in either of the following forms
e"
y/u>,
:
C du
dy /e"
The is
latter integral
J
y
f [I/log
it
a transcendental function which
is
u~\du
log
called the integral logarithm. 119.
Miscellaneous integrals. Let us consider an integral of the form
inx, cos
where
/
a;) eta,
an integral function of sin x and cos form
is
x.
Any term
of
this integral is of the
where
m and
n are positive
m product sin x
cos"x
of multiples of x.
two types
We
integers.
:
I
e
ax
cosbxdx,
I
Integrating each of these by parts,
C J I
e"*
cos bx
dx
e
=
ax
we
e
ax
sinbxdx.
find the formulae
a
sinbx
/pQZC ax
sin bx
dx
Hence the values of the e
ax
cos bx dx
=
b
QQg
Jjnf*
ft,
Tb
h
b
e"*
sin bx
dx
,
I
e
I
"
x
cos & x dx.
J
integrals under consideration are
=
x e"
(a cos bx -f b sin bx)L *
a*
e"*
T J I
;
b
e
Among
have seen above that the
be replaced by a sum of sines and cosines Hence it only remains to study the following
may
sm bx dx =
e
ax
+
(a sin bx 5 a2
+
r:
>
6*
b cos bx) i
2
the integrals which may be reduced to the preceding the following cases
types we may mention
:
I
/(log x) x dx
I
/(x) arc sin x dx
,
,
I
/(arc sin x) dx
I
/(x) arc tan x dx ,
,
INDEFINITE INTEGRALS
246 where
/
denotes any integral function.
[V,
In the
first
EM.
two cases we
should take log x or arc sin x as the new variable. In the last two we should integrate by parts, taking /(#) dx as the differential of another polynomial F(x), which would lead to types of integrals already considered.
EXERCISES Evaluate the indefinite integrals of each of the following functions
1.
- x 8 - 3x 2 2 8 (x + I)
z* (x*
+
2
x
I)
3
(x
+ 3
+
x
1
+ Vl +
x2
identities 2
(x
4.
a2 ) 2
P
/
x"
1
t*
+2
of the folium of Descartes
x3
+
fy dx,
y
3
-
ay2 (2y
+
=
3a)
Saxy
where x and y
-
y 2 (a
,
one of the following
satisfy
x)
=
x3
y
,
2
(x
+
y 2)
=
a (y 2
-
x2 )
Derive the formulae snv -^x cos(n
+
,
,
,
sin"
=
/sin /cos sm n -
l
x sin (n
l)xdx
x cos nx
+ l)xdx =
1
x sin (n
+
l)xdx
=
\-C,
n
cos- 1 xcos(n cos"-
C,
f-
n
n xsinnx
+ l)xdx= /cos"x
sinnx
nx cos xcosnx
+
C,
+
(7.
[EDLER.]
Evaluate each of the following pseudo-elliptic integrals
+
C
x2 )dx
/(l 6.
:
0.
:
/.
5.
cos2 x
1)
TV* tan T L til X
,
V a -f
Find the area of the loop
Evaluate the integral
x
x
+ Vx(x +
1
x2
3.
+ Vl + i _ Vx
I
1
Vx + Vx +
XC*COSX,
2.
x
.
+ VI + x i_vT^~x
1 1
3
I)
:
(l-x 2 )dx
:
.
,.>
Reduce the following
integrals to elliptic integrals
:
E(x)dx
Va(l
+ x6 + )
6x(l
+
x*)
+
cx 2 (l
+
x2)
R(x)dx 8 2 Va(l + x ) + 6x (1 +
where R(x) denotes a rational function.
x*)
+ ex*
+
dx*
.
V,
EXERCISES
E.] Let
7*.
Then there
a, 6, c,
247
d be the roots of an equation of the fourth degree P(x) form
=
0.
exist three involutory relations of the
+ Ni
Mix"
t
which interchange the roots by
<=
2,
1,
If the rational
pairs.
3,
function f(x) satisfies the
identity
VP
the integral f[f(x)/ (x)] dx matique, Vol. XV, p. 106). 8.
The
rectification of a curve of the type
a binomial differential. 9.
If
a
pseudo-elliptic (see Bulletin de la
is
>
1,
show
= Ax*
y
SocMM matM-
leads to an integral of
Discuss the cases of integrability.
that
+1 /_,
,
dx
-
(a
Vl - x 2
x)
Va2 -
1
Hence deduce the formula 1
X ^1C
-
J52
>
0,
dx
-
l
t
10. If
n
_ ~
+ 2Bz +
C)
[Apply the reduction formula of 11.
-
1)
2.4.6..-2n
show that
dx 2
(^lx x X*"*
1. 3. 5-- -(2re
_ ~
x2
1
.
3
.
5
(2ft
2
.
4
.
6
(2n
-
3) 2)
104.]
Evaluate the definite integral
C
J
o
sin 2 xdx 1
+
12. Derive the following formulae
C
2a cos x
+
1
a2
:
dx
1
Vl - 2ax + a 2 Vl -
JL LI
+
-
2/3x
ax)(l
-
/3x)
- 2ax + a 2)(l - 2^x +
dz
TT
p?) Vl
x2
2
2 1
-
13*. Derive the formula
f
x m ~ l dx
I/O ,
where
m
and n are positive integers
partial fractions.]
*
it
1
nsin
(m
M* n [Break up the integrand into
INDEFINITE INTEGRALS
248 From
14.
the preceding exercise deduce the formula
x n - dx }
/
Setting I
p
+
(p
(P
= ft q (t +
+
q
~
+
1
i/O
15.
[V, Exs
l)Ip
,
!)*-*,
q
l)
it
x
p
deduce the following reduction formulae
dt,
= + !( + !)*+!/_,, = i9 + (t + I) -* ~ (2 + q-p)I- P +
:
*
1
l
and two analogous formulae for reducing the exponent
i,
g
,
q.
16. Derive formulae of reduction for the integrals
_
7
xdx 2 V2x + 2Bx + C J
C
C
z _
~J
dx (x
-
2 a)* V^4z
+ 2Bx + C
17*. Derive a reduction formula for the integral
C
xn dx
J vnr^ Hence deduce a formula analogous
to that of Wallis for the definite integral 1
dx
vl-x*
o
Has
18.
the definite integral
dx
/
1+
Jo^ ^ a
finite
19.
value ?
Show
that the area of a sector of an ellipse
and a radius vector through the focus
^j
+
(1
tSQ
where p denotes the parameter W/a and
and show that the area A
=
ab
e the eccentricity.
A
where
is
20.
MNT,
=
t,
t
=
u V(l
Applying the gen
+
e)/(l
e)
succes
in question is
I
arc tan
u
may
=
the eccentric anomaly.
e
)
1
V Also show that this expression
focal axis
e cos w) 2
method, make the substitutions tan w/2
sively,
bounded by the
is
r
2 = P
eral
x 4 sin 2 x
+ M/
be written in the form
ab
.
(*
See
-
e sin
p. 406.
Find the curves for which the distance NT, or the area of the triangle is constant Construct the two branches of the curve. (Fig. 3, p. 31). [Licence, Paris, 1880; Toulouse, 1882.]
EXERCISES
V, Exs.j
21*. Setting
X 2n + l
An = 2
.
4
6
.
/.I
z 2)"cosxzdz
(1 ^
/
2n Jo
.
249
t
derive the recurrent formula
From
this
deduce the formulae
AZ P = Ut p sin x + Vzp cos z A.i p
+1
= Uz p +
1
sin
x
+
,
Vz p + 1 cos x
,
where UZ P V% p UV P + I, V^^ + are polynomials with integral coefficients, and where 72p and U^ p + \ contain no odd powers of x. It is readily shown that these formulae hold when n = 1, and the general case follows from the above ,
i
,
recurrent formula.
The formula
we assume
for
that
Ao p enables us 2
/4 relation of the form 7T
=
to show that n 2 is incommensurable. For if and then replace x by ir/2 in A^ p we obtain a
b/a,
a 2
,
.
4
.
6
f
4p Jo
V -*)
COB
^
2
where HI is an integer. Such an equation, however, is impossible, for the nand side approaches zero as p increases indefinitely.
right*
CHAPTER VI DOUBLE INTEGRALS I.
DOUBLE INTEGRALS METHODS OF EVALUATION GREEX S THEOREM
two variables. Let 2 = f(x, y} be a two independent variables x and y which is contin uous inside a region A of the plane which is bounded by a closed contour C, and also upon the contour itself. A number of proposi 70 for a continuous function tions analogous to those proved in variable can be shown to hold for this function. For of a 120. Continuous functions of
function of the
single A can be divided into instance, given any positive number c, the region the values of z at between the that a, such in difference way subregions
any
points (x, y), (x
tivo
We
,
y
)
shall always proceed
follows
:
same subregion is less than e. of successive subdivisions means by in the
as
A divided
into subregions by drawing to the two axes at equal dis parallels The corre tances 8 from each other.
Suppose the region
sponding subdivisions of
A
are either
squares of side 8 lying entirely inside C, or else portions of squares bounded in
Then, if the prop were untrue for the whole region A, it would also be untrue for at least one of the subdivisions, say A^. Sub dividing the subregion A l in the same
part by an arc of C. osition x FIG. 23
manner and continuing the process
indefinitely,
we would obtain a for An A n lies between
A lf sequence of squares or portions of squares A, which the proposition would be untrue. The region
,
,
,
= a n and x = b n which are parallel to the y axis, and the two lines y = c n y = dn which are parallel to the x axis. As n increases indefinitely a n and bn approach a common limit A, and c n and dn approach a common limit /A, for the numbers for example, never decrease and always remain less than a fixed
the two lines x
,
,
,
,
number.
It follows that all the points of 250
A H approach a limiting
VI,
INTRODUCTION
120]
GREEN
THEOREM
S
251
The rest of point (\, //,) which lies within or upon the contour C. the reasoning is similar to that in 70 if the theorem stated were ;
untrue, the function f(x, y) could be shown to be discontinuous at the point (A, /*), which is contrary to hypothesis. Corollary. Suppose that the parallel lines have been chosen so near together that the difference of any two values of z in any
one subregion is less than e/2, and let -^ be the distance between the successive parallels. Let (x, y*) and (x y ) be two points inside or upon the contour C, the distance between which is less than rj. 1
,
These two points will
two
either in the same subregion or else in which have one vertex in common. In
lie
different subregions
either case the absolute value of the difference
f(x,y}-f(x
,y<}
cannot exceed 2e/2 = c. Hence, given any positive number number 17 can be found such that
e,
another
positive
\f(x,
y}-f(x>,
y
)\<
whenever the distance between the two points (x, y*) and (x y ), which lie in A or on the contour C, is less than In other words, any func rj. ,
tion
which
is
continuous in
A and on
its
boundary C
is
uniformly
continuous.
From the preceding theorem it can be shown, as in 70, that every function which is continuous in A (inclusive of its boundary) is neces sarily finite in A.
If
M be the upper limit and m the lower limit of M m
the function in A, the difference is called the oscillation. The method of successive subdivisions also enables us to show that the
m
M
function actually attains each of the values and at least once inside or upon the contour C. Let a be a point for which z and b a point for which z M, and let us join a and b by a broken
=m
=
which
lies entirely inside C. As the point (x, y) describes this z is a function continuous of the distance of the point (x, y) line, from the point a. Hence z assumes every value p. between and
line
m
M at least
once upon this line
Since a and b can be joined ( 70). an infinite number of different broken by lines, it follows that the f unction assumes value between at an infinite and f(x, ?/) every
m
number
A
of points
which
lie
M
inside of C.
A of the plane is said to be less than I in all its a circle of radius I can be found which entirely
finite region
dimensions
if
encloses A.
A variable
region of the plane
is
said to be infinitesimal
DOUBLE INTEGRALS
252
[VI,
121
if a circle whose radius is arbitrarily preaswhich eventually contains the region entirely signed can be found a within it. For example, square whose side approaches zero or au axes approach zero is infinitesimal in all its ellipse both of whose On the other hand, a rectangle of which only one side dimensions. approaches zero or an ellipse only one of whose axes approaches zero
in all
its
dimensions
not infinitesimal in
is
all its
121. Double integrals.
into subregions a x , a 2
the subregion
a,-,
and
dimensions.
Let the region A of the plane be divided be the area of a n in any manner, and let
-,
,
M
{
u>,-
and
m,-
the limits of
/(a;,
y) in a
t
.
Consider
sums
the two
each of which has a definite value for any particular subdivision None of the sums are less than ml* where ft is the area of
of A.
m
is the lower limit of f(x, y) the region A of the plane, and where hence these sums have a lower limit /. Likewise, in the region A is the upper limit none of the sums s are greater than 3/ft, where ;
M
A hence
these sums have an upper limit / Moreover it can be shown, as in 71, that any of the sums S is greater than or equal to any one of the sums s; hence it follows of f(x, y) in the region
.
;
that />/
.
If the function f(x, y) is continuous, the sums S and s approach a common limit as each of the subregions approaches zero in all its
For, suppose that rj is a positive number such that the oscillation of the function is less than c in any portion of A which
dimensions.
less in all its
is
a2
,
M
i
less
,
dimensions than
a n be less in
nii
all its
will be less than
than
eft,
where S
ft
If each of the subregions a if each of the differences rj,
77.
dimensions than e,
s will be and hence the difference S But we have
denotes the total area of A.
-s =
S-I+I- /
where none of the quantities S
7,
+
/
/
,/
7
-*, s
can be negative.
7 <eft; and since e is an arbitrary posi Hence, in particular, / Moreover each of the numbers tive number, it follows that 7 = / / and / s can be made less than any preassigned number by S .
*If f(x, y)
TRANS.
is
a constant k,
M = m = Mf =
= m<
k,
and S
= s=
mft
= MQ.
VI,
a proper choice of /,
GREEN S THEOREM
INTRODUCTION
121]
which
Hence the sums
e.
>S
and
s
253
have a common limit
called the double integral of the function f(x, y\ extended
is
over the region A.
denoted by the symbol
It is
J -//
J(A)
and the region A If
is called the field of integration. be point inside or on the boundary of the subany 77,) it is evident that the sum 2/(, 77,-) to, lies between the two
(|,.,
region
sums S and
s or is equal to one of them. It therefore also approaches the double integral as its limit whatever be the method
of choice of the point (,-, 77,). The first theorem of the mean
may be extended without difficulty Let f(x, y) be a function which is continuous in A, and let y) be another function which is continuous and which has the same sign throughout A. For definiteness we shall to
double integrals. (x,
suppose that
<$>(x,
f(x, y] in A,
Adding
it is
y)
positive in A.
is
If
M and m are the limits of
evident that*
these inequalities and passing to the limit,
all
we
find the
formula I
I
J where
/x
lies
the value
/A
this in the
(1)
J(A)
Ax
>
y)
between
x y)dxdy >
= nl J
*(*
I
J(A)
M and m.
at a point (,
77)
Since the function f(x, y) assumes inside of the contour C, we may write
form
Jff J(A)
f(x,
y)4>(x,
y)dxdy =/(,
77)
\\ J
4>(x,
J
which constitutes the law of the mean for double integrals. If = 1, for example, the integral on the right, ffdx dy, extended y~)
<(a:,
over the region A, is evidently equal to the area In this case the formula (1) becomes
ff
(2)
*
If
f(x, y)
equations.
f(x, y) dx dy
J J(A) is
a constant k,
we
shall
The following formula
have
=
fl/(
M = m = k,
77)
O
of that region.
.
and these inequalities become M= k. TRANS.
holds, however, with
.
(
DOUBLE INTEGRALS
254
[VI,
122
To the
analytic notion of a double integral corre Let f(x, y) be notion of volume. sponds the important geometric contour C. a closed inside and continuous is which a function upon 122. Volume.
We
shall further suppose for definiteness that this function is posi Let S be the portion of the surface represented by the equa tive.
T whose projection =f(x, y) which is bounded by a curve denote shall C. We contour is the by E the por upon the xy plane the cylinder and the surface the bounded of tion S, xy plane, by space A of the The C. is section whose right region xy plane which is in C subdivided contour the bounded by any manner, let a,- be being the area of c and contour a bounded the one of by subregions c cuts curve is the section whose The this subregion. right cylinder and Let a curve s bounded 5 a the surface of p out by y portion tion z
t-,
o>
f
{
-
{
t
.
{
from the xy plane are a mini Pf be the points of s whose distances a mum and maximum, respectively. If planes be drawn through these two points parallel to the xy plane, two right cylinders are and whose altitudes are the obtained which have the same base {
o>
M
and m of the function /(cc, y) inside the contour c,, respec volumes Vt and v { of these cylinders are, respectively, The tively. The sums S and s considered above therefore repre and w,-m,-.* co, and ^v of these two types of cylin sent, respectively, the sums 2F limit of these two sums the volume common the call shall We ders. It E of the of may be noted, as was done in the case space. limits
-
t
{
M
f
f
t
portion of area ( 78), that this definition agrees with the ordinary concep tion of
meant by volume. S lies partly beneath the xy plane, the double integral to the the sign represent a volume if we agree to attach
what
is
If the surface will still
volumes of portions of space below the xy plane. It appears then that sum of volumes, just as every double integral represents an algebraic a simple integral represents an algebraic sum of areas. The limits of are replaced in the case of a integration in the case of a simple integral double integral by the contour which encloses the field of integration. 123. Evaluation of double integrals.
The evaluation
of a double
evaluations of two simple integral can be reduced to the successive the field of integration where the case consider us first Let integrals.
*By the volume of a right cylinder we shall understand the limit approached by in the volume of a right prism of the same height, whose base is a polygon inscribed a right section of the cylinder, as each of the sides of this polygon approaches zero. but is useful in showing that the [This definition is not necessary for the argument, definition of
volume
in general agrees with our ordinary conceptions.
TRANS.]
VI,
INTRODUCTION
123]
GREEN
THEOREM
S
255
R bounded by the straight lines x = x x = X, x X and y F. Suppose this rectangle where Y, y to be subdivided by parallels to the two axes x = x y = yk = = The of the small k area n rectangle 1, 2, TO). 1, 2, (i R ik bounded by the lines x = #,_ a; = y = yk _ y = y k is a rectangle
is
y = y
,
=
,
<
<
( ,
,
,
;
or,-,
Hence
the double integral
where
(,*,
inside or
sum
the limit of the
is
is
any point one of the upon r) ik )
R
sides of
l ,
ik
.
We shall employ the indetermination (it>
Vik)
i
the
of
n order
the calculation.
mark is
first
points
to simplify Let us re
of all that
if
/(a-)
a continuous function in
the interval
(a,
),
and
the interval
if
(a,
b~)
be subdivided in any such #,-)
can be found in each subinterval (x i _ l
manner, a value
,
that
For we need merely apply the law of the mean the subintervals
for integrals to each of
&) to find these values of
&. the portion of the sum S which arises from the row of rec = a:,-.! and x x is tangles between the lines x (a, Xj), (a^,
a- 2
),
,
(#_,,
Now
=
(
^ Let us take ^ a = |/2 = in such a way that the /(*,-!
i?n)(yi
=
^ im
= x^
l}
2
)G/2
-
//i)
+
and then choose
i;
n
-
2/0) +/(,-!> i7,-s)(y
!/
way
[ ,
for each of the
two consecutive parallels to the y (5)
S = *(or )(x
1
-
77,2,
- yO H
Y equal to the integral f f(x i _- y)dy, where the integral If evaluated under the assumption that x i _ l is a constant.
is
ceed in the same
,
sum
is
to be
we pro
rows of rectangles bounded by
axis,
we
finally find the equation
DOUBLE INTEGRALS
256 where we have
[VI,
123
set for brevity
*00 =
/
f(x y) dy>
JV*
This function
defined by a definite integral, where x
4>(x),
is
con
As all the sidered as a parameter, is a continuous function of x. x i _ approach zero, the formula (5) shows that S intervals x approaches the definite integral i
l
..r
(#)
dx
.
Jx.
Hence the double
(
6)
integral in question is given
f JfJ(,R)
/(*>
!/)
dxdy
= f J*t
dx
by the formula
Aff(x,
y) dy
.
In other words, in order to evaluate the double integral, the function f(x, y) should first be integrated between the limits y and Y, regard ing x as a constant and y as a variable ; and then the resulting func tion^ which is a function of x alone, should be integrated again between the limits x
If
and X.
we proceed
in the reverse order,
i.e. first
evaluate the portion
S which comes from a row of rectangles which lie between two consecutive parallels to the x axis, we find the analogous formula of
I
J/ J(R)
A comparison
/(*>
x;r
is
/>r
dx
Jx
An
=
I
dy
Jy
I
Jx9
f(x, y)dx.
of these two formulae gives the
I
which
y)dxdy
I
J
f(x,
V(S
^x
/->Y
= y) dy
I
dy
Jy
I
Jx
f(x,
y)dx
}
called the formula for integration under the integral sign. is that the limits x X, y , Y
essential presupposition in the proof
are constants, and that the function f(x, y} the field of integration.
Example. Let z
cc
new formula
^-
/JU>*
= xy/a.
Then the
,
is
continuous throughout
general formula gives
VI,
GREEN S THEOREM
INTRODUCTION
123]
In general, if the function f(x, alone by a function of y alone, /
J The two
/
(x)$(y)dxdy
=
shall
I
Jxn
J(R)
the product of a function of x
is
y*)
we
257
have
$(x)dx x
I
J
.I
integrals on the right are absolutely independent of each
other.
Franklin * has deduced from this remark a very simple demonstration of cer and f (x) be two functions
Let tain interesting theorems of Tchebycheff. which are continuous in an interval (a, b), where
a
extended over the square bounded by the
=
lines
x
(x)
<
b.
o,
x
Then
=
6,
y
the double integral
=
a,
y
=
b is equal
to the difference
2(6
-
a)
C Ja
2
(x)\l/(x)dx
C Ja
(p(x)dx
x C
\f/(x)dx.
*/a
But all the elements of the above double integral have the same sign if the two lunctions 0(z) and ^(z) always increase or decrease simultaneously, or if one of them always increases when the other decreases. In the first case the two func tions and \f/(x) ^(y) always have the same sign, whereas they have Hence we shall have opposite signs in the second case. (f>(x)
(f>(y)
(b
-a) C
Ja
<j>(x)t(x)dx
whenever the two functions <j>(x)
out the interval
(a, b).
(b-
a)
On f Ja
>
C x C ^(x)dx J a $(x)dx Ja
and \f/(x) both increase or both decrease through we shall have
the other hand,
(x)^(x)dx
<
f 0(z)dz x f Ja
Ja
whenever one of the functions increases and the other decreases throughout the interval.
=
The sign of the double integral is also definitely determined in case 0(z) ^(z), for then the integrand becomes a perfect square. In this case we shall have (b
-a]
whatever be the function 0(z), where the sign of equality can hold only when is
The
a constant. solution of an interesting problem of the calculus of variations may be and Q be two fixed points in a plane whose this result. Let
P
deduced from
coordinates are
(a,
A) and
(6,
any curve joining these two
Let y =/(z) be the equation of where /(z), together with its first derivative
B), respectively.
points,
* American Journal of Mathematics, Vol. VII, p. 77.
DOUBLE INTEGRALS
258
124
[VI,
The problem is to (a, b). which the integral f^ y 2 dx is a and noting minimum. But by the formula just found, replacing by that /(a) = A and f(b) = B by hypothesis, we have
/
(x), is
supposed
be continuous in the interval
to
find that one of the curves
y=f(x)
for
y"
(x)
*dx^(B(b-a) )Cy a The minimum value
of the integral
is
therefore (B
actually assumed when y is a constant, fixed points reduces to the straight line PQ. is
i.e.
A)*/(b a), and that value the curve joining the two
when
124. Let us now pass to the case where the field of integration is bounded by a contour of any form whatever. We shall first suppose that this contour is met in at most two points by any parallel to the y axis. We may then suppose that it is composed of two straight lines x = a and x = b (a 6) B and two arcs of curves APB and A QB whose equations are A re~ YI = : (cc) and F2 = $2 spectively, where the functions and are continuous be 2 tween a and b. It may happen that the points A and A coin cide, or that B and B coin <
<
(#)>
<
!
FIG. 25
cide, or both.
This occurs, for
a convex curve like an ellipse. Let us again subdivide the field of integration R by means of parallels to the axes. Then we shall have two classes of subregions regular if instance, if the contour
is
:
they are rectangles which
lie
wholly within the contour, irregular
they are portions of rectangles bounded in part by arcs of the contour. Then it remains to find the limit of the sum if
where
o>
is
the area of any one of the subregions and (,
rj)
is
a point
in that subregion.
Let us
first
evaluate the portion of S which arises from the row
=
=
xi _ l x x of subregions between the consecutive parallels x These subregions will consist of several regular ones, beginning with a vertex whose ordinate ordinate
is
point (,
77)
is
y ^
Y and t
,
.
going to a vertex whose
Y and
several irregular ones. Choosing a suitable in each rectangle, it is clear, as above, that the portion
y"
^
2
,
S which comes from these regular rectangles may be written the form of
(
in
VI,
GREEN S THEOREM
INTRODUCTION
124]
(,-_ i,
259
y)dy.
and 2 (a;) Suppose that the oscillation of each of the functions x ) is less than 8, and that each of the l} differences yk yk _ is also less than 8. Then it is easily seen that <
in each of the intervals (x { _
t
l
=
=
x _ 1 and x the total area of the irregular subregions between x xt is less than 48(x ;_,), and that the portion of S which arises i
-
t
x _^) in absolute value, from these regions is less than 4:HB(x where H is the upper limit of the absolute value of f(x, y) in the whole field of integration. On the other hand, we have {
/(*<-i>
y)dy
and since \Yl
/(,-i,
I
y}dy+
Jf\
y
\
and |F2
y"\
ny
s*Yi
f*Yt
=
XV"
t
I
Jy
+
,
/
J Y*
are each less than 28,
we may
write
f/fa-u y)dy= JC f(xM)
Jy
}
,
The portion of S which arises from the row of subregions under consideration may therefore be written in the form
where
0,-
between
lies
1
+ 1.
and
The sum SH8 2,O
i
(x
i
ar
_j)
than 87/8(6 a) in absolute value, and approaches zero with which may be taken as small as we please. The double integral
less
therefore the limit of the
is 8,
is
sum
where
Hence we have the formula
(
7)
In the
f JfJ(R)
f(*, V)
first
the limits constants.
dxd = fdx f Ja JY
integration x are l
Y and F2
l/
f(x, y) dy.
l
is
to be regarded as a constant, but
themselves
functions
of
x and not
DOUBLE INTEGRALS
260
[VI,
124
Example. Let us try to evaluate the double integral of the function xy/a over the interior of a quarter circle bounded by the axes and the circumference
+
X2
The limits for x are Hence the integral
and R, and
The
x
is
_
_
#2
.
constant, y
may vary from
to
VR
2
z2
is
p r^
Jo
if
yl
Jo
_.
, 2
Jo
<
value of the latter integral
is
i(fe
L
easily
=
Jo
/.(. ^ J
shown
2
be R*/8a.
to
When the field
of integration is bounded by a contour of any form whatever, may be divided into several parts in such a way that the boundary of each part is met in at most two points by a parallel it
We
might also divide it into parts in such a way that y axis. the boundary of each part would be met in at most two points by any line parallel to the x axis, and begin by integrating with respect to x. Let us consider, for example, a convex closed curve which lies to the
= a, x = b, y = c, y = d, the four which lie A, B, C, D, respectively, for which x points upon and y2 = 2 (cc) or y is a minimum or a maximum.* Let y = be the equations of the two arcs ACS and ADB, respectively, and = 2 (y) be the equations of the two arcs CAD let o^ = (y) and x 2 and are continu and CBD, respectively. The functions ous between a and b, and i/^ (y) and i/^ (y) are continuous between c and d. The double integral of a f unction /(x, y), which is continuous inside this contour, may be evaluated in two ways. Equating the inside the rectangle formed by the lines x
<
v
1/^1
<#>i()
\j/
i(x~)
values found,
we
obtain the formula
~& I
(8)
Jo,
fa(x")
~y 2
dx
I
f(x,
~fi
y}dy
=
\
t/c
i/i/j
dy
-.T, I
/(or,
y)dx.
lyj-,
It is clear that the limits are entirely different in the
two
integrals.
Every convex closed contour leads to a formula of this sort. For example, taking the triangle bounded by the lines y = 0, x = a, y = x as the field of integration, we obtain the following formula,
which
is
due
to
/
Jo
Lejeune Dirichlet
dx
I
Jo
f(x,
*The reader
:
y)dy=\dy\ Jo Jy is
f(x, y)dx.
advised to draw the figure.
VI,
GREEN S THEOREM
INTRODUCTION
125]
261
125. Analogies to simple integrals. The integral JJf(t)dt, considered as a There exists an analogous theorem for function of x, has the derivative /(x). double integrals. Let f(x, y) be a function which is continuous inside a rec
bounded by the straight
tangle
lines
x
= a,
x
A, y
The double integral of /(x, y) extended over a x = a, x = X, y = b, y = F,(a A, b Y
X X and Y of the variable corner, that
<
<
<
nates
<
b,
y
=
J5,(a
<
A,
b
<
B).
rectangle bounded by the lines .B), is a function of the coordi
is,
F(X,Y)= JCa dxCf(x,y)dy. Jb Setting *(x)
= fb
/(x, y) dy, a first differentiation with respect to
= *(X) = f
A
second differentiation with respect to 2
/<-*,
X gives
V)dy.
F leads to
the formula
F
(9)
The most general function u(X, Y) which satisfies the equation (9) is evi dently obtained by adding to F(X, Y) a function z whose second derivative It is therefore of the form d 2 z/dXdY is zero. u(X, Y)
(10)
where
and (X)
functions
function
\1<(Y)
= C
A
Ja
Y dx C f(x, y) dy Jb
X=a
Y= b
and then
V(Y) = whence we
+ f (F)
,
are two arbitrary functions (see 38). The two arbitrary such a way that u(X, Y) reduces to a given
may be determined in V(Y) when X = a, and
Setting conditions
+
to another given function
in the preceding equation,
+ *(F)
0(a)
U(X) when Y b. we obtain the two
U(X) - t(X) + f (6)
,
,
find
= F(F) - *(a) and the formula
=
(10) takes the
u(X, Y)
(11)
*(&)
,
= ( a
F(6)
- 0(a)
(X)
= U(X) - F(6)
form
X
*/
,
dx ( /(x, Jb
y)
dy
+ U(X) + F(F) - F(6)
.
Conversely, if, by any means whatever, a function u(X, Y) has been found which satisfies the equation (9), it is easy to show by methods similar to tne above that the value of the double integral is given by the formula
(12)
f dx f /6
/(x,
y)dy
=
u(X, Y)
-
u(X,
b)
-
u(o,
F)
+
u(a, b).
i/a
analogous to the fundamental formula (6) on page 156. is in a sense analogous to the formula for integration Let A be a finite region of the plane bounded by one or more curves
This formula
is
The following formula by
parts.
DOUBLE INTEGRALS
262
[VI,
126
A function /(a;, y) which is continuous in A varies between its and its maximum V. Imagine the contour lines /(x, y) = v drawn where v lies between v and F, and suppose that we are able to find the area of the portion of A for which /(x, y) lies between v and v. This area is a func tion F(v) which increases with u, and the area between two neighboring contour of
any form.
minimum
lines
is
t
F(v
+ A) -
F(v)
= AvF
+
(v
If this area
0Av).
be divided into
infinitesi
mal portions by lines joining the two contour lines, a point (, 77) may be found in each of them such that /(, i}) v + 6A.v. Hence the sum of the elements of the double integral / ffdxdy which arise from this region is (V
It follows that the
that
is
double integral
is
equal to the limit of the
sum
to say, to the simple integral
v v
This method
by two contour
is
F
(v)
dv
= VF( F) -
especially convenient
when
rv I
the
F(v) dv
.
field of
integration
/(x, y)
=
For example, consider the double the interior of the circle x 2
+
2 j/
v
/(x,
,
y}=
bounded
V.
// Vl + x 2 + y 2 dx dy extended over we set v = Vl + x 2 + y 2 the field of contour lines v = 1 and v = \/2, and the
integral
=
1.
If
,
bounded by the two function F(v), which is the area of the circle of radius Vv 2 Hence the given double integral has the value I). 7[(v*
integration
is
lines
is
/v/iT |
The preceding formula
2itv 2 dv
=
2ir
3
J\ is
(2V5-1).
1, is
equal to
*
readily extended to the double integral
where F(o) now denotes the double integral y)dxdy extended over that portion of the field of integration bounded by the contour line v =f(x, y). ff(x,
126. Green s theorem.
tive of a
known
integrations integration.
may
If the function f(x, y}
is
the partial deriva
function with respect to either x or y, one of the be performed at once, leaving only one indicated
This very simple remark leads to a very important is known as Green s theorem.
formula which *
Numerous
applications of this method are to be found in a series, Vol. IV, p. 233).
(Journal de Liouville, 1st
memoir by Catalan
VI,
GREEN
INTRODUCTION
126]
Let us consider
first
S
a double integral
THEOREM
263
// cP/dy dx dy extended
over a region of the plane bounded by a contour C, which is met in at most two points by any line parallel to the y axis (see Fig. 15, p. 188).
A and B
Let
be the points of
maximum, Bb
respectively. meets C in two points
A m
at
which x
is
a
minimum and a Aa and
m
Then the double
respectively. to y may be written
and y z
,
integral after integration with respect
b
CCcP J
dxd>
JJ
C
parallel to the y axis between x and z whose ordinates are y-^
~fy
= C dx C"^P dy= C J j ^y j
But the two integrals fa P(x, y\)dx and fa P(x, y^)dx are line 2 B, respectively; hence l B and integrals taken along the arcs the preceding formula may be written in the form
Am
Am
(13)
where the line integral is to be taken along the contour C in the direction indicated by the arrows, that is to say in the positive In order to extend sense, if the axes are chosen as in the figure. the formula to an area bounded by any contour we should proceed as above ( 94), dividing the given region into several parts for each
which the preceding conditions are satisfied, and applying the for mula to each of them. In a similar manner the following analogous
of
form
u
is
easily derived
:
// 1?
>
<
where the
line integral is
always taken in the same sense. we find the formula
Sub
tracting the equations (13) and (14),
(15)
where the double integral This is Green s formula ;
extended over the region bounded by C. Just applications are very important.
is
its
merely point out that the substitution Q = x and P = y gives the formula obtained above ( 94) for the area of a closed curve as a line integral.
now we
shall
DOUBLE INTEGRALS
264
AREA OF A SURFACE
CHANGE OF VARIABLES
II.
127
[VI,
In the evaluation of double integrals we have supposed up to the present that the field of integration was subdivided into infinitesimal
We are now going
rectangles by parallels to the two coordinate axes. to suppose the field of integration subdivided by any
two systems of
curves whatever.
Let u and v be the coordinates of a point
127. Preliminary formula.
with respect to a set of rectangular axes in a plane, x and y the coor dinates of another point with respect to a similarly chosen set of rectangular axes in that or in some other plane.
x =f(u, v),
(16)
=
y
(u,
The formulae
v)
establish a certain correspondence between the points of the shall suppose 1) that the f unctions /(w, v) and planes.
We
two
(,
v),
together with their first partial derivatives, are continuous for all points (u, v) of the uv plane which lie within or on the boundary of
A l bounded by a contour C l 2) that the equations (16) transform the region AI of the uv plane into a region A of the xy plane bounded by a contour C, and that a one-to-one correspond ence exists between the two regions and between the two contours a region
;
in such a way that one and only one point of A 1 corresponds to any point of A 3) that the functional determinant A D(f, v) does not change sign inside of C lt though it may vanish at certain points of A i.
=
;
Two
<)/Z>(w,
When
the point (u, v) describes the con may arise. in the positive sense the point (x, y) describes the contour C either in the positive or else in the negative sense without ever
tour
C
cases
l
We
shall say that the corre reversing the sense of its motion. spondence is direct or inverse, respectively, in the two cases. The area fl of the region A is given by the line integral
Q=
I
J(.C
taken along the contour
new
variables
u and
C
in the positive sense. In terms of the by (16) this becomes
v defined
ft
=
I
f(u, v)
d<j>(u,
v)
,
Ac,)
where the new integral is to be taken along the contour C l in the should be taken positive sense, and where the sign -f- or the sign
VI,
CHANGE OF VARIABLES
127]
265
according as the correspondence is direct or inverse. Green s theorem to the new integral with x = u, v = y, P
Q
=/
we
Applying
=
fd/du,
find
d/dv,
c/u
cv
D(u, v)
whence
A\
dudv or,
,
applying the law of the mean to the double integral, D(f, *)
n=
(17)
where (, rf) is a point inside the contour C l} and n l is the area of the region A v in the uv plane. It is clear that the sign -f or the should be taken according as A itself is positive or negative. sign
Hence
the correspondence is direct or inverse according as
A
is
positive
or negative.
The formula (17) tional determinants
moreover establishes an analogy between func and ordinary derivatives. For, suppose that the
region A i approaches zero in all its dimensions, all its points approach ing a limiting point (u, v~). Then the region A will do the same, and the ratio of the two areas O and f^ approaches as its limit the abso lute value of the determinant A. Just as the ordinary derivative is the limit of the ratio of two linear infinitesimals, the functional determinant is thus seen to be the limit of the ratio of two infinites
imal areas.
From this point of view the formula (17) mean for derivatives.
is
the analogon
of the law of the
Remarks. The hypotheses which we have made concerning the correspondence between A and AI are not all independent. Thus, in order that the correspond ence should be one-to-one, it is necessary that A should not change sign in the regional of the uv plane. For, suppose that A vanishes along a curve 71 which divides the portion of AI positive from the portion
where A where A
is
is
Let us consider a small arc negative. mini of -yi and a small portion of AI
which contains the arc mini. This portion is composed of two regions a\ and a\ which are separated by mini (Fig. 26).
When the point (u, v) describes the region a\, where A is positive, the point (x, y) describes a region a bounded by a contour mi HI pi mi and
mnpm
Fio. 26
mnpm, and
the two contours
are described simultaneously in the positive sense. the point (, v) describes the region af, where A is negative, the point
When (x,
y)
DOUBLE INTEGRALS
266
[VI,
128
whose contour nmqr is described in the negative sense as described in the positive sense. The region a must therefore Hence to any point (x, y) in the common part cover a part of the region a. nrm correspond two points in the uv plane which lie on either side of the describes a region a
n\m\q^n\
line
is
mini.
X
= x, Y = y 2 for which A = 2 y. which encloses a segment a& of the x axis, it is evident that the point (X, Y) describes two regions both of which axis and both of which are bounded by the same segment AB of lie above the A sheet of paper folded together along a straight line drawn upon it that axis. gives a clear idea of the nature of the region described by the point (X, Y}. The condition that A should preserve the same sign throughout AI is not suf = x2 y 2 Y = 2 xy, In the example ficient for one-to-one correspondence. 2 is 2 = if But 4 the Jacobian A always positive. (r, 6) and (.R, w) are the (x + y ) and the formulae of of the coordinates respectively, points F), (x, y) (X, polar transformation may be written in the form R = r2 u = 2 0. As r varies from a varies from OtO7T + a(0
consider the transformation
,
If the point (x, y) describes a closed region
X
X
,
,
<
.
itself.
overlaps
128. Transformation of double integrals. First method. Retaining the hypotheses made above concerning the regions A and A l and the formulae (16), let us consider a function F(x, y) which is continuous To any subdivision of the region A l into subregions in the region A.
a n corresponds a subdivision of the region A into subbe the areas of the two corre an Let to, and sponding subregions a, and a,., respectively. Then, by formula (17), a lf a 2
,
,
regions a l} a 2
,
,
.
(I),-
=
CTf
D(ui} vj and v are the coordinates of some point in the region a,. a point x, =/(,-, v,), y,- = ^.-) v,-) corresponds of the region a,-. Hence, setting *(M, v) = -F[/(w, v), v)], we where
To
,
{
this point (,-,
<(w,-,
<(w,
may
write
D(f,
D(u i}
<#
Vi )
whence, passing to the limit, we obtain the formula
(18)
ff
J
J(A)
F(x, y) dx dy
=
F[f(u,
I JI */Ui)
v},
(u,
dudv.
v)
D(u, v)
VI,
CHANGE OF VARIABLES
128]
Hence
to
267
perform a transformation in a double integral x and y should new variables u and v, We have seen already
be replaced by their values as functions of the and dx dy should be replaced by A du dv. |
how
the
new
|
field of integration is
determined.
In order to find the limits between which the integrations should be performed in the calculation of the new double integral, it is in general unnecessary to construct the contour C\ of the new field of integration A lf For, let us consider u and v as a system of curvilinear coordinates, and let one of the variables u and v in the
formulae (16) be kept constant while the other varies. in this way two systems of curves u const, and v
=
=
We
obtain
const.
By
the hypotheses made above, one and only one curve of each of these families passes through any given point of the region A.
Let us suppose for ness that a curve
family v
=
const,
definite-
of
the
meets the
C in at most two MI and MI which cor
contour points
respond to values w t and u z of u (HI w 2 ), and that each <
(<
7
of the (v) curves which meets
the contour
C
between the two curves v a and v
b
lies
^//^^f^T^
In this case
(a
Fia. 27
we should
integrate first with regard to u, keeping v constant and letting u vary from to w 2 where u l and u z are in general functions of v, and then inte
^
,
grate this result between the limits a and
The double
integral
is
f do f \Ju.
V[/(
Ua
b.
therefore equal to the expression
,
V),
A
change of variables amounts essentially to a subdivision of the by means of the two systems of curves (u) and (v). Let w be the area of the curvilinear quadrilateral bounded by the field of integration
+
curves
(it), (u du), (v), (v -f dv ), where du and dv are positive. this quadrilateral corresponds in the uv plane a rectangle whose sides are du and dv. Then, by formula A(, 77)) du dv, where (17), w lies between u and u v dv. The expres and between and v du, 77
To
=
|
+
sion |
b.(u, v)
du dv |
is
+
called the element of area in the system of
DOUBLE INTEGRALS
268
[VI,
=
129
+
c du dv, The exact value of is \ \\ A(M, v) be infinitesimal dv. This du and with zero may approaches for since limit of the sum the in w, A(M, v) finding ^,F(x, y) neglected is continuous, we may suppose the two (u) curves and the two
coordinates
where
o>
u>
(u,
v~).
\
c
(y) curves
taken so close together that each of the
less in
e s is
ab
Hence the abso may be made less than
preassigned positive number.
solute value than
any
lute value of the
sum 2F(x, y^tdudv
itself
any preassigned positive number. 129. Examples. 1) Polar coordinates. Let us pass from rectangu p cos w, by means of the transformation x varies as of the the all obtain GO. sin xy p plane points y p the hence from zero to 2?r. Here A oo and from zero to p
=
lar to polar coordinates
=
We +
=
u>
;
Let dp, p us try first to evaluate a double integral extended over a portion of the plane bounded by an arc AB which intersects a radius vector in element of area
is
which
da>
also evident geometrically.
is
most one point, and by the two straight lines OA and OB which Let angles ^ and to 2 with the x axis (Fig. 17, p. 189). R = w ) be the equation of the arc AB. In the field of integration Hence the double inte varies from ^ to 2 and p from zero to R.
at
make
<(
o>
o>
gral of a function f(x, y) has the value
r
2
I
If the arc
AB
p sin
GO,
CD)
p dp
.
a closed curve enclosing the origin,
is GO X
c /O cos
i/O
i/ojj
take the limits
CR
M
=
and
co 2
= 2?r. Any
we should
field of integration
can
be divided into portions of the preceding types. Suppose, for C of a given of the contour outside lies the that origin instance,
convex closed curve.
Let
OA and OB
be the two tangents from and 7? 2 =/2( w ) be the 1
=/
the origin to this curve, and let RI and equations of the two arcs
ANB
between o^ and given value of the value of the double integral o>
o>
2
,
(
A MB, p varies
respectively. from RI to
is
/* 6?(0 ,
2) Elliptic coordinates.
A I
/(p cos
GO,
p sin w) p dp.
Let us consider a family of confocal conies
For a and
7? 2 ,
CHANGE OF VARIABLES
269
where X denotes an arbitrary parameter. Through every point of the plane pass an ellipse and an hyperbola, two conies of this family, for the equation (19)
FIG. 28 c2 ,
and another positive root p. less than c 2 for any (19) and from the analogous equation where X is
has one root X greater than
From
values of x and y.
replaced by p
we
,
find
V(X
y
(20)
-
To avoid ambiguity, we
shall consider only the first quadrant in the xy plane. This region corresponds point for point in a one-to-one manner to the region of the X/u plane which is bounded by the straight lines
X It is
=
c2 ,
M
=
0,
n
=
cz .
evident from the formulae (20) that when the point (X, /u) describes the this region in the direction indicated by the arrows, the point (a;, y)
boundary of
describes the two axes
transformation
is
Ox and Oy
in the sense indicated
therefore inverse, which
=
D(x, y)
is
verified
by the arrows. by calculating A
The
:
=
D(X, M )
130. Transformation of double integrals.
Second method.
We
shall
now
derive the general formula (18) by another method which depends solely upon the rule for calculating a double integral.
We
shall retain, however, the hypotheses made above concerning the correspondence between the points of the two regions A and A.
If the formula
correct for
is
x
= f(n,
v)
two particular transformations
,
v
,
it is
evident that
it
is
)
,
also correct for the transformation obtained
by carrying out the two transformations in succession. This follows at once from the fundamental property of functional determinants
(30)
_ D(u
,
v
)
,
D(u, v}
D(u
,
v
)
DOUBLE INTEGRALS
270 Similarly, to
if
130
[VI,
the formula holds for several regions A, B, C, -, L, B 1) C lf L 19 it also holds for ,
which correspond the regions A l9
the region A -{transformation
x
Here
B is
-\-
C
+
+
Finally, the formula holds
L.
a change of axes
= x + x cos a y sin a, A = 1, and the equation
J ff
if
the
:
y
=y +x
sin
a
+y
cos a.
F(x, y} dx dy
J(A)
F(x is satisfied,
We
+x
since the
cos
two
y
sin a,
y
-f
x sin a
+y
cos a) dx
1
dy
integrals represent the same volume. formula for the particular trans
shall proceed to prove the
formation
x
(21^
=
1
is
d>(x
}
11
=
?/
which carries the region A into a region A which is included between the same parallels to the x axis, y = y and y = y. We shall sup A to one of of A and that given point corresponds any point just pose conversely. If a paral to the x axis meets
lel
the boundary
A
C
of the
most two points, the same Avill be true for the boundary region
/in
.
C"
x
of the region
A
.
any pair of points and m 1 on C whose
To
m or-
dinates are each y cor
FIG. 29
respond two points ?tt But the correspondence may be direct or To distinguish the two cases, let us remark that if is x increases with x and the points m and m and m and shown in Fig. 29 hence the correspondence is direct. On
and m{ of the contour C inverse.
in at
.
c/dx
positive,
m{
lie
as
l
,
;
the other hand, if is negative, the correspondence is inverse. Let us consider the first case, and let x , Xi, x n x[ be the abscissae d/dx
,
m^
of the points ra , m. , m{, Then, applying the for respectively. mula for change of variable in a simple integral, we find
f Jr
F(x,
(*
,
y ), y
]
VI,
CHANGE OF VARIABLES
130]
A
where y and y are treated as constants.
271
single integration gives
the formula
r I
r*
tfi
dy\ J^O
J
ry c x( F(x,y)dx=\ dy J F^(
1
<
I
J*
u
But the Jacobian A reduces in this case to preceding formula may be written in the form
d/dx
Jff J(A)
F(x,
=
y}dxdy
Jff J(A
F[A(x y }, y ,
and hence the
,
*\dx dy
]
.
)
This formula can be established in the same manner if negative, and evidently holds for a region of any form whatever.
is
d/dx
In an exactly similar manner
can be shown that the trans
it
formation
x
(22)
=z
y
,
=
t(x
y
,
)
leads to the formula
F(r, Jff J(A)
= ff J
y)dxdU
F[.r
,
t(x
,
y )]
A dx dy
I
|
,
J{4")
where the new
field of integration .1 corresponds point for point the region A. Let us now consider the general formulae of transformation
x
(23)
=f(x l} y
y =f,
x ),
(x,, y,)
tc
,
where for the sake of simplicity (a-, y) and (x l} y^) denote the coor dinates of two corresponding points m and MI with respect to the same system of axes. Let A and A l be the two corresponding regions
bounded by contours C and C 1} respectively. Then a third point m whose coordinates are given in terms of those of m and l by the A which = relations x = x l} y y, will describe an auxiliary region each to for for the moment we shall assume corresponds point point x x The six quantities x, y, lf y l} of the two regions A and A^ y ,
M
,
,
satisfy the four equations
whence we obtain the x
relations
=
y =/i(*nyi), which define a transformation of the type (22). From the equation hence relation of the form y = TT(X y ) y yj) we find a (24)
sr l
,
=/!(# we may write
l
,
(25)
x
=f(x yO = ,
4>(x
,
y
},
y
=y
,
.
;
DOUBLE INTEGRALS
272
[VI,
131
The given transformation (23) amounts to a combination of the two transformations (24) and (25), for each of which the general formula Therefore the same formula holds for the transformation (23).
holds.
We
Remark. point
Av
assumed above that the region described by the corresponds point for point to each of the regions A and
m
At least, this can always be brought about. For, let us con sider the curves of the region A l which correspond to the straight If these curves meet a parallel to lines parallel to the x axis in A. .
the y axis in just one point,
m
evident that just one point of of A. Hence we need any given point merely divide the region A t into parts so small that this condition If these curves were parallels to the is satisfied in each of them.
A
y
it is
m
will correspond to
axis, AVO
should begin by making a change of axes.
Let S be a region of a curved sur from singular points and bounded by a contour F. Let S be subdivided in any way whatever, let s be one of the subregions bounded by a contour y i} and let m^ be a point of s Draw the tan taken so gent plane to the surface S at the point m i} and suppose small that it is met in at most one point by any perpendicular to 131. Area of a curved surface.
face free
{
t
.
s,-
The contour
this plane.
we by
y,-
projects into a curve
y-
upon
this plane
shall denote the area of the region of the tangent plane yl
by
o- f
in such a
the
sum
.
way 2o\-
As the number of subdivisions is increased indefinitely that each of them is infinitesimal in all its dimensions,
approaches a limit, and this limit
S of the given
the region
;
bounded
is
called the area of
surface.
Let the rectangular coordinates x, y, z of a point of S be given in terms of two variable parameters u and v by means of the equations
x =/(M,
(26)
= (u,
v),
z
=
if/(u,
v)
,
the region S of the surface corresponds point for of the uv plane bounded by a closed contour C. shall assume that the functions /, and if/, together with their
in such a
way that
point to a region
We
y
v),
R
<j>,
Let R be partial derivatives, are continuous in this region. subdivided, let ri be one of the subdivisions bounded by a contour c,-,
first
and let w be the area of r To r corresponds on S a subdivision s bounded by a contour y Let a- be the corresponding area upon the tangent plane defined as above, and let us try to find an expression -
{
t
f
.
.
{
t
t
for the ratio o^/o^.
Let a
{ ,
(3^
at a point
m
y be the direction cosines of the normal to the surface S z( of st which corresponds to a point (ui} Vf) f (ic -, yt) f
t
~)
VI,
CHANGE OF VARIABLES
131]
273
Let us take the point m as a new origin, and as the new axes m and two perpendicular lines m X and m Y in the direction cosines with respect to the old axes are whose tangent plane of rf
.
t
the normal at
{
{
t
a ft y and respectively. Let X, Y, Z be the coordinates of a point on the surface S with respect to the new axes. Then, by the well-known formulae for transformation of coordinates, we ,
a",
,
shall
ft",
y",
have
X= F= Z=
a
-
(x
t)
{
a"(x
or,
+ ?(y- y +?*(*- *,) y) + ) + (y y (z
*,.)
-x) +
(x
Xi)
t
ft"(y
f
y"(z
?/,) -f-
fti
{
,
,
.
,-)
is the area of that portion of the A F plane which The area bounded by the closed curve which the point (X, F) describes, the point (u, v) describes the contour c Hence, by 127, tr,-
is
as
.
p(x, y)
^where
u\
and
v\
easy calculation
or,
Hswr^r
are the coordinates of
now
some point inside of
ci
.
An
leads us to the form
by the well-known relations between the nine direction cosines, =s 3.
<
D(z,x)
Oi ><X>
O
Applying the general formula
D(u[, v
(17),
we
t
T* )
D(x,y) D(u[, v
therefore obtain the equation
*D(4O where
u\
and
uv plane. the point T)fii
>y\
iJ\Jh Z)
v\
are the coordinates of a point of the region r{ in the
If this region is very small, the point (?* f , v,.),
(M,-,
v)
is
very near
and we may write
T)fii f^\J[t
v\ Z)
.
f JT)(\ z
f\ >
)
_
Dff z u \
i
*r\ c)
i
\
D(Ui, Vt)
u,
~r
f C>
)
D(y,
where the absolute value of 6 does not exceed unity. Since the and ^ are continuous in the derivatives of the functions /, <,
DOUBLE INTEGRALS
274
[VI,
i;;i
region R, we may assume that the regions rt have been taken so small that each of the quantities e,-, e-, e- is less than an arbitrarily preassigned number rj. Then the supplementary term will certainly
be less in absolute value than 3^0, where O is the area of the Hence that term approaches zero as the regions st region R.
(and
Tf)
sum
So-,
approach zero in the manner described above, and the approaches the double integral
all
y)
(x,
()
where
a,
ft,
du dv
,
D(u, v)
D(u, v)
y are the direction cosines of the normal to the surface S
at the point (u,
v~).
Let us calculate these direction cosines. tangent plane
39)
(
The equation of the
is
whence a
B
D(u, v)
D(u, v)
1
y D(x,
_
, (
D(u, v)
Choosing the positive sign in the
D
(y> ,
Q P
we
obtain the formula
-
7 D(u, v)
D(u, v)
D(u, v)
last ratio,
.D(1
The well-known identity (aft
-
ba
2
)
+
1
(be
-
c&
2
)
+
f
(ca
-
ac
2
)
which was employed by Lagrange, enables us to write the quantity F 2 where under the radical in the form EG ,
(27)
S
the symbol indicating that a; is to be replaced by y and z succes and the three resulting terms added. It follows that the area sively of the surface S is given by the double integral
(28)
A
-
== / J/ J(B (B)
F*dudv.
VI,
CHANGE OF VARIABLES
132]
275
E, F, and G play an important part in the theory Squaring the expressions for dx, dy, and dz and adding
The functions of surfaces.
the results,
we
ds
(29)
2
find
=
dx 2
+ dy* + dz* = E du + 2Fdu dv + G dv 2
2 .
It is clear that these quantities E, F, and G do not depend upon the choice of axes, but solely upon the surface S itself and the inde pendent variables u and v. If the variables u and v and the sur
face
5 are
evident that
all real, it is
EG
F
2
must be
The expression V EG the surface S in the system
132. Surface element.
positive.
F 2 du dv
is
called the
of coordinates (u, v). element of area of The precise value of the area of a small portion of the siirface bounded F 2 t)dudv, by the curves (u), (u du), (v~), (v -\- dv) is (j\EG
+
+
where
e
approaches zero with
that the term
e
du dv
is
du and
dv.
It is evident, as above,
negligible.
Certain considerations of differential geometry confirm this result. For, if the portion of the surface in question be thought of as a small curvilinear parallelogram on the tangent plane to S at the point (u, v), its area will be equal, approximately, to the product of the lengths of its sides times the sine of the angle between the two curves (u) and (v~). If we further replace the increment of arc by the differ
by formula (29), are ^/Edu and of du and dv are taken ^/Gdv, positive. The direction parameters the tangents to the two curves (u) and (v) are dx/du, dy/du, dz/du and dx/dv, dy/dv, dz/dv, respectively. Hence the angle a between them is given by the formula ^ dx dx ential ds, the lengths of the sides, if
COS
a
^
= iV
F
cu Gv
(
\du VI" 2 a = V EG .F /V EG. Forming the product mentioned, we find the same expression as that given above for the element of area. The formula for cos a shows that F = when and only when the two families of curves (u) and (y~) are orthogonal to each other. When the surface S reduces to a plane, the formulae just found
whence
reduce
we
sin
to
find
the formulae
found in
128.
For,
if
we
set
if/(u,
v)
= 0,
DOUBLE INTEGRALS
276
[VI,
132
whence, by the rule for squaring a determinant,
dx ex
E F
du dv
F G
= EG - F
2 .
du dv
Hence
^EG
F
2
reduces to |A|.
To find the area of a region of a surface whose equa y} which projects on the xy plane into a region R in which the function f(x, y), together with its derivatives p = df/dx and q = df/dy, is continuous. Taking x and y as the independent vari 2 F pq, G = 1 + q 2 and the area in ques ables, we find E = \ + p Examples.
tion is z
1)
= f(x
}
,
,
tion
is
given by the double integral
=
(30)
where y
is
=
Jff J(R)
Jff J(R (R)
COS y
the acute angle between the z axis and the normal to the
surface. 2) To calculate the area of the region of a surface of revolution between two plane sections perpendicular to the axis of revolution. Let the axis of revolution be taken as the z axis, and let z f(x)
=
be the equation of the generating curve in the xz plane. Then the coordinates of a point on the surface are given by the equations
z=f(p), where the independent variables p and are the polar coordinates of the projection of the point on the xy plane. In this case we have o>
F=0,
p-),
To
G=
P *.
find the area of the portion of the surface
bounded by two plane whose radii are p and should be allowed to from p 2 respectively, p vary Pl to p 2 p 2 ) and from zero to 2?r. Hence the required area is given by the integral sections perpendicular to the axis of revolution ,
t
(pi<
)
and can therefore be evaluated by a single quadrature. the arc of the generating curve, we have ds*
=
rf
2
+
=
<
If s denote
VI,
IMPROPER INTEGRALS
133]
277
and the preceding formula may be written in the form
A
= f I
p*
27Tp ds
.
Jpl
The geometrical interpretation of this result is easy 2jrp ds is the lateral area of a frustum of a cone whose slant height is ds and whose mean radius is p. Replacing the area between two sections :
whose distance from each other is infinitesimal by the lateral area of such a frustum of a cone, we should obtain precisely the above formula for
A.
For example, on the paraboloid of revolution generated by revolv 9 ing the parabola x = 2pz about the z axis the area of the section between the vertex and the circular plane section whose radius is r is
III.
GENERALIZATIONS OF DOUBLE INTEGRALS IMPROPER INTEGRALS SURFACE INTEGRALS
Let f(x, y) be a function which is con whole in the tinuous region of the plane which lies outside a closed double The F. contour integral of f(x, y) extended over the region 133. Improper integrals.
between F and another closed curve C outside of F has a finite value. If this integral approaches one and the same limit no matter how C varies, provided merely that the distance from the origin to the nearest point of C becomes infinite, this limit is defined to be the value of the double integral extended over the whole region outside F.
Let us assume for the moment that the function f(x, y] has a In this case the limit of the constant sign, say positive, outside F. double integral is independent of the form of the curves C. For, Cn be a sequence of closed curves each of which let Ci, C z ,
,
,
encloses the preceding in such a way that the distance to the nearest / extended point of Cn becomes infinite with n. If the double integral over the region between F and C n approaches a limit /, the same will
which Cm , be true for any other sequence of curves C{, C^, , double of the I the value if be conditions. the same m For, satisfy integral extended over the region between F and C m n may be chosen so large that the curve Cn entirely encloses C m and wa ,
,
shall
have
/
<
/
<
/.
Moreover
/
increases with m.
Hence
Im
DOUBLE INTEGRALS
278 has a limit /
[VI,
133
I. It follows in the same manner that I I Hence the two limits are equal. As an example let us consider a function f(x, y), which outside a circle of radius r about the origin as center is of the form
=
/
<
<
.
I, i.e.
where the value of the numerator \f/(x, y~) remains between two posi numbers m and M. Choosing for the curves C the circles
tive
concentric to the above, the value of the double integral extended over the circular ring between the two circles of radii r and R is given by the definite integral
C
\l/(p
/, J,
cos
to,
p sin
o>)p
dp
between the values of the two expressions
It therefore lies
By 90, the simple integral involved approaches a limit as increases indefinitely, provided that 2a 1 1 or a 1. But >
becomes
infinite
with
R
if
a
>
R it
1.
<
If no closed curve can be found outside
which the function /(a;, y) can be shown, as i 89, that the integral ffffa y)dxdy approaches a limit if the integral // f(x, y) \dxdy itself approaches a limit. But if the latter integral becomes infinite, has a constant sign,
the former integral to Cayley,
is
is
it
grate this function first
and the two
r I
Jo
a
dx
r I
lines
x
The following example, due Let f(x, y) = sin (x 2 + if), and let us inte over a square of side a formed by the axes
indeterminate.
interesting.
= a,
y
= a.
The value
of this integral
is
a
sin
Jo
2
(a;
= I Jo
+ y }dy z
sinx^dx x
I
Jo
cosy
2
dy+
I
Jo
cosx*dx x
I
Jo
siny*dy.
As a increases indefinitely, each of the integrals on the right has a limit, by 91. This limit can be shown to be V?r/2 in each case hence the limit of the whole right-hand side is TT. On the other ;
hand, the double integral of the same function extended over the quarter circle bounded by the axes and the circle x 1 + y 2 = R 3 is equal to the expression
/I,
IMPROPER INTEGRALS
134]
279
7T
r* r du \
\
yo
*/0
which, as R becomes infinite, oscillates between zero and 7r/2 and does not approach any limit whatever.
We should define in a similar manner the double integral of a function f(x, y) which becomes infinite at a point or all along a line. First, we should remove the point (or the from the field of line)
integration by surrounding it by a small contour (or by a contour very close to the line) which we should let dimmish
For example,
_
indefinitely.
the function f(x, y) can be written in the form
if
f(x
)
=
^(*>
y)
in the neighborhood of the point (a, b), where \}/(x, y) lies between two positive numbers m and M, the double integral of f(x, y) extended over a region about the point (a, b) which contains no other point of discontinuity has a finite value if and only if a is less
than unity.
134. The function B(p, q). We have assumed above that the contour C n recedes indefinitely in every direction. But it is evident that we may also sup pose that only a certain portion recedes to infinity. This is the case in the above
example of Cayley
s
and
also in the following example.
=
/(x, y)
where p and q are each first
quadrant.
and the
lines
1
2
-1
2
e-*
-!
Let us take the function
2 ,
This function is continuous and positive in the over the square of side a bounded by the axes we find, for the value of the double integral,
positive.
Integrating
a and y
x
4x P- y 2
=
first
a,
a
C 2x2
2 J>-ie-*
Jo
dx x C 2y*i-ie-*dy. Jo
Each of these integrals approaches a limit as a definition of the function T(p) in 92,
becomes
infinite.
For, by the
+C
f T(p)= Jo whence, setting
t
= x2 we ,
V-ie-<(K,
find
+
r(p)= C
(31)
2x*P-*e-**dx.
Jo
Hence the double
integral approaches the limit T(p) T(q) as a becomes infinite. Let us now integrate over the quarter circle bounded by the axes and the circle z2 + y 2 = R 2 The value of the double integral in polar coordinates is .
f*R I
Jo
2p
-P
2<
+ i)- l e-P dp x
/* I
J
DOUBLE INTEGRALS
280 As R becomes
u
product approaches the limit
infinite this
T(p
where we have
[VI,
+
q)B(p, q),
set rr
B(p,q)
(32)
= C
2
2cos?P-
2
l sin
d<}>.
Jo
Expressing the fact that these two limits must be the same, we find the equation T(p)T(q)
(33)
The it
is
called Euler
integral B(p, q} be written in the
may
form
(34)
B(p,
The formula
(33)
r(p
+
q)B(p,
s integral
q)= JQ f
t9-i(l
= vV.
-
For example,
Setting
Hence the formula
=
1
= sin 2
<,
p
q
=
1/2,
we
q) to the calcu
find
(31) gives
2
p and
T(p)T(l-p) =
t
1 t)v- dt.
setting
f. In general, setting q
taking
t
and
p between
l-p) =
f
Jo
We shall
q).
of the first kind.
reduces the calculation of the function B(p,
lation of the function T.
whence F(l/2)
=
see later that the value of this integral
1
,
we
find
(~ \
l
is jr/sin pit.
The definition of surface integrals is analogous to that Let S be a region of a surface bounded by one or more curves F. We shall assume that the surface has two distinct sides in such a way that if one side be painted red and the other blue, for instance, it will be impossible to pass from the red side to the blue side along a continuous path which lies on the sur 135. Surface integrals.
of line integrals.
and which does not cross one of the bounding curves.* Let us think of S as m and m be two points m let us draw that half of The direction the normal mn to the surface which does not pierce the surface. thus defined upon the normal will be said, for brevity, to correspond to that side of the surface on which m lies. The direction of the normal which corresponds face
a material surface having a certain thickness, and let near each other on opposite sides of the surface. At
to the other side of the surface at the point
m
will
be opposite to the direction
just defined.
Let z
= (x,
in at
y) be the equation of the given surface, and let -S be a region of shall assume that the surface is met
bounded by a contour F. We most one point by any parallel to the
this surface
z axis,
and that the function
0(z, y)
* It is very easy to form a surface which does not satisfy this condition. We need only deform a rectangular sheet of paper by pasting the side B C to the side in such a way that the point C coincides with A and the point B with D.
ABCD
AD
VI,
is
SURFACE INTEGRALS
135]
continuous inside the region
into
which T
projects.
It is
281
A
of the xy plane which is bounded by the curve C evident that this surface has two sides for which
the corresponding directions of the normal make, respectively, acute
and obtuse
We shall call that side whose angles with the positive direction of the z axis. corresponding normal makes an acute angle with the positive z axis the upper Now
side.
let
P(x, y, z) be a function
ofi
the three variables x, y, and z which
continuous in a certain region of space which contains the region S of the sur face. If z be replaced in this function by y), there results a certain function is
<(x,
P [x,
y,
y)] of
x and y alone
;
and
it is
to call the double integral of this function
P [x,
vf f (A)
(35)
natural by analogy with line integrals extended over the region A,
dx dy
y, 0(x, y)]
,
<s
the surface integral of the function P(x, y, z) taken over the region S of the given Suppose the coordinates x, y, and z of a point of S given in terms of two auxiliary variables u and v in such a way that the portion S of the surface corre
surface.
sponds point for point in a one-to-one manner to a region R of the uv plane. Let da- be the surface element of the surface S, and 7 the acute angle between the posi tive z axis and the normal to the upper side of S. Then the preceding double integral, by 131-132, is equal to the double integral
rr
(36)
P(x,
y,
z)cos7d
This new expression x, y, and z are to be expressed in terms of u and v. however, more general than the former, for cos 7 may take on either of two
where is,
values according to which side of the surface is chosen. When the acute angle 7 is chosen, as above, the double integral (35) or (36) is called the surface integral
(37)
extended over the upper side of the surface
But
if 7 be taken as the obtuse changed in sign, and the new / / Pdxdy extended over the lower side of In general, the surface integral// Pdx dy is equal to the double integral (35) according as it is extended over the upper or the lower side of S. This definition enables us to complete the analogy between simple a^id double
S.
angle, every element of the double integral will be double integral would be called the surface integral <S.
Thus a simple integral changes sign when the limits are interchanged, while nothing similar has been developed for double integrals. With the gen eralized definition of double integrals, we may say that the integral///(x, y) dx dy integrals.
previously considered is the surface integral extended over the upper side of the xy plane, while the same integral with its sign changed represents the surface integral taken over the under side. The two senses of motion for a simple inte gral thus correspond to the two sides of the xy plane for a double integral.
The expression (36) for a surface integral evidently does not require that the surface should be met in at most one point by any parallel to the z axis. In the same manner we might
define the surface integrals
ff Q(z,
y, z)
dy dz,
ff R x (
v* z )
dz dx
DOUBLE INTEGRALS
282 and the more general
f fp(x, This latter integral
[VI,
136
integral
y,
z)dxdy +
z)dydz
Q(x, y,
also be written in the
may
+
R(x, y, z)dzdx.
form
C f [Pcos? + Qcosa + JJcos/3]d
,
7 are the direction angles of the direction of the normal which cor
responds to the side of the surface selected. Surface integrals are especially important in Mathematical Physics. 136. Stokes* theorem.
P(x, y,
z),
Q(x, y,
z),
Let
R(x, y,
L z)
be a skew curve along which the functions
are continuous.
Then
the definition of the line
integral
C Pdx + Qdy
+Rdz
J((L)
taken along the line L is similar to that given in 93 for a line integral taken along a plane curve, and we shall not go into the matter in detail. If the curve L is closed, the integral evidently may be broken up into the sum of three line inte grals taken over closed plane curves. Applying Green s theorem to each of these, it is
we may replace the line integral by the sum of The introduction of surface integrals enables us to state
evident that
integrals.
three double this result in
very compact form. Let us consider a two-sided piece S of a surface which we shall suppose for To each side of the surface definiteness to be bounded by a single curve P. corresponds a definite sense of direct motion along the contour r. We shall of the contour let us draw assume the following convention At any point
M
:
consideration,
and
let
Mn which
corresponds to the side of the surface under us imagine an observer with his head at n and his feet at
that half of the normal
M
we
;
the positive sense of motion which the observer must take in shall say that that
is
order to have the region
Thus
S
at his left hand.
two sides of the surface corre spond two opposite senses of motion along to the
the contour F.
Let us
first
face which
any
is
consider a region
met
S
of a sur
most one point by axis, and let us suppose
in at
parallel to the z
the trihedron Oxyz placed as in Fig. 30, where the plane of the paper is the yz plane and the x axis extends toward the observer.
To
F
the boundary
C
of
S
will correspond a
and these two curves are described simultaneously in Let the sense indicated by the arrows. z = /(x, y) be the equation of the given surface, and let P(x, y, z) be a function which is continuous in a region of space which contains S. Then the line inte closed contour
gral
fr
P(x, y,
z)
dx
is
in the
identical with the line integral
xy plane
;
VI,
SURFACE INTEGRALS
136]
P[x,
/
y,
283
y)]dx
/(C)
Let us apply Green
taken along the plane curve C. latter integral.
s
theorem
(
126) to this
Setting
= P[z,
P(x, y)
we
for definiteness,
y,
y)]
<(>(x,
find
,
_ dP
y)
y
_8P_ S
dP_ cz cy
d
cy
cz
cy
cosy
where
or, /3, 7 are the direction angles of the normal to the upper side of Hence, by Green s theorem,
-D,
-
^j
P(x, y)dx
=
S.
dP
CC J J (A I
cz
cos y
dV
where the double integral is to be taken over the region A of the xy plane bounded by the contour C. But the right-hand side is simply the surface integral
cos
cos 7
/3
dz
extended over the upper side of S
f
P(x,
?/,
z)dx
dy
=
/
I
This formula evidently holds also
when
is
it
be.
formulae
Green
s
dxdy. dy
the surface integral is taken over the taken in the other direction along F. And theorem, no matter what the form of the surface
the line integral
also holds, as does
may
--dzdx
J J(S) cz
other side of
if
do-
and hence we may write
;
J(F)
<S,
)
I
By cyclic permutation of
x, y,
and
z
we
obtain the following analogous
:
^ Q(x, y,z)dy=
Jf
C R(x,y,z)dz= J
Adding the
three,
we
C i dydzJ J(S) sy
obtain Stokes theorem in
its
dzdx. dx
general form
:
/
P(x, y, z)dx
J
+
+
Q(z, y, z)dy
R(x, y, z)dz
dP\, ^(dE ^ ----dQ\. 2 }dydz + --- \dzdx. )dxdy + /d
(
(
ty/
The sense
in
which T
double integral
is
is described and the side of the surface over which the taken correspond according to the convention made above.
DOUBLE INTEGRALS
284
IV.
[VI,
137
ANALYTICAL AND GEOMETRICAL APPLICATIONS
Let us consider, as above, a region of space bounded surface S above that plane, and a cylinder whose a the xy plane, by We shall suppose that the generators are parallel to the z axis. 137. Volumes.
is a contour similar to section of the cylinder by the plane z that drawn in Fig. 25, composed of two parallels to the y axis and two
APB and A QB
curvilinear arcs
If %
.
surface S, the volume in question
V
rb
=
\
Jo.
Now
is
f(x, y)
is
given, by
the equation of the by the integral
124,
rvt
dx
I
f(x, y)dy.
Jy,
2 the integral f /(o:, y}dy represents the area A of a section of a this volume by plane parallel to the yz plane. Hence the preceding in the form be written formula may "
b
A dx. V=f U
(39)
a
The volume of a solid bounded in any way whatever is equal For to the algebraic sum of several volumes bounded as above. convex closed a bounded of a solid volume the find by instance, to should circumscribe the solid by a cylinder whose gen erators are parallel to the z axis and then find the difference between two volumes like the preceding. Hence the formula (39) holds for surface
we
= a and x = b any volume which lies between two parallel planes x surface bounded which is and whatever, where A by any (a I) a made a section of denotes the area by plane parallel to the two the interval the Let us (a, ) subdivided by suppose given planes. be the areas A let A x and A x x u n _ l} b, points a, l} z of the sections made by the planes x = a, x = x lt -, respectively. <
,
Then the
definite integral
,
,
,
$*k dx
is
; ,
the limit of the
sum
The geometrical meaning of this result is apparent. For A,-_! (#,-
a;,-_i),
whose base is whose height Hence the volume
for instance, represents the volume of a right cylinder and the section of the given solid by the plane x a-,-_i
=
is
the distance between two consecutive sections.
of the given solid is the limit of the sum of such infinitesimal cylin This fact is in conformity with the ordinary crude notion of ders.
volume.
VI,
APPLICATIONS
138]
A be known as a fnnction of
If the value of the area
ume
285 x,
the vol
be found by a single quadrature. As an example let us try to find the volume of a portion of a solid of revo lution between two planes perpendicular to the axis of revolution. to be evaluated
may
=
Let this axis be the x axis and let z f(x) be the equation of the generating curve in the xz plane. The section made by a plane par Hence the required allel to the yz plane is a circle of radius f(x). 2 given by the integral TT^ [/(x)] er. Again, let us try to find the volume of the portion of the ellipsoid
volume
is
n
n
n
^4.^4. f! = a*
c
b*
i
2
bounded by the two planes x = x x = X. The section made by a is an ellipse whose semiaxes are plane parallel to the plane x = x 2/a 2 Hence the volume sought is x 2 /a 2 and c Vl b Vl ,
.
V=
cA
To
l-
Trbc
I
Y8
/
a-A
*
-
\
find the total
volume we should
x
set
=
a and
X=
which
a,
gives the value %irabc. 138. Ruled surface.
Prismoidal formula.
When
the area
A
is
an integral
function of the second degree in z, the volume may be expressed very simply in terms of the areas B and B of the bounding sections, the area 6 of the mean section,
and the distance h between the two bounding
section be the plane of yz,
V = JC But we h
also
=
whence n
If
the
mean
+ 2
(te
+ 2mx +
n)
dx
=
+
21
2na.
3
a
have b
2a,
=
sections.
we have
&,
a
=
=
B=
n,
2 h/2, 2Za
Itf
+ 2ma + n
=B+B
26.
B =
la*
- 2ma + n
These equations lead
+B + F=^[B o
(40)
,
to the
,
formula
4&],
which
is called the prismoidal formula. This formula holds in particular for any solid bounded by a ruled surface and two parallel planes, including as a special case the so-called prismoid.* For, let y = ax + p and z = bx + q be the equations of a variable straight line, where
a, 6, p,
and q are continuous functions of a variable parameter t which resume when t increases from t Q to T. This straight line describes
their initial values
* lel
A
prismoid
and contain
is
all
a solid bounded by any number of planes, two of which are paral the vertices.
TRANS.
DOUBLE INTEGRALS
286
[VI,
a ruled surface, and the area of the section made by a plane x = is given, by 94, by the integral
=
139
parallel to the plane
/T
+
(ax
I
p)(b x
+
q )dt,
Jto
where a
,
&
derivatives
and
T,
,
c
,
d denote the derivatives of a, 6, c, d with respect to t. These be discontinuous for a finite number of values between t
may even
which
be the case when the lateral boundary consists of portions of The expression for A may be written in the form
will
several ruled surfaces.
A
=
x2
rr
ab dt
/
+
x
rr I
Jt
Jt n
rr pq
+ pb )dt +
(aq
\
Jt
dt,
where the integrals on the right are evidently independent of x. Hence the formula (40) holds for the volume of the given solid. It is worthy of notice that the same formula also gives the volumes of most of the solids of elementary geometry .
139. Viviani s problem. Let C be a circle described with a radius OA (= R) of a given sphere as diameter, and let us try to find the volume of the portion of the sphere inside a circular cylinder whose right section is the circle C.
Taking the origin at the center of the sphere, one fourth the required volume given by the double integral
is
-
-
extended over a semicircle described on dinates p and w, the angle u varies from
we
-
z
OA
2
dxdy
2/
as diameter.
to it/2,
and
Passing to polar coor
from
p
to
R cos w.
Hence
find
v= 4 If this
rim Rm Bin* w)\*du = RS i*---2 \
l
- I (R 3 3 Jo
3
(
3 \2
)
3/
volume and the volume inside the cylinder
which
is symmetrical to this one with respect to the z axis be subtracted from the volume of the
whole sphere, the remainder
is
3
3
Again, the area
\2
of the portion of the sur face of the sphere inside the given cylinder is
FIG. 31
ft
dxdy. Replacing p and q by their values polar coordinates,
we
find
x/z and
y/z, respectively,
and passing to
VI,
APPLICATIONS
140]
f
ftRcoiui
*
*
,
r>
=
Rf>d(>
I
V^ 2 -,
Jo
=
tt
287
2
2
4 f Jo
-
f 2 (l-sinw)dw = 4R2 (- -
4fl 2
a
Jo
Subtracting the area enclosed by the two cylinders from the whole area of the sphere, the remainder
is it
^ The theorems estab
140. Evaluation of particular definite integrals.
lished above, in particular the theorem regarding differentiation under the integral sign, sometimes enable us to evaluate certain defi nite integrals without proceed to give a
We
knowing the corresponding indefinite integrals few examples.
Setting
= F(a\ = r* log
A
(
1
+
aX ^
dx
the formula for differentiation under the integral sign gives
dA _ ~
log
da
(1+a
1
+
f
2
)
a2
xdx
Breaking up this integrand into partial fractions, we find x
1
+
Ix
a
a
1+ax whence x dx
log (1
+
2 <*
)
fai .
-
It follows that
1+a a
2
rfa
whence, observing that
A
r
=
\
Jo Integrating the
,4
vanishes
-
2
log(l4-
arc tan
o/-i
2(1+ a when a
i
=
a ^ + Jr 1+a
0,
) 2
)
we may
write
g
log 0/1
(i+<>
2(1+ a
first
I
2 2\
)
.,
I
of these integrals
A =
T~i
by
- arc tan a log (1 m
2
"2
parts,
+
a 2)
.
we
finally find
DOUBLE INTEGRALS
288
[VI,
140
v This function is continuous Again, consider the function x and 1 and y between any two positive when x lies between .
numbers a and
Hence, by the general formula of
b.
/-&
/i
dx
I
/*
/?>
x v dy
=
dy
I
Ja
t/a
123,
\
x y dx.
I
UQ
But
r
l
>i
ipy
flf I
,, Ly
Joo
+ IJ 1
I
,,
I
y
+L 1
i
hence the value of the right-hand side of the previous equation
-= On
the other hand,
is
log
we have r log
a;
whence log x
= log
In general, suppose that P(x, y) and Q(x,
two functions
are
?/)
which satisfy the relation dP/dy = dQ/dx, and that x x^ y y^ are Then, by the general formula for integration given constants. under the integral sign, we shall have ,
c?o;
p ap -dy=\p
I
ty
J*
J dy
J,Jo
,
p0Q -^-dx, ^ \
J*
or
r^
r*i
(41)
1
[P(x,
y,}-P(x, y )-]dx=
^^o
I
[Q(^ 15
y)- Q
^"o
Cauchy deduced the values of a large number of definite inte It is also closely and simply related to grals from this formula. Green s theorem, of which it is essentially only a special case. For it may be derived by applying Green s theorem to the line
fPdx + Qdy taken along the boundary of the rectangle formed by the lines x = X Q x x^ y = y y = y l In the following example the definite integral is evaluated by a
integral
special device.
The
.
,
,
integral
=
I
Jo
log (1
2a
cos x
+
n-
2
)
dx
VI,
APPLICATIONS
110]
has a
value
finite
if
This function
has the following properties.
F(<x)
F(-
1)
a)
=
For
F(a).
F(or,
from unity.
different
is
\a\
289
a)
=
I
log (1
Jo
=
making the substitution x
F(2
F(a
2)
)
= 2F(a).
2a cos x
TT
Jo
+
a-
+
a 2) dy
2
)
dx,
y,
2a
log (1
I
a)
+
For we may
cos y
=
F(a)
.
set
2F() = F() + F(-a), whence
=
2F(a)
I
Jo
=
I
Jo If
2a cos x
[log (1
log (I
we now make
+
a2)
- 2a* cos 2x +
= -1 i
+Making a second
= y,
COS x
+
a 2 )] dx
becomes
this
C"
i
\og(l-2a cosy
I
* Jo
2a
a*)dx.
the substitution 2x
2F(a)
+ log (1 +
+ a )dy 4
2*
r
log (1
I
substitution y
- 2a
=
2
cos y
+
a 4 ) dy.
z in the last integral,
lit
we
find f*n 2
\og(l-2a cosy
X2rr
which leads
From
to the
this result
a is The same
=
\ F(a
2
)
\
if
<
|
-
2
2
cos z
+
a*)dz,
=
F(a")
than unity, a2 approaches zero as n becomes infinite. 2 for the logarithm approaches zero. is true of F(a = 0. we have 1, F(a) a| "),
Hence,
log (1
Jo
=
=
less
|
I
have, successively,
"
If
|
a*)dy=
formula
we
F(a)
+
DOUBLE INTEGRALS
290 a
If |
greater than unity, let us set a
is |
F(a)
=
2 cos x
1
Jo
=
F(a) Finally,
find
1
Hence we have 2
TT
log
ft
=
TT
log
can be shown by the aid of Ex. 6, continuous for all values of a.
it
hence F(a)
Then we
log(l-2/?cos;r
than unity.
is less
\ft\
I//?.
f Jo
= where
=
141
[VI,
in this case
a2
p.
.
205, that F(
1)
=
;
is
A
141. Approximate value of logF(n + l). great variety of devices may be to find either the exact or at least an approximate value of a definite
employed
We proceed
integral.
We
an example.
to give
T(n
+ l) =
have, by definition,
n +00
xn
I
e- x dx.
Jo
The function xn e~ x assumes its maximum value nn e~ n for x = n. As x increases from zero to n, xn e~ x increases from zero to nn e~ n (n>0), and when x increases from n to + oo, x n e~ x decreases from nn er n to zero. Likewise, the function nn er n e- tt increases from zero to n n er n as t increases from oo to zero, and decreases from nn e~ n to zero as t increases from zero to + oo. Hence, by the substitution n
(42)
-*
=
the values of x and to
+ It
t correspond in such a way that as t increases from oo x increases from zero to + oo. remains to calculate dx/dt. Taking the logarithmic derivative of each side
oo,
of (42),
we
find
2tx
dx,
x
dt
We
have
also,
by
(42), the
n
equation tfl
=
x
n
n log
W f
-
j
For simplicity let us set x = n + z, and then develop log (1 + z/n) by Taylor s theorem with a remainder after two terms. Substituting this expansion in the value for
2 <
,
we
find
nz2
n
[2 where
6 lies
between zero and unity.
x
-
From
this
we
find, successively,
VI,
APPLICATIONS
142]
291
whence, applying the formula for change of variable,
Y(n
The
+
first
1)
= 2nn
e-"
*/- C \ / I/
"e-^dt
+
2n"er
As
2
C V
*V<
(1
00
is
integral
e-
not
n
00
<2
dt
,_ = Vnr.
for the second integral, though we cannot evaluate it exactly, since we do know 0, we can at least locate its value between certain fixed limits. For
and zero, and they are all positive elements are negative between +cc is ^ ess m between zero and + oo. Moreover each of the integrals f_ n / + ter^dt = 1/2. It follows that absolute value than / all its
<x>
,
F(n
(43)
+
= \/2nnn e~ n (
1)
-
V*n/ 1 and + 1where u lies between If n is very large, w/V2n is very small.
=
Hence,
if
we take
n"e-
as an approximate value of T(n + 1), our error is relatively small, though the actual error may be considerable. Taking the logarithm of each side of (43), we find the formula
log r(n
(44)
+
1)
=
(n
+
1) logn
-
n
+ 1 log(27r) +
e,
we have an
expression very small when n is very large. Neglecting e, This formula is inter called the asymptotic value of logT(n 1). esting as giving us an idea of the order of magnitude of a factorial.
where which
e is
+
is
142.
s theorem. The formula for integration under the integral function any /(x, y) which is continuous in the rectangle of inte Hence, if two different results are obtained by two different methods
D
Alembert
sign applies to
gration. of integrating the function /(x, y), we may conclude that the function /(x, y) is Gauss deduced discontinuous for at least one point in the field of integration. this fact an elegant demonstration of d Alembert s theorem. Let F(z) be an integral polynomial of degree m in z. We shall assume for definiteness that all its coefficients are real. Replacing z by p(cosw + isinw), and separating the real and the imaginary parts, we have
from
F(z)
where
If
we
set
P=A
p
Q=A
p
V=
m m smmw
cosm<,}
a/T
and
it is
+ Aip m + A\pm ~
arc tan (P/Q),
8V =
-
we
l
shall
aP_ p aQ dp
dp
P*+
l
Q2
=P+ cos(m sin(m
iQ, !)&>
1)
w
have
8V ~ ~^
q^-P^ du 8u P2
+
Q2
evident, without actually carrying out the calculation, that the second
derivative
is
of the
form
&V
M
DOUBLE INTEGRALS
292 where
[VI, Exs.
M
is a continuous function of p and u. This second derivative can only be discontinuous for values of p and for which P and Q vanish simultaneously, that is to say, for the roots of the equation F(z) = 0. Hence, if we can show that
the two integrals
r\du I
(4o)
R
c -dp, w* d du
I
P
Jo
Jo
C
I
R *
dp
c"*r* du
I
d P Cu
Jo
Jo
are unequal for a given value of R, we may conclude that the equation F(z) = has at least one root whose absolute value is less than R. But the second inte is
gral
always zero, for ^ au
=
I
Jo
f"
and dV/dp is a periodic function of w, of period 2x. gral in a similar manner, we find
Calculating the
first
inte
X and
it is
easy to
show
that
dV/du
of the
is
dV _
form
m+
mA?) 2m -f p"
du
Ag
-
p
where the degree of the terms not written down is less than 2m in p, and where the numerator contains no term which does not involve p. As p increases indefi nitely, the right-hand side
approaches
m.
m
fQ
hence the
first of
R may be chosen so large m + where e is less than m
Hence
that the value of cV/dw, for p = R, is equal to in absolute value. The integral 2n ( + e)
e,
du
is
evidently negative, and
the integrals (45) cannot be zero.
EXERCISES 1.
At any point
of the catenary defined in rectangular coordinates
by the
equation
y
- ea + = 0/5 I
-
e
2 \
us draw the tangent and extend it until it meets the x axis at a point T. Revolving the whole figure about the x axis, find the difference between the areas described by the arc of the catenary, where A is the vertex of the catenary, let
AM
and that described by the tangent point
M,
(2)
MT
(I)
as a function of the abscissa of the
as a function of the abscissa of the point T. [Licence, Paris, 1889.]
Using the usual system of trirectangular coordinates, let a ruled surface The plane zOA revolves about the x axis, while the gen erating line -D, which lies in this plane, makes with the z axis a constant angle whose tangent is X and cuts off on OA an intercept OC equal to Xa0, where a is a given length and 6 is the angle between the two planes zOx and zOA. 2.
be formed as follows
:
EXERCISES
VI.Exs.]
293
the solid bounded by the ruled surface and the planes 1) Find the volume of xOy, zOx, and zOA, where the angle 6 between the last two is less than 2n. the portion of the surface bounded by the planes xOy, 2) Find the area of
zOx, zOA. [Licence, Paris, July, 1882.] 3.
&2 X2
Find the volume of the solid bounded by the xy plane, the cylinder a 2&2 anc[ the elliptic paraboloid whose equation in rectangular a2y2
_|_
?
coordinates
is
2z
_
c
&
y*
p2
q*
[Licence, Paris, 1882.] 4.
Find the area of
the curvilinear quadrilateral bounded by the four con-
focal conies of the family
which are determined by giving X the values
c 2 /3,
2c 2/3, 4c a /3, 5c 2/3, respectively. [Licence, Besan^on, 1885.]
5.
Consider the curve
y
= \/2 (sin x
cos x)
,
where x and y are the rectangular coordinates of a point, and where x varies from 7T/4 to 5?r/4. Find :
1)
the area between this curve and the x axis
2) the
volume of the
3) the lateral
;
by revolving the curve about the x axis
solid generated
;
area of the same solid. [Licence, Montpellier, 1898.]
6.
In an ordinary rectangular coordinate plane let A and B be any two be any curve joining A and B which, together y axis, and let whose area is a preAB, forms the boundary of a region
AMB
points on the with the line
AMBA
assigned quantity S. over the curve
AMB
of the following definite integral taken
Find the value :
my] dx
m is a constant,
where
+
x
[
(y}e
and where the function
- m]dy,
(y),
together with [Licence,
7.
By
>
+ 00
/ ft
Jo /O
e- x
different ways,
show
!
Jo that,
sinaxdydx
provided that a
+ 00
8.
Nancy, 1895.]
+00
I
two
derivative
calculating the double integral ft
in
its
continuous.
(y), is
.
sin
ax
,
is
not zero,
ft ,
Find the area of the lateral surface of the portion of an ellipsoid of revo an hyperboloid of revolution which is bounded by two planes perpen
lution or of
dicular to the axis of revolution.
DOUBLE INTEGRALS
294 To
9*.
area A
find the area of
[VI, Exs.
Half of the total
an ellipsoid with three unequal axes.
given by the double integral
is
1-
X2
-
-
C"
dxdy
extended over the interior of the ellipse 62 x2 + a2 ?/ 2 = a 2 ft2 Among the methods employed to reduce this double integral to elliptic integrals, one of the simplest, 125. due to Catalan, consists in the transformation used in Denoting the .
integrand of the double integral by v, and letting v vary from 1 to + easy to show that the double integral is equal to the limit, as I becomes
cc,
it is
infinite,
of the difference il
7tabl(P-l)
This expression
is
nab
an undetermined form
;
but we
may
write
--
-
>
and hence the
limit considered above is readily seen to be +00
/.2
/
Ttab
ab
dv 1,2
i
10*. If
from the center of an
dicular be let fall
which
is
ellipsoid
ellipsoid
upon the tangent plane
J \
(V
2_l +
V
U
\
whose semiaxes are
a, 6, c
C2
\ &*/ -
a perpen
to the ellipsoid, the area of the surface
the locus of the foot of the perpendicular
whose semiaxes are
^W 2_1 +
a2
is
equal to the area of an
be/a, ac/b, ab/c.
[WILLIAM ROBERTS, Journal de
Liouville, Vol. XI, 1st series, p. 81.]
EXERCISES
VI, EXB.]
11.
295
Evaluate the double integral of the expression
-
(*
V)"f(v)
extended over the interior of the triangle bounded by the straight lines y = i = x, and x = in two different ways, and thereby establish the formula
X
y
dx
From
this result
(x
-
n
y) f(y)
dy
=
y
I
f(y)
^
dy
,
.
deduce the relation X
x-
l
C f(x)dx= f\x J*9 (n-l)!^x
- y)f(y)dy.
In a similar manner derive the formula X
f(x)
dx
= 2
.
4
.
\ 6
-
f 2n Jr
*(x*
-
2 2/
)
and verify these formulae by means of the law for differentiation under the integral sign.
CHAPTER
VII
MULTIPLE INTEGRALS INTEGRATION OF TOTAL DIFFERENTIALS
I.
CHANGE OF VARIABLES
MULTIPLE INTEGRALS
143. Triple integrals. Let F(x, y, z) be a function of the three variables x, y, z which is continuous for all points M, whose rec tangular coordinates are (cc, y, z), in a finite region of space bounded by one or more closed surfaces. Let this region be sub
()
divided into a number of subregions (e^, (e 2 ~), (e n ), whose vol v be coordinates of any and let the umes are v l9 v z , n (., 17,-, ) ,
,
point
m,-
,
;
of the subregion
(e,-).
Then
the
sum
(i)
approaches a limit as the number of the subregions (e.) is increased indefinitely in such a way that the maximum diameter of each of
them approaches
zero.
the function F(x, is
y,
This limit
called the triple integral of
is
z) extended throughout the region (), and
represented by the symbol
JJJ
(2)
The proof that
F(x,
dx dy dz
y, z}
.
this limit exists is practically a repetition of the
proof given above in the case of double integrals. Triple integrals arise in various problems of Mechanics, for instance in finding the mass or the center of gravity of a solid filled with a heterogeneous sub Suppose the region body. stance, and let p.(x, y, z) be the density at any point, that is to say,
()
the limit of the^ratio of the mass inside an infinitesimal sphere about and /x 2 the point (x, y, ) as center to the volume of the sphere. If
^
are the it is
and
maximum and
the
minimum
value of
/t
in the subregion
evident that the mass inside that subregion fujv,-;
hence
it
is
equal to
v,./i(,.,
suitably chosen point of the subregion 296
17.-,
(e ). (
,.),
lies
where
(e^),
between ^v,
(,-,
7;,-,
,-)
is
a
It follows that the total
VII,
CHANGE OF VARIABLES
INTRODUCTION
143]
297
mass is equal to the triple integral fffp dx dy dz extended through out the region (). The evaluation of a triple integral may be reduced to the suc Let us suppose
cessive evaluation of three simple integrals.
first
a rectangular parallelepiped bounded by the six planes x x x z z = Z. Let (E) X, y F, z y y be divided into smaller parallelepipeds by planes parallel to the that the region
()
=
three
=
,
yk _
l
)
=
,
,
The volume of one and we have to ,_!),
coordinate planes.
o^j) (yk
(x f
is
(z t
of
the
latter
is
find the limit of
sum
the
S
(3)
=
where the point (, w
any point inside the corresponding first that part of S which arises parallelepiped. from the column of elements bounded by the four planes ij ikl
,
lW ) is
,
Let us evaluate
the points
all
taking
y
= y/c-i(x {
(
ikl
yM
,
iw )
,
upon the straight
This column of parallelepipeds gives
-x_ i
l
)(yt
-
yt_i)[^(*,-_i,
y^i, Ci)(i
line
rise to
-
)
the
H
x
=
ar.-.j,
sum ],
s may be chosen in such a way that the 123, the inside the bracket will be equal to the simple integral quantity
and, as in
*(**-! It only
But
yt-i)
=
F( x
I
Jz
i-i>
yt-u
*) d*-
remains to find the limit of the sum
this limit is precisely the double integral
$(x,
?/)
dx dy
extended over the rectangle formed by the lines x y = 2/o y = Y. Hence the triple integral is equal to >
dx
I
or,
pX I
J
*(x, y)dy,
its value,
replacing &(x, y) by (4)
I
Jv
J*,
*,*
f*Y
dx
f*Z
dy Jvn
*J *a.
F(x,y,z)dz.
=x
,
x
= X,
MULTIPLE INTEGRALS
298
[VII,
144
The meaning of this symbol is perfectly obvious. During the first integration x and y are to be regarded as constants. The result will be a function of x and y, which is then to be integrated between the limits and F, x being regarded as a constant and y as a variable. The ?/ result of this second integration is a function of x alone, and the last step is the integration of this function between the limits x and X.
There are evidently as many ways of performing this evaluation on three letters, that is, six. For instance,
as there are permutations
the triple integral
I
equivalent to
is
rZ
f*Y
r>X
dz
I
J*
J*
dx
I
r>
F(x,
J
y,
z)dy =
I
J*a
*() denotes the double integral of F(x, y, z) extended over Y. We x x the rectangle formed by the lines x X, y y y of the with the this formula rediscover part might by commencing
where
=
,
=
=
,
=
arises from the layer of parallelepipeds bounded by the two planes z = z _ l z = z Choosing the points (, 77, ) suitably, the part of S which arises from this layer is
sum S which
l
and the
,
(
.
rest of the reasoning
is
similar to that above.
Let us now consider a region of space bounded in any manner whatever, and let us divide it into subregions such that any line parallel to a suitably chosen fixed line meets the surface which 144.
bounds any subregion in at most two points. We may evidently restrict ourselves without loss of generality to the case in which a line parallel to the z axis meets the surface in at most two points. The points upon the bounding surface project upon the xy plane into the points of a region A bounded by a closed contour C.
To every FIG. 32
=
and
(x, y)
inside
C
cor
We
shall suppose that the functions fa( x y}Let us now are continuous inside C, and that l t >
i
point
respond two points on the bounding surface whose coordinates are .
<j>
<<j>
VII,
CHANGE OF VARIABLES
INTRODUCTION
144]
299
divide the region under consideration by planes parallel to the coor Some of the subdivisions will be portions of paral dinate planes. The part of the sum (1) which arises from the column lelepipeds. of elements
y = yk
is
bounded by the four planes x = by 124, to the expression
Xf_ l}
x
=
x
{ ,
y
= yk _i,
equal,
(x f
,._!>
(yk
- yt _,)
where the absolute value of
I
F(x
J
{
_
l
yt _ lt
,
may be made
z)
dz
+ fk ,
J
,
than any preassigned parallel planes sufficiently near together. e ik
number c by choosing the The sum
less
approaches zero as a limit, and the triple integral in question
is
therefore equal to the double integral
y) dx dy
<(x,
extended over the region (J) bounded by the contour C, where the function y) is denned by the equation 3>(z,
=
r**
F
I
(*>
y>
Jz.
*)**
y axis meets the contour C in at most two are y = ^ (x) and y = fa (x), respectively, whose coordinates points while x varies from x l to a; 2 the triple integral may also be written in the form If a line parallel to the
,
rt
rx
t
/
(5)
dx
i/Xj
The
limits z 1
and
z2
dy
\
J
y^
rz
t
I
F(x,
Jz l
y,
z)dz.
depend upon both x and y, the limits y t and yz and finally the limits x l and a? 2 are constants.
are functions of x alone,
We may
invert the order of the integrations as for double inte grals, but the limits are in general totally different for different orders of integration. Note.
If ^(x) be the function of
rz
/
*()
=| Jv,
x given by the double integral
dy
I
J*,
*
F(x, y, z) dz
MULTIPLE INTEGRALS
300
[VII,
145
extended over the section of the given region by a plane parallel to the yz plane whose abscissa is x, the formula (5) may be written
This
is
the result
we should have
obtained by starting with the
=
Xf_ l} x = x layer of subregions bounded by the two planes x this contributes to the Choosing the points (, r), ) suitably, layer total sum the quantity
Example. Let us evaluate the out that eighth of the sphere x2
triple integral
+
y*
+
z2
=
t
.
fffz dx dy dz extended through
fi*
which
lies in
the
first
octant.
If
integrate first with regard to z, then with regard GO y, and finally with regard to x, the limits are as follows x and y being given, z may vary from zero to 2 x 2 and x itself x2 x being given, y may vary from zero to V.R2 V-R 2 y
we
:
;
;
may
vary from zero to B.
/*// I
I
I
J J J
whence we
Hence the
zdxdydz =
~V2-:r2
(*R I
integral in question has the value
dx
Jo
/>V
dy
I
Jo
ff2-x2-y4
zdz,
I
Jo
find successively
f
zdz
Jo
=
-x2 -y ),
2
2
(
if 2 Jo and it merely remains by the substitution x
Hence the value
to calculate the definite integral
= B cos
,
x2)^dx, which,
is,
by
116,
x
Let
= y= z = $(u, x
(6)
(R*
takes the form
of the given triple integral
145. Change of variables.
K
\f
/(?/,, v,
<(?*,
w),
v,
w},
v,
w)
,
be formulae of transformation which establish a one-to-one corre
spondence between the points of the region (E~) and those of another region (^i). We shall think of u, v, and w as the rectangular coor dinates of a point with respect to another system of rectangular
VII,
CHANGE OF VARIABLES
INTRODUCTION
145]
301
If F(x, y, ) is a coordinates, in general different from the first. continuous function throughout the region (), we shall always have
mF(x, dx dy dz mF[f(u, y, 2)
;>
(7)
D(f, v,
dudv dw,
w),
D(u,
i>
v,
where the two integrals are extended throughout the regions (E) and (E-i), respectively. This is the formula for change of variables in triple integrals. In order to show that the
commence by remarking that
formula
always holds, we shall
(7)
holds for two or more particular will hold also for the transformation obtained by if it
transformations, it carrying out these transformations in succession, by the well-known properties of the functional determinant ( 29). If it is applicable to several regions of space, it is also applicable to the region obtained
We shall now proceed to show, as we did for double integrals, that the formula holds for a transformation which leaves all but one of the independent variables unchanged, for by combining them.
example, for a transformation of the form
x
(8)
We
=x
1
y
=
a
?/
=
1
y z
\!/(x
}
M
two points M(x, y, z) and (x y ,z } are same system of rectangular axes, and that a parallel to the z axis meets the surface which bounds the region (E) in at most two points. The formulae (8) establish a corre spondence between this surface and another surface which bounds the region (E ). The cylinder circumscribed about the two sur shall suppose that the
,
referred to the
faces with its generators parallel to the z axis cuts the plane z along a closed
=
curve C.
Every point
inside the contour
C
m. of the region
A C
the projection of of the first surface,
is
FIG. 33
two points m^ and m 2 whose coordinates are z 1 and z2 respectively, and also of two points m[ and m 2 of the second surface, whose coordinates are z[ and z 2 respectively. Let us choose the notation in such a way ,
,
and z[
that z l
,
.
m
.
.
MULTIPLE INTEGRALS
302
[VII,
145
m
m
and the points l and 2 go into the On the other hand, if c\(//dz is respectively. t and 2 go into negative, z decreases as 2 increases, and s and m{, respectively. In the previous case we shall have
positive, z increases with %
points m{ and
,
ra^,
m
I */Zi
F(x, y, z)dz
=
m
m
^ dz
F^x,
y, t(x, y, 2 )]
F\x,
y,
$(x, y, **)] jfc dz
V|>,
y,
t(x, y, z )]
I
Jzi
,
whereas in the second case /"**>
F(x, y, z)
I
dz
=-
/* 2 I
9
Jz{
Jz^
/
.
In either case we may write
f
(9)
*F(x, y,
z)dz
=
f
If
dz
.
*) ZJ
C/Zj
we now consider the double
two
integrals of the
sides of this
equation over the region A, the double integral of the left-hand side,
dxdy I JII Je J(A) is
F(x,
y,
z)dz,
t
precisely the triple integral fffF(x, y, z) dx dy dz extended through Likewise, the double integral of the right-hand
out the region (E). side of (9)
is
equal to the triple integral of
F[x y ,
,
which readily follows when extended throughout the region (E x and y are replaced by x and y respectively. Hence we have in ~),
,
this particular case
J J J (E)
F(x, y, z) dx dy dz
dx
di
dz
.
1
But
in this case the determinant D(x, y, z)/D(x y z } reduces to Hence the formula (7) holds for the transformation (8). d(j//dz ,
,
.
Again, the general formula (7) holds for a transformation of the type (10)
*=/(*
,
y ,*%
y
= (x
,y<,zi),
VII,
CHANGE OF VARIABLES
INTRODUCTION
145]
We
where the variable z remains unchanged.
303
shall suppose that
the formulae (10) establish a one-to-one correspondence between the points of two regions (E) and ( ), and in particular that the
R and R made in (E) and ( ), respectively, by any plane parallel to the xy plane correspond in a one-to-one manner. Then by the formulae for transformation of double integrals we sections
shall have
(11)
=
I
RT
/
/
i/
fi-r.
?.
A*(r
\
.
-j/
j-
Y
^ dx dy
l
f.
,
D(x y
.
)
,
The two members z
=z
of this equation are functions of the variable Integrating both sides again between the limits z v
alone.
and 2 between which z can vary formula
JJ
F (x
y,
>
z)
(), we
in the region
,
find the
dx dv dz
J(E)
Jill =
F^^^,^^,^,^ rrr J(E
JJ
)
D(x
dx dy dz
.
,y<)
= D(x, y)/D(x y But in this case D(x, y, z)/D(x y Hence ) the formula (7) holds for the transformation (10) also. We shall now show that any change of variables whatever ,
(13)
x
= f(x
l}
y
yl} zj,
,
= <j>(
Xl ,
,
z
y l} zj,
~).
= ^(x
lt
y 1}
j)
be obtained by a combination of the preceding transformations. For, let us set x = x l} y = y lt z = z. Then the last equation of (13) may be written z = if/(x y z^, whence z v = 7r(x , y z Hence the equations (13) may be replaced by the six equations
may
,
(14) x (15)
=/[*
,
x
y
,
=
TT(X<,
xi,
y}
y
)],
y
=
,
yi,
,
= ftx *
,
V
<,
ir(x
= <j>(xi,
t
yi,
y
,
)],
z
~).
=z
,
,).
The general formula (7) holds, as we have seen, for each of the transformations (14) and (15). Hence it holds for the transforma tion (13) also.
We
might have replaced the general transformation (13), as the reader can easily show, by a sequence of three transformations of the type (8).
MULTIPLE INTEGRALS
304 146.
we
Element
of volume.
Setting F(x,
[VII,
=I
y, K)
146
in the formula (7),
find
//***// The
du dv dw
left-hand side of this equation
Applying the law of the mean
.
the volume of the region ("). on the right, we find
is
to the integral
the relation
(16)
D(u,
where Vi
is
v,
the volume of (^i), and
This formula
point in (#1).
Chapter VI.
are the coordinates of
rj,
some
exactly analogous to formula (17), that the functional determinant is the limit
shows two corresponding
It
of the ratio of
,
is
infinitesimal volumes.
w
in (6) be assigned a constant value, v, while the others are allowed to vary, we obtain three families of If one of the variables u,
surfaces,
u
=
const., v
= const., w =
const.,
by means
of
which the
region (T) may be divided into subregions analogous to the parallelopipeds used above, each of which is bounded by six curved faces.
The volume (u),
(u
+
of one of these subregions
+ dv),
du), (v}, (v
(w),
AF =
.
D(w,
v,
bounded by the surfaces is, by (16),
(w + dw} c f
-j-
w)
c?<
c?v
dw
,
J
dv, and div are positive increments, and where c is infini tesimal with du, dv, and dw. The term e du dv dw may be neglected, as has been explained several times ( 128). The product
where du,
dV =
(17)
D(u,
v,
w}
the principal part of the infinitesimal AF, and is called the element of volume in the system of curvilinear coordinates (u, v, w). Let c?s 2 be the square of the linear element in the same system of is
coordinates.
o/
dx
= ~- du cu
Then, from ^i /*
^
vv
(6),
-P
- dw.
dy
Ow
=
%
~JL
-r-
du
dz
= -^cu
du
whence, squaring and adding, we find
=H du*+H. dv*+H dw +2F dvdw+2F. dudiv+2F dudv, 2
l
t
3
1
t
3
VII,
CHANGE OF VARIABLES
INTRODUCTION
14fi]
305
the notation employed being
(19) (7 IT
C OC G OT
dw
du dw
(JOT
^
O *y
du dv
where the symbol AJ means, as usual, that x is to be replaced by y and z successively and the resulting terms then added. The formula for dV is easily deduced from this formula for ds*. For, squaring the functional determinant by the usual rule,
F
l
we
find
= M,
F,
whence the element
volume
of
is
equal to
\M du dv dw.
Let us consider in particular the very important case in which the coordinate surfaces (u), (?;), (w) form a triply orthogonal system, say, in which the three surfaces which pass through any The tangents to in point space intersect in pairs at right angles. the three curves in which the surfaces intersect in pairs form a tri-
that
is to
=
= 0, 0, F2 rectangular trihedron. It follows that we must have FI The formulae for sufficient. these conditions are also and 0;
^=
dV and (20)
ds*
ds
2
then take the simple forms
=H
a.
l
du*
These formulae
+H
may
2
dv
2
+ H dw s
dV
2 ,
also be derived
=
from certain considerations of
Let us suppose du, dv, and dw very small, infinitesimal geometry. and let us substitute in place of the small subregion defined above a small parallelopiped with plane faces. Neglecting infinitesimals of higher order, the three adjacent edges of the parallelopiped may be taken to be \fn\ dii, ^/If2 dv, and \/^ dw, respectively. The for mulae (20) express the fact that the linear element and the element of
volume are equal to the diagonal and the volume of this parallelo faces repre piped, respectively. The area ^/H1 H2 du dv of one of the sents in a similar
As an example (21)
x
manner the element
of area of the surface (w).
consider the transformation to polar coordinates
= p sin0cos<,
y
= p sin
z
= p cos 6,
MULTIPLE INTEGRALS
306
[VII,
146
where p denotes the distance of the point M(x, y, z) from the origin, 6 the angle between OM and the positive z axis, and the angle which the projection of OM on the xy plane makes with the positive x axis. In order to reach all points in space, it is sufficient to let p from zero to TT, and from zero to 2?r. vary from zero to +
<
<x>,
From
(21)
we
find
ds 2
(22)
=
+P
2
dp
2
+P
d6 2
2
sin 2
2 0d<j>
,
whence
dV =
(23)
These formulae
The three
may
2 p sin 6 dp dO
. d
be derived without any calculation, however.
families of surfaces
are concentric spheres (0), about the origin, cones of revolution about the z axis with their vertices
(/a),
(<)
and planes through These respectively. surfaces evidently form a triply orthogonal system, and the dimen at the origin,
the
z
axis,
sions of the elementary subregion are seen from the figure to be dp,
y
p dO, p sin
d<j>
the formulae (22)
;
and (23) now follow immediately.
To
FIG. 34
calculate in terms of the va
riables
p, 0,
and
<
a triple integral
extended throughout a region bounded by a closed surface S, which contains the origin and which is met in at most one point by a radius vector through the origin, p should be allowed to vary from zero to R, where R is the equation of the surface 6 from zero to TT /(0,
=
;
)
and
<
from zero
to 2?r.
V=
I
I
d<j>
Jo
The
first
Jo
dO
I
Jo
w
- c
d+
i/O
is
r
2
we may
write
R 3 sin dQ.
t/0
made of
and z defined cylindrical coordinates r, r sin w, z z. It is evident that y
by the equations x
is
P sinOdp.
integration can always be performed, and
Occasional use
;
For example, the volume of such a surface
= r cos
to,
=
=
and
dV = r dw dr dz
.
VII,
147. Elliptic coordinates.
The surfaces represented by the equation
(24)
a
X
where X conies.
an
CHANGE OF VARIABLES
INTRODUCTION
147]
is
-1=0,
X
b
a variable parameter and a
c
b
>
>
0,
c>
form a family of confocal
in space there pass three surfaces of this family,
Through every point
ellipsoid,
X
307
a parted hyperboloid, and an unparted hyperboloid.
For the equa
tion (24) always has one root \i which lies between b and c, another root X 2 between a and 6, and a third root X 3 greater than a. These three roots \i, X2 , Xg
are called the elliptic coordinates of the point whose rectangular coordinates are Any two surfaces of the family intersect at right angles if X be given (x, y, z). :
the values Xi
and X 2
tracted, a division
,
by
and the resulting equations be sub
for instance, in (24),
X2 gives
Xi
-
(25)
-
a)
-
6)(X 2
-
b)
c)(X 2
-
= 0, c)
which shows that the two surfaces (Xi) and (X 2 ) are orthogonal. In order to obtain x, y, and z as functions of Xi X 2 X 3 we may note that the ,
,
,
relation (X
is
-
o)(X
-
6)(X
-
c)
-
x2 (X
-
=
Setting X obtain the values
identically satisfied.
tion,
we
-
6)(X
-
c)
=
X
a,
y2
(X
-
=
X
6,
c)(X
c,
- a) - z 2 (X - a)(\ - b) = (X - Xi)(X - X 2 )(X - X 3 )
successively, in this
equa
- a)(a-X )(q-X 2 )
(X 3
1
b)(a
(a
(26)
(a
_
(X 8
_
c)
-
b)(b
c)
C )(X,
(a
c)(b
c)
whence, taking the logarithmic derivatives, ,
dx
=
x 2 V Xi
-
a
b
X2
b
X2
c
-p
I
2 \Xi
Forming the sum
of the squares, the terms in
4 L(Xi
a)
X3
b
Xs
c/
2
(\!
-
I
(X,
-
,
,
2
6)
XQft,
*
dXidX 2 dX 2 dX 8 dX 3 dXi must dis Hence the coefficient of dXj is
(X!
-
by their values and simplifying, 1
(27)
-
^)
d\ 3
~T
c
appear by means of (25) and similar relations.
or, replacing x, y, z
X3
+
*-5 UZ
a
X2
- XQ
4(X 1 -a)(X 1 -6)(X 1 -c)
2
c)
MULTIPLE INTEGRALS
308 The
[VII,
148
and 3f8 of d\\ and d\g, respectively, may be obtained from The element of volume is cyclic permutation of the letters.
coefficients 3f2
this expression
therefore
by
-VMi M2 Mz
d\i d\2 d\ a
.
Consider the triple integral
148. Dirichlet s integrals.
xpyiz r (l
x
y
dxdydz
z)
taken throughout the interior of the tetrahedron formed by the four planes x = 0, 2/ = 0, z = 0, x + y + z = 1. Let us set
x
where
and
,
+
=
z
+
y
,
z
=
new variables. These
f are three
r/,
+
y
TJ
z
,
formulae
=
|r;f ,
may be written in the form
the inverse transformation is
When
x, y, and z are all positive and x + y + z is less than unity, 17, and f all between zero and unity. Conversely, if 17, and f all lie between zero and The tetra unity, x, y, and z are all positive and x + y + z is less than unity. hedron therefore goes over into a cube. ,
lie
,
In order to calculate the functional determinant,
transformation z
=
Z.
X=f, F=)j,
Z=
?;f ,
f
D(x,
y, z)
D(,
T,,
=
dg
f
rV +
*i
/0
D(x,
77,
D(X,
f)
triple integral
t/O
z)
D(JT, r, Z)
Y, Z)
f
Jo
p
=
X
=F
F, y
Z
,
+r+2 (i-
_ lyfy
O
i 7+r+i
l)
+
q
Canceling the found to be
+
r
+
+ 4)
s
common
of
a- ^ra- f)df.
f,
a function of
77,
and a func
written in the form
X C 7 9 + + l(l -
-
T? )?dr,
/o
X f /o
introducing T functions (see (33),
T(p
=
^
becomes
/0
+ 9 + r + 2(l
r,,
J>(|,
The integrand is the product of a function Hence the triple integral may be tion of f.
(28)
us introduce the auxiliary
Hence the functional determinant has the value
and the given
or,
let
which gives x
T(q
p. 280),
+ r + 2)T(p+l) + q + r + 3)
Y(p
T(q
+r+
2)
factors, the value of the given triple integral is finally
r(p+i)r( g +i)r(r+pr(+i)
VII,
149]
CHANGE OF VARIABLES
INTRODUCTION A
149. Green s theorem.*
derived for triple integrals.
309
formula entirely analogous to (15), 126, may be Let us first consider a closed surface S which is
met in at most two points by a parallel to the z axis, and a function R(x, y, z) which, together with dR/dz, is continuous throughout the interior of this surface. All the points of the surface S project into points of a region of the xy plane
A
bounded by a closed contour C. To every point of A inside C corre = 2 (x, y). spond two points of S whose coordinates are z\ = (x, y) and z 2 The surface S is thus divided into two distinct portions /Si and 83 We shall which
is
\
<j>
.
suppose that z\ is less than z 2 Let us now consider the triple integral .
dz
dxdydz
taken throughout the region bounded by the closed surface S. A first integra may be performed with regard to z between the limits z\ and z 2 ( 144),
tion
-
which gives R(x, y, z 2 ) R(x, equal to the double integral
J J[Jfc(,
The given
y, z t ).
y, z 2 )
-
triple integral is therefore
R(x, y, Zi)]dxdy
over the region A. But the double integral surface integral ( 135)
f f R(x,
y, z 2 )dxdy is
equal to the
R(x, y, z)dxdy
taken over the upper side of the surface /S2 Likewise, the double integral of R(x, y, Zi) with its sign changed is the surface integral .
R(x,y,z)dxdy Jff J(sj taken over the lower side of
/ / /
where the surface
<Si
aT*
integral
is
Adding these two
.
6
^* = ff R ^ y
integrals,
Z)
we may
write
dxd 2/
s
to be
extended over the whole exterior of the sur
face S.
By the methods already used several times in similar cases this formula may be extended to the case of a region bounded by a surface of any form whatever. Again, permuting the letters
and
z,
we
III
^ dxdy
III
-Qdxdydz= JCJC
JJJ** *
z, y,
obtain the analogous formulae
dz
Q(x, y, z)dzdx.
<^
Occasionally called Ostrogradsky s theorem. The theorem of 126 is sometimes Riemann s theorem. But the title Green s theorem is more clearly established and seems to be the more fitting. See Ency. der Math. Wiss., II, A, 7, b and c. called
TRANS.
MULTIPLE INTEGRALS
310 Adding these three triple integrals
formulae,
we
150
[VII,
Green
finally find the general
s
theorem for
:
(29)
= C J
P(x,
I
+
dy dz
y, z)
Q(x, y, z) dz dx
+
R(x, y, z)dxdy,
J(S)
where the surface integrals are to be taken, as before, over the exterior of the bounding surface. = R = QorQ = y, P R = Q or R = z, If, for example, we set P = x, Q P = Q = 0, it is evident that the volume of the solid bounded by S is equal to any one of the surface integrals (29
CC
)
J
CC
xdydz,
J J(S)
J(S)
JCC J(S)
ydzdx,
zdxdy.
150. Multiple integrals. The purely analytical definitions which have been given for double and triple integrals may be extended to any number of vari ables. We shall restrict ourselves to a sketch of the general process.
We
shall say for brevity Let Xi x2 %n be n independent variables. t x n of these variables represents a point in that a system of values x\ x% x ) = 0, whose first member , space of n dimensions. Any equation F(x\, x 2 , ,
,
,
,
tl
a continuous function, will be said to represent a surface; and if F is of the Let us consider the first degree, the equation will be said to represent a plane. totality of all points whose coordinates satisfy certain inequalities of the form is
ti(xi,X3,-.-,x H
(30)
i
)<0,
=
.-., k.
l, 2,
D
We shall say that the totality of these points forms a domain in space of n dimensions. If for all the points of this domain the absolute value of each of is the coordinates x,- is less than a fixed number, we shall say that the domain
D
finite.
If the inequalities
(31)
we
xJ^X!<x},
shall call the
which define
D are
x<;x2<x 2
of the
-,
,
domain a prismoid, and we
form
<<x
n
^xj,,
shall say that the
xf are the dimensions of this prismoid. lies on the frontier of the point of the domain functions fr in (30) vanishes at that point. tities x\
Finally,
D
domain
if
we
n
positive
quan
shall say that
a
at least one of the
x n) be a function which Now let Z) be a finite domain, and let f(x\ x 2 continuous in that domain. Suppose D divided into subdomains by planes = (t = 1, 2, n), and consider any one of the prisparallel to the planes x t ,
,
,
is
-
,
moids determined by these planes which lies entirely inside the domain D. Ax n be the dimensions of this prismoid, and let i 2 Let Axi Ax 2 n be the coordinates of some point of the prismoid. Then the sum ,
,
S=
(32)
formed
,
,
for all
S/(fc,
2
,--,) AX! Ax 2
the prismoids which
lie
Ax
entirely inside the
a limit I as the number of the prismoids
is
,
,
,
domain
Z),
approaches
increased indefinitely in such a
way
VII,
that
CHANGE OF VARIABLES
INTRODUCTION
150]
them approach
of the dimensions of each of
all
I the n-tuple integral of f(x\ x 2 denote it by the symbol limit
,
1
The evaluation
=ff-
*2,
dx n
z n )dzidz 2
,
We
zero.
shall call this
taken in the domain
z,,)
,
,
311
D and shall
.
"fffai
be reduced to the evaluation of this in general, we need only (n l)-tuple integral, it will also be true for an For this purpose let us consider any point (xi x2 xn ) n-tuple integral. of D. Discarding the variable z n for the moment, the point (x\ x 2 xn _ i) evi We shall suppose dently describes a domain D in space of (n 1) dimensions. of an n-tuple integral
n successive simple integrals. show that if it is true for an
may
show
In order to
,
,
,
,
,
,
that to any point (x 1? X2, x n _i) inside of JK there correspond just two , xn _i; x^) and points on the frontier of Z), whose coordinates are (xi, X2, , x n _i x^2) ), where the coordinates x^ and x^2) are continuous func , (xi, x 2 , tions of the n 1 variables x\ x 2 x n -i inside the domain IX. If this con ;
,
dition were not satisfied,
,
,
we should
that the condition would be
met by each
(xi
treated in
124, that the part of
x2
,
,
,
xn -i)-
It is easy to
S which
r Ax 1 Ax 2 ---Ax n _ i where
e |
|
may
(i)
it is
*(xi, x 2
,
which correspond
we
to the (n
1
(34)
in the
domain
integral,
it
is
to the
did in the similar case
column
this
--^x^dXn +
,
-,
xn
we now
If
of prismoids
is
~\ e
,
set
r
-i)=J
(1)
clear that the integral I will be equal to the limit of the
is,
domains so small Let us now
be made smaller than any positive number whatever by choos
S*(xi, x 2 that
from
arises
/(x 1 ,x2
ing the quantities Ax* sufficiently small.
(33)
D
show, as
r *w
J
into
of the partial domains.
consider the column of prismoids of the domain
same point
D
domain
divide the
,
,
x n _i)AxiAx 2
-
sum
Ax n _i,
l)-tuple integral
=
fff- --J^Xi,
x2
,
,
xn -i)dxi---dxn -i,
The law having been supposed
ZK.
evident,
to hold for an (n l)-tuple it holds in general. Consider the totality of points
by mathematical induction, that
We
might have proceeded differently. Then the Xn) for which the coordinate x n has a fixed value. xn _i) describes a domain 8 in space of (n point (xi, x 2 1) dimensions, and it is easy to show that the n-tuple integral I is also equal to the expression (xi
,
za
,
,
,
(35)
,
I
where 0(x n) is the (. l)-tuple integral /// -ffdxi dxn-i extended through out the domain 5. Whatever be the method of carrying out the process, the limits for the various integrations depend upon the nature of the domain D, and
MULTIPLE INTEGRALS
312 vary
general for different orders of integration. by inequalities of the form
in
150
[VII,
An
exception exists in case
D is a prismoid denned The multiple
integral
and the order
in
is
then of the form
which the integrations are performed may be permuted in any altering the limits which correspond to each of the
way whatever without variables.
The formula
for change of variables also
may
be extended to n-tuple integrals.
Let Xi
(36)
=
0,-(zi, zg,
,
x n ),
t
= l,
-,n,
2,
be formulae of transformation which establish a one-to-one correspondence between the points (xi z 2 .,--, xn ) of a domain jy and the points (xi domain D. Then we shall have ,
,
x2
xn ) of a
,
,
(37)
The proof argument
similar to that given in analogous cases above. that we shall attempt here.
is
A
sketch of the
is all
1) If (37)
holds for each of two transformations,
it
also holds for the trans
formation obtained by carrying out the two in succession. 2) Any change of variables may be obtained by combining two transforma tions of the following types (38)
Xi
(39)
zi
=
x2
xf,
=
:
Xa,
,
xn _i
=
x;,_i,
xn
n (xi, xg,
x;,),
,
The formula
3) (37) holds for a transformation of the type (38), since the given n-tuple integral may be written in the form (34). It also holds for any transformation of the form (39), by the second form (35) in which the multiple
integral
These conclusions are based on the assumption that by mathematical
be written.
may
The usual reasoning l)-tuple integral. induction establishes the formula in general. (37) holds for
an (n
As an example I
let
= CC-
us try to evaluate the definite integral
f xf x?2
x"
(1
-
xi
-
x2
-----
x n f dxi dx 2
---
dx n
where ai, cr 2 an are certain positive constants, and the integral extended throughout the domain D defined by the inequalities ,
G<X!,
,
,
0
The transformation
,
,
0<x n
,
xi
+
x2
+
+x n
<
1.
,
is
to be
VII,
TOTAL DIFFERENTIALS
151]
D into
carries
new domain IX
a
it is
easy to show as in ,
The new integrand +-+ .+
^
1
I
II.
1
integral
(40)
=
--,
<
148 that the value of the functional determinant
X2
-,
,
Xn)
_
t n-i
fc
is
-2
therefore of the form
is
-i^
and the given
defined by the inequalities
0<&<1,
0
and
313
+-+
may
.
+
-*...^ (1 _ f y (1 _ &)
1
...(1
_&,)-.,
be expressed, as before, in terms of T functions
:
T(
INTEGRATION OF TOTAL DIFFERENTIALS
151. General method. Let P(x, y) and Q(x, y~) be two functions of the two independent variables x and y. Then the expression
Pdx + Qdy is not in general the total differential of a single function of the variables x and y. For we have seen that the equation
du
(41) is
two
= Pdx + Qdy
equivalent to the two distinct equations
=
(42)
=
*<-,),
(*,). .
Differentiating the first of these equations with respect to y and the second with respect to x, it appears that u(x, y) must satisfy each of the equations C2 u
cxdy
A
_
dP(x, y)
82 u
dy
Cydx
_
dQ(x, y) dx
necessary condition that a function u(x, y) should exist which requirements is that the equation
satisfies these
(43)
dx Cy _.! should be identically satisfied. This condition is also sufficient. For there exist an infinite
number
of functions u(x, y) for which the first of equations (42) All these functions are given by the formula
is satisfied.
= / Jx n
P(x,y)dj.
MULTIPLE INTEGRALS
314
[VII,
151
where x is an arbitrary constant and Fis an arbitrary function of y. In order that this function u(x, y) should satisfy the equation (41), it is necessary and sufficient that its partial derivative with respect to
x should be equal to Q(x,
f_
that the y), that is,
dp
+
fe
<
Jxn should be
But by the assumed
satisfied.
ftx
7\T>
I
f*
=
dx
I
_=^
x
I
Cy
si
f)
=
-z-dx CX
I
The right-hand there are an
y)
relation (43)
Q(x, y)
whence the preceding relation reduces
equation
-
Q(x
y)
,
we have ,
to
Hence side of this equation is independent of x. number of functions of y which satisfy the
infinite
formula equation, and they are all given by the
= f An
Q(o,
an arbitrary value of y, and C is an arbitrary constant. It follows that there are an infinite number of functions u(x, y) which satisfy the equation (41). They are all given by the formula
where y
is
u
(44)
and
differ
=
P(x, y} dx
f Jx
+
I
J
Q(x
,
/Q
from each other only by the additive constant
C.
Consider, for example, the pair of functions
~
a;
+ my + y*
which
mx
y x2
x2
+y
2
=
Setting x
satisfy the condition (43).
and y
= 1,
formula for u gives
C -^dy C*x + my. + 2 dx+ ,
u=l Jo
,
x*
I
+y
J
l
C,
y
whence, performing the indicated integrations, we find
u or,
= -1 [log(a; + y )]* + m * 2
2
F
L
arc tan
x~\ -
x
+
log y
yjo
simplifying,
u
=
1 *
2
log(cc
+
y
x
2
)
-f
m arc tan - + y
C.
+
C,
the
VII,
TOTAL DIFFERENTIALS
151]
The preceding method may be extended
R
any number of inde
to
We shall
give the reasoning for three variables. be three functions of x, y, and z. Then the total
variables.
pendent Let P, Q, and
315
differential equation
du
(45) is
= Pdx +
Q dy
+ R dz
equivalent to the three distinct equations
=
= P,
(46)
= R.
Q,
2 Calculating the three derivatives d^u/dxdy, d u/dydz, d^u/dzdx in two different ways, we find the three following equations as neces
sary conditions for the existence of the function u
d_P_d_Q, dx dy
2Q
= fL^
dz
dy
:
= dJL.
dx
dz
Then, by the Conversely, let us suppose these equations satisfied. there exist an infinite number of functions u(x, y, z) whose
first,
partial derivatives with respect to x and y are equal to respectively, and they are all given by the formula
u where
=
P(x,
I
Z denotes an
y,
z)dx
+
Q(x
I
arbitrary function of
tive du/dz should be equal to R, the equation
it is
y,
,
z)dy
R(x,
- R(x
,
y, z}
+ R(x
,
y, z)
=
Q,
+ Z,
necessary and sufficient that
satisfied. Making use of the relations (47), to hold, this condition reduces to the equation
y, z)
and
In order that the deriva
z.
should be
assumed
P
- R(x
,
y
,
z)
+
which were
= R(x,
y, z}
,
R(x ,y ,z).
an infinite number of functions u(x, y, z) exist which satisfy the equation (45). They are all given by the formula It follows that
f* x
(48)
u
=
where x
I
JxK ,
P(x, y, z)dx
+
f* z
pv /
Jy
Q(x
,
y,
z)dy+
\ c>
R(x ,y z)dz ,
yn z a are three arbitrary numerical values, and C ,
arbitrary constant.
+ C,
za
is
an
MULTIPLE INTEGRALS
316
[VII,
152
Pdx + Qdy. The same subject may be The integral treated from a different point of view, which gives deeper insight Let P(x, y) and Q(x, y} into the question and leads to new results.
^
x>y)
152.
be two functions which, together with their first derivatives, are continuous in a region A bounded by a single closed contour C. It may happen that the region A embraces the whole plane, in which case the contour C would be supposed to have receded to
The
infinity.
line integral
I
taken along any path
D which
ties (x
lies in
A
will
depend
in general
upon
try to find the conditions under this integral depends only upon the coordinates of the extremi be any two points and y ) and (x i} yv ) of the path. Let
Let us
the path of integration.
which
Pdx + Qdy
first
M
,
N
be any two paths which connect these two points without intersecting each other between the extremities. Taken together they form a closed contour. In order that the values of the line integral taken along these two paths L and L should be of region A, and let
L and L
it is evidently necessary and sufficient that the integral taken around the closed contour formed by the two curves, proceeding
equal,
always in the same sense, should be
zero.
issue is exactly equivalent to the following under which the line integral
:
Hence the question at What are the conditions
fPdx + Qdy taken around any closed contour whatever which
should vanish
The answer theorem
to this question
Pta + a Ay
where C
is
integral
is
if
A
in the region
an immediate result of Green
is
s
:
(49)
that
lies
?
-
=//(|f
g
double any closed contour which lies in A, and where the whole interior of C. It is clear
to be extended over the
the functions
P and Q
satisfy the equation
the line integral on the left will always vanish. also necessary.
For,
if
dP/dy
This condition
is
dQ/dx were not identically zero
VII,
TOTAL DIFFERENTIALS
152]
317
in the region A, since it is a continuous function, it would surely be possible to find a region a so small that its sign would be constant But in that case the line integral taken around the inside of a.
boundary of a would not be If the condition (43
)
is
by
zero,
(49).
identically satisfied, the values of the
integral taken along two paths L and and are equal provided the points
V
between the same two two paths do not intersect and N. It is easy to see that the same thing is true when the two paths intersect any number of times between even and N. For in that case it would be necessary only to compare
M between M
N
M
the values of the integral taken along the paths L and L with its value taken along a third path which intersects neither of the L",
M
and N. preceding except at Let us now suppose that one of the extremities of the path of integration is a fixed point (x 0) y }, while the other extremity is a variable point (x, y) of A. Then the integral V)
X(*, o-
Pdx + Qdy
Vo>
taken along an arbitrary path depends only upon the coordinates The partial derivatives of this (x, y*) of the variable extremity. function are precisely P(x, y) and Q(x, y).
For example, we have
s*(x + A:r, y)
F(x
+ As,
y)
=
F(x, y}
+
I
P(x, y} dx,
/(*
for
we may suppose
that the path of integration goes from (x y ) to (x -f Ace, y) along a line parallel to y~) ,
to (x, y), and then from (x, the x axis, along which dy
may
write
F(x v
-
?/)
u
F(L
Taking the limit when
Fy =
= 0.
-
+ Ace,-*
v)
Applying the law of the mean, we
= P(x + Ace
0Az, y},
0<0<1.
approaches zero, this gives
Fx = P.
The
line integral F(x, y}, therefore, satisfies the Similarly, total differential equation (41), and the general integral of this Q.
equation is given by adding to F(x, y) an arbitrary constant. This new formula is more general than the formula (44) in that the path of integration is still arbitrary. It is easy to deduce (44)
from the new form.
To avoid ambiguity,
let (x y ) and (a^, yi) be two extremities, and let the path of integra be the two straight lines x = x y = y^ Along the former,
the coordinates of the tion
,
,
MULTIPLE INTEGRALS
318
=x = y y
x
is
,
l}
= 0, and = 0, and x
dx
dy
y varies from y
[VII,
153
Along the second, Hence the integral (50)
to y t
varies from x to
.
equal to
r, I
J;/
which But
differs
from (44) only
in notation.
might be more advantageous to consider another path of be the equations of a curve Let x = /(), y = integration. and let t be supposed to vary con y ) and (x lt y^, joining (x t t as the to point (x, y) describes the curve tinuously from Then we shall have between its two extremities. it
<}>()
,
l
l
+
pdx
f
Qdy
= I Jf
>
/<%,
If the path be #), example, we should set x = x + tfa to 1. and we should let t vary from y = y + t(y y,,), (x, y) of the equation (41) Conversely, if a particular integral be known, the line integral is given by the formula
where there remains but a single quadrature. a straight
for
line, l
(x.
y)
=
L which
is
3>(x,
y)
3>(x
,
yc),
analogous to the equation (6) of Chapter IV.
153. Periods. first place,
contours.
More general
cases
may
be investigated.
In the
Green s theorem applies to regions bounded by several Let us consider for defmiteness a region A bounded by an exterior contour C and two contours and which lie inside the first (Fig. 35). Let P and Q be two functions which, together with C"
C"
their first derivatives, are continuous in this
region.
and
C"
(The regions inside the contours C should not be considered as parts of
the region A, and no hypothesis whatever
made regarding Let the contours
C and
C"
P
and Q inside these
is
regions.)
be joined to the contour C by trans
and cd. We thus obtain a closed contour abmcdndcpbaqa, which may be described at one stroke. Applying Green s theorem to the region bounded by this contour, the line integrals versals ab
or F,
VII,
TOTAL DIFFERENTIALS
153]
319
which arise from the transversals ab and cd cancel out, since each It follows that of them is described twice in opposite directions.
fPdx + Qdy
f/Y-JJ \&;
r/0
u
line integral is to be taken along the whole boundary of in the senses the region A, i.e. along the three contours C, C , and indicated by the arrows, respectively, these being such that the
where the
C",
region j\ always lies on the left. If the functions P and Q satisfy the relation dQ/dx dP/dy in the region A, the double integral vanishes, and we may write the
=
resulting relation in the
I
(51)
form
Pdx + Qdy =
I
Pdx + Qdy +
J(C )
J(C)
where each of the
I
Pdx
+
J(C")
line integrals is to
be taken in the sense desig
nated above.
Let us now return to the region A bounded by a single contour C, and let P and Q be two functions which satisfy the equation
= dQ/dx, and which, together with their continuous except at a finite number of points of A, at which at least one of dP/dy
the functions
P
or
Q
is
first
derivatives, are
discontinuous.
We
shall suppose for definiteness that there are three points of discontinuity a, b, c in
Let us surround each of
A.
these points by a small circle, and then join each of these circles to the contour
C by a cross cut (Fig. 36). Then the integral j Pdx -\-Qdy taken from a fixed point (x T/O) to a variable point (.r, y) along a curve which does not cross any ,
of these cuts has a definite value at every point.
FIG. 36
For the contour C,
the circles and the cuts form a single contour which may be described shall call at one stroke, just as in the case discussed above.
We
such a path taken along
direct,
from
and shall denote the value of the
M
line integral
y ) to M(x, y) by F(x, y}. to We shall call the path composed of the straight line from a point a whose distance from a is infinitesimal, the circumference a loopof the circle of radius aa about a, and the straight line a it
(x
,
M
,
M
circuit.
The
line integral
fPdx
-f
Qdy taken along a
,
loop-circuit
MULTIPLE INTEGRALS
320
153
[VII,
reduces to the line integral taken along the circumference of the circle. This latter integral is not zero, in general, if one of the functions P or Q is infinite at the point a, but it is independent of the radius of the circle.
It is a certain constant
two senses
A, the
double
which the circumference be described. we shall denote B and C the may Similarly, by values of the integral taken along loop-circuits drawn about the two singular points b and c, respectively. and Any path whatever joining may now be reduced to a sign corresponding to the
M
in
M
M
combination of loop-circuits followed by a direct path from to M. For example, the path be reduced to a combination mdefM may
M M mdM M deM M efM
M
and fM. The path then be reduced to a about the singular M^mdM^ may loop-circuit and for the other two. the point a, similarly Finally, path fM of the paths
,
,
,
M
is
equivalent to a direct path.
of integration, the value of the line (52)
F(x, y)
where m,
n,
is
y)
+ mA +
+ pC
nB
,
and p may be any positive or negative
C
integers.
The
are called the periods of the line integral. evidently a function of the variables x and y
quantities A, B, integral
= F(x,
whatever be the path will be of the form integral
It follows that,
admits of an infinite number of different origin of this indetermination
is
That which determinations, and the
apparent.
Remark. The function F(x, y} is a definitely defined function whole region A when the cuts aa, b/3, cy have been traced. But it should be noticed that the difference F(m) between
in the
m
the values of the function at two points and of sides a cut does not opposite necessarily vanish. /""
A- /: + i/a;
/"
F(m )
which lie on For we have
r*o
/+/
J,,i
m
,
Jmf
which may be written
But
is j"^
zero
;
hence J^-F<X)= A.
It follows that to
A
all
the cuts.
the difference F(ni)
along aa.
F(m
)
is
constant and equal for each of
The analogous proposition holds
VII,
TOTAL DIFFERENTIALS
154]
Example.
The
321
line integral "
xdy
/
ijdx ~f~
i/fll
y In order to find the corre
has a single critical point, the origin. let
sponding period,
x
= p cos
integrate
2 along the circle x
+
y
2
=p
2 .
we have
this circle
Along
us
y
o,
whence the period
is
=p
sin w,
xdy
ydx
= p^dto,
= 2?r. It is easy to verify equal to / is the total differential of arc tan y/x. "dw
for the integrand
this,
X
Common roots of two equations. Let and Y be two functions of the and y which, together with their first partial derivatives, are con tinuous in a region A bounded by a single closed contour C. Then the expres 154.
variables x
sion
(XdY
2 YdX)/(X* + Y
the derivative of arc tan
Y/X.
satisfies the
)
Hence the
L
(53)
X
2
condition of integrability, for
it is
line integral
+
taken along the contour C in the positive sense vanishes provided the coeffi dx and dy in the integrand remain continuous inside (7, i.e. if the two = 0, Y = have no common point inside that contour. But if these curves
cients of
X
of common points a, 6, c, inside C, the value of the integral will be equal to the sum of the values of the same integral taken as along the circumferences of small circles described about the points a, b, c,
two curves have a certain number
centers. Let (a, 0) be the coordinates of one of the common points. We shall suppose that the functional determinant D(X, Y)/D(x, y) is not zero, i.e. that = are not tangent at the point. Then it is pos the two curves and sible to draw about the point (a, /3) as center a circle c whose radius is so small
X
that the point (JT,
which circle c
is (
Y
Y)
describes a small plane region about the point
bounded by a contour 7 and which corresponds point 25 and 127).
(0, 0)
for point to the
As
the point (x, y) describes the circumference of the circle c in the positive sense, the point (X, Y) describes the contour y in the positive or in the negative sense, according as the sign of the functional determinant inside the circle c is
But the definite integral along the circumference of c is positive or negative. in one revolution, that is, 2x. Similar equal to the change in arc tan
Y/X
reasoning for
where
P
all of
the roots shows that
denotes the number of points
-^(^, Y)/D(x, y) is positive, and determinant is negative.
common
to the
two curves
at
which
N the number of common points at which the
MULTIPLE INTEGRALS
322 The
on the
definite integral
around
in going
c,
that
is,
describes the contour C.
contour
C
is
composed
left is also
155
[VII,
equal to the variation in arc tan Y/X Y/X as the point (x, y)
to the index of the function
the functions
If
X and Y are polynomials, and
if
the
number of arcs of unicursal curves, we are led more rational functions, which involves only
of a finite
to calculate the index of one or and Y, elementary operations ( 77). Moreover, whatever be the functions we can always evaluate the definite integral (54) approximately, with an error less than 7t, which is all that is necessary, since the right-hand side is always a
X
multiple of
2it.
The formula (54) does not give the exact number of points common to the two curves unless the functional determinant has a constant sign inside of C. Picard s recent work has completed the results of this investigation.*
The results of the preceding paragraphs without essential alteration to line integrals in space. Let P, be three functions which, together with their first partial derivatives,
155. Generalization of the preceding.
may be extended Q,
and
-B
are continuous in a region (E) of space bounded by a single closed surface S. Let us seek first to determine the conditions under which the line integral
(55)
(v
j/
z
,
)
yo Zo) and (x, y, z) of the path of inte depends only upon This amounts to inquiring under what conditions the same integral vanishes when taken along any closed path T. But by Stokes theorem ( 136) the above line integral is equal to the surface integral the extremities (x
,
,
gration.
CC/ j
I
JJ
(
\ dx
--p\j}dx dy +T /^ R --t>
f>Q\^
(
\a*
dy/
I
dy dz
dz/
SR ---
5P +/ ^\cz ,
(
\
7
}dz dx
ex/
extended over a surface S which is bounded by the contour T. In order that should be zero, it is evidently necessary and sufficient that
this surface integral
the equations
8P_BQ dy should be satisfied.
dx
^Q =
^,
dz
dy
<3jR
= eP
dx
If these conditions are satisfied,
P
dz
U is a function of the
dx + Q dy + ables x, y, and z whose total differential is valued in the region (E). In order to find the value of
R dz,
and which
U at any
vari
is
single point, the path
be chosen arbitrarily. If the functions P, Q, and E satisfy the equations (56), but at least one of them becomes infinite at all the points of one or more curves in (E), results of integration
may
analogous to those of 153 may be derived. If, for example, one of the functions P, Q, R becomes infinite at all the points will admit a period equal to the value of the of a closed curve 7, the integral line integral taken along a closed contour which pierces once and only once a
U
surface
bounded by
7.
We may also consider questions relating to
surface integrals which are exactly
analogous to the questions proposed above for line integrals. Let A, B, and C be three functions which, together with their first partial derivatives, are *
TraM
d Analyse, Vol.
II.
VII,
TOTAL DIFFERENTIALS
155]
323
continuous in a region (E) of space bounded by a single closed surface S. Let 2 be a surface inside of (E) bounded by a contour r of any form whatever. Then the surface integral
I
(57)
= C
A dy dz + B dz dx + C dx dy
f
in general upon the surface S as well as upon the contour r. In order that the integral should depend only upon F, it is evidently necessary and suffi cient that its value when taken over any closed surface in (E) should vanish.
depends
Green s theorem ( 149) gives at once the conditions under which this is true. For we know that the given double integral extended over any closed surface is equal to the triple integral
dB
dA
dC\,
\dxdydz
1
1
dz /
dy
extended throughout the region bounded by the surface. In order that this latter that the integral should vanish for any region inside (E), it is evidently necessary functions A, B, and C should satisfy the equation
+
dx
This condition
is
~
+ ~dz~
Ity
also sufficient.
Stokes theorem affords an easy verification of this fact. For if A, J5, and C are three functions which satisfy the equation (58), it is always possible to deter mine in an infinite number of ways three other functions P, Q, and R such that
~ In the
first
place,
if
dx
dx
dz
dz
dy
number, for they remain unchanged
if
P, Q,
dx
f *B(x,
y, z)
dz
+
R
and
be replaced by
dz
dy
respectively, where X is an arbitrary function R = 0, the first two of equations (59) give
P=
dy
these equations admit solutions, they admit an infinite
(x,
of x, y,
Q=- C
y),
A(x,
and
z.
y, z)
dz
y) and \f/(x, y) are arbitrary functions of x and y. values in the last of equations (59), we find
where
(x.
dA
d h
~ or,
making use
B\ )
dx
d\b
,
dx
=
+
$(x, y)
,
Substituting these
_,.
C7(j y, z)
,
dy
of (58),
= dx
One of The
d
dz H
dy
Again, setting
the functions
or
C(x, y, z
)
dy
still be chosen at random. having been determined, the surface integral, by
^ may
functions P, Q, and
R
Stokes theorem, is equal to the line integral evidently depends only upon the contour F.
f(r) Pdx + Qdy + Rdz, which
MULTIPLE INTEGRALS
324
[VII, Exs.
EXERCISES Find the value
1.
of the triple integral
(
x -y) 2
+ 3az-
4a*]dxdydz
extended throughout the region of space defined by the inequalities x2
+
2
x2
-az<0,
7/
+
+
2 2/
-2a 2
z2
<0.
[Licence, Montpellier, 1895.]
Find the area
2.
of the surface
2
and the volume
of the solid
& 2 y2
bounded by the same
= C
F(X, F, Z) considered as a function of
xo
surface.
JT,
dx C dy C /(z, z
Jo
F, and Z.
(z
which
lies in
the
Reduce
first
2
+
y
C C I
C x,a x a * y
I
J
<
,
[Proceed as in
Reduce
)
solid
= 3a
s
bounded by the surface
xyz
octant.
x a F(x\ "
Zj
,
+
x2
D defined
extended throughout the domain Xi
+
z2 8
125.
to a simple integral the multiple integral
J J
<
2
y, z)dz
o
Generalize the results of
Find the volume of the portion of the
4.
6.
+
Investigate the properties of the function
3.
5.
X2
,
+
+
by the
^ xn
,
xn ) dx\ dx% inequalities
x\
+
x2
+
148.]
to a simple integral the multiple integral
extended throughout the domain
D defined
by the
inequalities
7*. Derive the formula n
f f f--- JCdx 1 dx---dxn = JJJ
dxn
iri -
+
xn
^a
.
EXERCISES
VII. Exs.]
where the multiple
integral
is
325
extended throughout the domain
D denned
by the
inequality
Derive the formula
8*.
C
n C F(a cos 6 r27T
"de
+
b sin
cos
$
+
sin 0) sin 6
c sin
d<j>
- 2n C
Jo
/0
where
a,
F(uR) du
,
I
&,
and
c
are three arbitrary constants,
and where
B = Va2 +
b2
+
c 2.
[POISSON.]
double integral is equal to a certain surface inte [First observe that the given 2 2 2 = 1. Then take the plane gral taken over the surface of the sphere x + y + z
ax
+
by
+
cz
=
&s the plane of xy in a
new system
of coordinates.]
be the equation in polar coordinates of a closed surface. Let p = F(6, that the volume of the solid bounded by the surface is equal to the double
9*.
)
Show
integral
J
(a)
j
ip cos y do-
extended over the whole surface, where
da-
represents the element of area, and 7 exterior normal.
makes with the
the angle which the radius vector
10*. Let us consider an ellipsoid
whose equation
is
us define the positions of any point on its surface by the elliptic coordi and that is, by the roots which the above equation would have if /* The application of the formulae (29) to were regarded as unknown (cf. 147).
and
let
nates
v
/>,
---
the volume of this ellipsoid leads to the equation "
r"
dv
_
m
Jo
M
Likewise, the formula (a) gives
- p2
>&
i/O
11.
Jb
V(62-p
2
)(
C2
)
dv
-p 2 )(,2 -62 )(c 2 -,2
2 )
Determine the functions P(z, y) and Q(x, y) which, together with their and for which the line integral
partial derivatives, are continuous,
x
+ a,y +
/3)dx
+
Q(z
+
a,
y
+
p)dy
taken along any closed contour whatever is independent of the constants and depends only upon the contour itself.
a and
/3
[Licence, Paris, July, 1900.]
MULTIPLE INTEGRALS
326
12*. Consider the point transformation defined
As
[VII, Exs.
by the equations
z ) describes a surface S the point (x, y, z) describes a sur y be the direction angles of the normal to S a -/ the direction angles of the corresponding normal to the surface S and and the corresponding surface elements of the two surfaces. Prove the formula
the point (z
y
,
Let a,
face S.
,
,
/3,
^
M
=
y)
coscr-
^
+r
y)
+
cos
,
,
;
d
;
d
^
13*. Derive the formula (16) on page 304 directly. may be expressed by the surface integral [The volume
V
V=
I
z cos
7
d
,
J(S)
and we may then make use D(f, *, *)
D(x ,y ,z which
is
)
d
dx
(
T
easily verified.]
of the identity
D(/,
)
)
8
J>(/,
j
D(y ,z
)
dy
Y
0)
D(z ,x
)
)
a
3z
(
D(x
,
y
)
it,
CHAPTER
VIII
INFINITE SERIES
SERIES OF REAL CONSTANT TERMS I. GENERAL PROPERTIES TESTS FOR CONVERGENCE 156. Definitions and general principles. Sequences. The elementary in all texts on College Algebra properties of series are discussed
We shall
and on Elementary Calculus.
review rapidly the principal
discussions. points of these elementary First of all, let us consider an infinite sequence of quantities SQ
(1)
SZ
Si,
,
,
,
Sn ,
which each quantity has a definite place, the order of precedence if s n approaches being fixed. Such a sequence is said to be convergent a limit as the index n becomes infinite. Every sequence which is in
not convergent of
two ways
:
is
sn
This may happen in either become and remain larger than any may approach no limit even though it
said to be divergent.
may
finally
preassigned quantity, or does not become infinite.
sn
In order that a sequence should be convergent, sufficient that, corresponding
positive integer n should less
than
In the a limit
e
to
first place,
positive
number
e,
a
sn is
any positive integer p.
the condition
n becomes
and
necessary
exist such that the difference sn+p
in absolute value for
s as
any preassigned
it is
infinite,
a
is
necessary.
For
number n always
if s n
approaches
exists for
which
is less than n+p sn It follows that the absolute value of sn+p in value. absolute e/2 will be less than 2 e/2 = c for any value of p. In order to prove the converse, we shall introduce a very impor tant idea due to Cauchy. Suppose that the absolute value of each
each of the differences
s
sn , s
s n+l ,
,
s
s
,
of the terms of the sequence (1) is less than a positive
N
number N.
N
and -fThen all the numbers between may be separated into number two classes as follows. We shall say that a belongs to the class
A
if
there exist an infinite
number 327
of terms of the sequence (1)
INFINITE SERIES
328
156
[VIII,
which are greater than the given number. A number belongs to the class R if there are only a finite number of terms of the It is sequence (1) which are greater than the given number. evident that every number between N and + N belongs to one of the two classes, and that every number of the class A is less than any number of the class B. Let S be the upper limit of the numbers of the class A, which is obviously the same as the lower limit of the numbers of the class B. Cauchy called this number the grande des limites) of the terms of the This number S should be carefully distinguished
greatest limit (la plus
sequence (1).*
from the upper limit of the terms of the sequence
(1) (
For
68).
instance, for the sequence
11
1 "
11
2
n
3
the upper limit of the terms of the sequence limit is 0.
is 1,
while the greatest
The name given by Cauchy is readily justified. There always number of terms of the sequence (1) which lie and S + e, however small e be chosen. Let us then between S consider a decreasing sequence of positive numbers t l} c 2 en where the general term ^ approaches zero. To each num of the sequence (1) ber of the sequence let us assign a number S and We which lies between S shall thus obtain a + e, e^ an to the suite of numbers a 1? a 2 belonging sequence (1) which approach S as their limit. On the other hand, it is clear from the very definition of S that no partial sequence of the kind just mentioned can be picked out of the sequence (1) which approaches a limit greater than S. Whenever the sequence is convergent its limit is evidently the number S itself. Let us now suppose that the difference s n+p s n of two terms of the sequence (1) can be made smaller than any positive number c for any value of p by a proper choice of n. Then all the terms of the sequence past s n lie between s Let S be the e and s n + e. exist an infinite
,
,
,
,
.
,-
,
,
,
lt
greatest limit of the terms of the sequence. By the reasoning just given it is possible to pick a partial sequence out of the sequence (1)
which approaches 5
as its limit.
sequence, after a certain one, lies *
The
Since each term of the partial
between
sn
e
and
sn
+
c,
it
is
Resumes analytiques de Turin, 1833 (Collected Works, 2d series, Vol. X, p. 49). definition may be extended to any assemblage of numbers which has an upper
limit.
CONSTANT TERMS
VIII, Ufi7]
329
s n is at most equal to e. S Now be any term of the sequence (1) whose index is greater than n. Then we may write
clear that the absolute value of
m
let s m
and the value of the right-hand side is surely less than 2c. Since e is an arbitrarily preassigned positive number, it follows that the general term s m approaches S as its limit as the index m increases indefinitely.
If S is the greatest limit of the terms of the sequence (1), number greater than S belongs to the class B, and every num every ber less than S belongs to the class A. The number S itself may
Note.
belong to either
class.
Given any
157. Passage from sequences to series.
infinite
sequence
the series formed from the terms of this sequence,
is
+U
!
H-----\- U n
2
----
-\
,
said to be convergent if the sequence of the successive
S is
+
U
(2)
=
?/
,
Sl
=
U
+
1( 1
Sn
,
,
=
+
U
!
+
sums
+ Un
,
Let 5 be the limit of the latter sequence, i.e. the convergent. which the sum Sn approaches as n increases indefinitely:
limit
v=n
S
Then S
is
called the
=
lim S
= lim
?/.
sum of the preceding
series,
and
this relation is
indicated by writing the symbolic equation
S
A
= MO + MI
H-----h
un
H
which is not convergent is said to be divergent. evident that the problem of determining whether the series convergent or divergent is equivalent to the problem of determin series
It is is
ing whether the sequence of the successive sums S0) Slf convergent or divergent. Conversely, the sequence S 0)
s li
S
" 2>
)
s n)
will be convergent or divergent according as the series SQ
+ Oi - s + (s )
*i)
H-----1- (*
---a.-i) H
52
,
is
INFINITE SERIES
330
For the sum Sn of the
convergent or divergent.
is
of this series
sequence.
The
is
We
[VIII,
first
precisely equal to the general term shall apply this remark frequently.
series (2) converges or diverges
S + Vn +
(3)
*
sn
n
+
1
157
terms
of the given
with the series
+ *++
obtained by omitting the first p terms of (2). For, if S n (n p) denote the sum of the first n + 1 terms of the series (2), and 2 n _ p >
sum
the
of the
p+
n
1 first terms of the series (3),
i.e.
2 n - P = uo + u \ H-----h Mp_i is independent of w. Hence the sum 2 n - P approaches a limit if Sn approaches a limit,
the difference Sn
and conversely.
It follows that in
determining whether the series many of the terms at the
converges or diverges we may neglect as beginning of a series as we wish.
Let S be the
n
+1
first
sum
terms, and
Ti
+1
Rn
of a convergent series, Sn the sum of the first sum of the series obtained by omitting the
the
terms,
=U
RH It is evident that
we
n
+l
shall
+
-----h +2 H
Un + p
-\
---- .
always have
It is not possible, in general, to find the sum S of a convergent If we take the sum 5 of the first n -+- 1 terms as an approxi
series.
mate value of S, the error made is equal to R n Since Sn approaches S as n becomes infinite, the error R n approaches zero, and hence the number of terms may always be taken so large at least theoret that the error made in replacing S by Sn is less than any ically In order to have an idea of the degree of preassigned number. .
approximation obtained,
Rn
it
is sufficient
to
know an upper
limit
It is evident that the only series which lend themselves readily to numerical calculation in practice are those for which
of
.
R n approaches zero rather rapidly. of properties result directly from the definition of con shall content ourselves with stating a few of them.
the remainder
A
number
vergence.
We
1) If each of the terms of a given series be multiplied by a constant k different from zero, the new series obtained will converge or diverge with the given series; if the given series converges to a sum S, the sum
of the second series
is
kS.
VIII,
2)
CONSTANT TERMS
158]
If there
two convergent series
be given
wo io
331
+ +
+w H + wH
MI
2
vi
whose sums are S and S
,
h
MH H
1-
vn H
respectively, the
,
,
new
series obtained by
adding the given series term by term, namely,
Oo and
+
*
o)
sum
+ (MX + Vj) + S
-f-
S
the term-by -term addition of
p
converges,
its
is
+ (M +
)
+
,
The analogous theorem holds
.
for
series.
convergent
3) The convergence or divergence of a series is not affected if the values of a finite number of the terms be changed. For such a change would merely increase or decrease all of the sums S n after a certain
one by a constant amount. 4)
The
test for
convergence of any infinite sequence, applied to * gives Cauchy s general test for convergence
series,
:
In order that a series be convergent
and sufficient number that, corresponding any preassigned positive e, an integer n should exist, such that the sum of any number of terms what is less than c in absolute value. For ever, starting with u ll+l Sn + p Sn = U n + + U n + 2 H ?+ it is
necessary
to
,
l
\~
In particular, the general term zero as n becomes infinite.
Cauchy apply
it
s
u n+l
=
Sn+l
Sn must approach
test is absolutely general, but it is often difficult to It is essentially a development of the very
in practice.
We shall proceed to recall the practical rules most used for frequently testing series for convergence and divergence. None of these rules can be applied in all cases, but together they notion of a limit.
suffice for
the treatment of the majority of cases which actually arise.
158. Series of positive terms.
We
shall
commence by
investigating
a very important class of series, those whose terms are all posi tive. In such a series the sum Sn increases with n. Hence in
sum Sn should The most for the convergence of such a series is based upon com the given series with others previously studied. The
order that the series converge
remain
less
general test parisons of
it is
sufficient that the
than some fixed number for
all
values of n.
following propositions are fundamental for this process * Exercices
de Mathtmatlques, 1827.
:
(Collected Works, Vol. VII, 2d series, p. 267.)
INFINITE SERIES
332
159
[VIII,
If each of the terms of a given series of positive terms is less than or at most equal to the corresponding term of a known convergent 1)
of positive terms, the given series is convergent. For the sum S n of the first n terms of the given series is evidently less than the sum S of the second series. Hence Sn approaches a limit S which
series
is
than S
less
2)
.
If each of the terms of a given
series of positive terms is greater
than or equal to the corresponding term of of positive terms, the given series diverges.
a known divergent series For the sum of the first
n terms of the given series is not less than the sum of the first n terms of the second series, and hence it increases indefinitely with
n.
We may lemma.
compare two
also
series
by means of the following
Let
(U)
MO
(V)
v
+ +
!
Vl
+ MJ H + Va +
h
+
...
un ,
H
+
,
...
If the series (7) converges, and if, v n+l /v n 5: u n+l /u n , the series (V) we have a certain term, always after series also converges. the If (7) diverges, and if, after a certain have u we term, always n+ i/u n ^v n+} /v n , the series (F) also diverges.
be two series of positive terms.
In order to prove the first u n+i/ u n whenever n
statement, let us suppose that Since the convergence of a series is not affected by multiplying each term by the same con stant, and since the ratio of two consecutive terms also remains v n+i/ v n^
>
2).
unchanged, we may suppose that vp should have vp + ^up + 1 vp+2 ^up + 2
<
l
,
,
up and ,
etc.
it is
evident that
Hence the
must converge. The proof of the second statement Given a series of positive terms which is known to diverge,
we may make use
is
series
we
(F)
similar.
to converge or
of either set of propositions in order
determine in a given case whether a second series of positive terms converges or diverges. For we may compare the terms of to
the two series themselves, or consecutive terms.
we may compare
the ratios of two
159. Cauchy s test and d Alembert s test. The simplest series which can be used for purposes of comparison is a geometrical progression
The is r. It converges if r 1, and diverges if r ^ 1. of of a series terms with a geometrical given comparison positive progression leads to the following test, which is due to Cauchy: whose ratio
<
VIII,
CONSTANT TERMS
159]
If the nth
333
\u n
of the general term u n of a series of positive is constantlyjess than a fixed number less than unity, the series converges. If ~\/un after a certain term is con stantly greater than unity, the series diverges. root
terms after a certain term
For
in the first case ~\/un
whence u n
of the terms of the series after a certain one
n
is less
Hence each than the corre
.
sponding term of a certain geometrical progression whose ratio is than unity. In the second case, on the other hand, ~\/u n whence wn Hence in this case the general term does not
less
>l
)
>l.
approach zero. This test is applicable whenever fact,
the following proposition
may
V^
approaches a limit.
be stated
In
:
If -Vu n approaches a limit I as n becomes infinite, the series will converge if I is less than unity, and it will diverge if I is greater than unity.
=
A
doubt remains if I 1, except when ~^fu n remains greater than unity as it approaches unity, in which case the series surely diverges.
two consecutive terms of a given series two consecutive terms of a geometrical progression, we obtain d Alembert s test:
Comparing the
ratio of
of positive terms with the ratio of
If in a given
of positive terms the ratio of any term to the a certain term remains less than a fixed number preceding after less than unity, the series converges. If that ratio after a certain series
term remains greater than unity, the series diverges.
From
this
theorem we
may deduce
the following corollary
:
un+l /un approaches a limit I as n becomes infinite, the series converges if I 1, and diverges ifl>l. The only doubtful case is that in which 1 = 1; even then, ifu n If the ratio
<
remains greater than unity as
it
n+l /u approaches unity, the series is divergent.
General commentary. Cauchy s test is more general than d Alembert s. For suppose that the terms of a given series, after a certain one, are each less than the corresponding terms of a decreasing geometrical progression, i.e. that the general term u n is less than Arn for all values of n greater than a fixed integer p, where is a certain constant and r is less than rvl 1 and unity. Hence Vu n the second member of this inequality approaches unity as n becomes infinite.
A
<
/",
a fixed number between r and 1, we shall have after a cer Hence Cauchy s test is applicable in any such case. But it may happen that the ratio u n + \/Un assumes values greater than unity, however far out in the series we may go. For example, consider the series
Hence, denoting by term \/M n k.
tain
A;
<
1
+
r sin |
a + |
r2 sin 1
2a + \
+
r n sin \
na
-{ \
,
INFINITE SERIES
334
159
[VIII,
where r 1 and where a is an arbitrary constant. In this case whereas the ratio <
v un = r v
sin net |
<
\
r,
sin(n + l)
number
assume, in general, an infinite increases indefinitely.
may
Nevertheless,
advantageous to retain d Alembert For instance, for the series
it is
many
convenient in
of values greater than unity as s test, for it is
n
more
cases.
x ~
A H~
~
x2
x3 ~
~
"
~
I
i
xn ~T~
T
1.2.3
1.2
1
1.2---W,
the ratio of any term to the preceding is x/(n + 1), which approaches zero as n becomes infinite whereas some consideration is necessary to determine inde ;
n as n becomes infinite. pendently what happens to Vun = x/\/l 2 After we have shown by the application of one of the preceding tests that each of the terms of a given series is less than the corresponding term of a decreasing 2 it is easy to find an upper Ar n geometrical progression A, Ar, Ar of the first m terms is taken in place of when the sum made the error limit of For this error is certainly less than the sum of the the sum of the series. .
,
,
,
,
geometrical progression
Ar m + Ar m +
When two .
Ar m l
each of the two expressions
limits are necessarily the same.
+ uix +
(4)
-\iun and Un + i/u H approaches a limit, the For, let us consider the auxiliary series
u 2 z2 H-----h u n
x"
H----
,
In this series the ratio of any term to the preceding Hence the approaches the limit Ix, where I is the limit of the ratio u n + \/u n I /I. series (4) converges when x Denoting the 1/Z, and diverges when x
where x
is
positive.
.
>
<
limit of
by
-\/Un
*
,
the expression
the series (4) converges
two and
tests
x
if
<
1/i
,
-\/u n x
n
also approaches a limit
and diverges
should not give contradictory results,
if
x
it is
>
1/i
.
Z
x,
and
In order that the
evidently necessary that
I
were greater than T, the series (4) would be convergent, by Cauchy s test, for any number x between 1/Z and 1/i whereas the same series, for the same value of x, would be divergent by d Alembert s test. I
should be equal.
for instance,
If,
I
,
Still
limit.*
more generally, if Un + \/u n approaches a limit Z, n approaches the same For suppose that, after a certain term, each of the ratios >Xu
Un +
lies
between
as small as
I
we
e
and
I
U,,
\
+
e,
+2
U,,
+
,,
where e is a positive number which may be taken n sufficiently large. Then we shall have
please by taking
or
*Cauchy, Cours d Analyse.
CONSTANT TERMS
Vlll,
160]
As
number p
the
335
two terms on and left of this double inequality approach I -f e and I e, Hence for all values of m greater than a suitably chosen number increases indefinitely, while n remains fixed, the
the extreme right respectively.
we
shall
have
and, since the
e is
number
It
I
an arbitrarily assigned number,
as
should be noted that the converse
follows that
not true.
is
sequence
b are
two
a2 62
2
a6,
a,
1,
where a and
it
Vu^
approaches
its limit.
a"
6,
,
,
The
different numbers.
ing is alternately a and as n becomes infinite.
6,
an bn
Consider, for example, the
~l
a"6
,
n ,
-,
any term to the preced approaches the limit Va6
ratio of
whereas the expression
-s/u^
The preceding tain expressions
proposition may be employed to determine the limits of cer which occur in undetermined forms. Thus it is evident that 2 1 n increases indefinitely with n, since the ratio n \/(n 1)1
the expression v increases indefinitely with .
the expressions
n.
manner it may be shown that each of approaches the limit unity as n becomes infinite.
In a similar
v n and v^ogn
limit. Cauchy formulated the preceding test a more general manner. Let an be the general term of a series of positive terms. Consider the sequence
160. Application of the greatest
in
on,
(5)
1
I
a;,
ajj,
I
-,
a*,
.
If the terms of this sequence have no upper limit, the general term an will not approach zero, and the given series will be divergent. If all the terms of the sequence (5) are less than a fixed number, let w be the greatest limit of the terms
of the sequence.
The
series
Sa n
is
convergent if
u
is less
than unity, and divergent if u
is
greater
than unity.
a be a number between first part of the theorem, let 1 Then, by the definition of the greatest limit, there exist but a finite It follows a. number of terms of the sequence (5) which are greater than 1 1 a for all values of n that a positive integer p may be found such that 3/a n In order to prove the
w and
1.
<
Hence the series 2a converges. On the other hand, if 1, let 1 + a be a number between 1 and w. Then there are an infinite number of terms of the sequence (5) which are greater than 1 + a, and hence there are an It follows that infinite number of values of n for which a n is greater than unity. the series Sa n is divergent in this case. The case in which w = 1 remains in doubt. greater than p.
re
>
Cauchy s theorem. In case un+l /un and \u n both approach without remaining constantly greater than unity, neither unity d Alembert s test nor Cauchy s test enables us to decide whether the series is convergent or divergent. We must then take as a 161.
comparison series some series which has the same characteristic
INFINITE SERIES
336
[VIII,
161
is known to be convergent or divergent. The following proposition, which Cauchy discovered in studying definite integrals, often enables us to decide whether a given series is convergent or
but which
when
the preceding rules fail. be a function which is positive for values of x greater than a certain number a, and which constantly decreases as x
divergent
Let
(x)
increases past x Then the x axis
=
approaching zero as x increases an asymptote to the curve y =
a,
is
indefinitely.
and the
(%),
definite integral
1 or
may
The
may
(x~)dx
not approach a finite limit as
I
increases indefinitely.
series
(6)
4>(
a)
+
4,(a
+
+!)+
<(>
+
+
)
converges if the preceding integral approaches a limit, does not.
and
diverges if
it
For, let us consider the set of rectangles whose bases are each -, unity and whose altitudes are 1), n), respec (a),
(a
+
+
(a
Since each of these rectangles extends beyond the curve the sum of their areas is evidently greater than the area
tively.
y between the x axis, the curve y x = a + n, that is, (#),
= <j>(x),
and the two ordinates x
= a,
+
Xa On
the other hand,
altitudes <j>(a
+1),
<j>(a
we consider the rectangles constructed
if
common
inside the curve, with a
+
2),
,
base equal to unity and with the -f- n), respectively, the sum of
<j>(a
the areas of these rectangles the curve, and we may write
is
evidently less than the area under +n
Xa the integral fj indefinitely, the sum
Hence,
if
<j>(z)
-\
a+n
<j>(x)dx
is
sum
in question approaches a limit convergent. On the other hand, if the inte increases beyond all limit as n increases indefinitely,
hence the series (6) the same
(a
It follows that the
-f L.
gral f
dx approaches a limit L as I increases + + n) always remains less than
----
(a)
+(x)dx.
true of the
;
is
-f
sum a
+1 +
+
<
a
+
",
VHi,
CONSTANT TERMS
161]
as is seen
from the
337
Hence
of the above inequalities.
first
case the series (6) diverges. Let us consider, for example, the function
= l/x*,
<(ce)
in this
where
/u,
= 1.
This function satisfies all the requirements and a of the theorem, and the integral // [1/a^] dx approaches a limit as It is greater than unity. I increases indefinitely if and only if is
positive
/u,
follows that the series
111 T 2*
1*
_! +
.
n*
3*
if p. is greater than unity, and diverges if /x ^ 1. 1 consider the function I/ [x(logo;) ], where log a; A.gain, 2. denotes the natural logarithm, p. is a positive number, and a
converges
=
<(#)
=
Then,
if /*
=
1,
we
shall have
1
/
i ^-[(log7 ) -^-(log2)
c/2
1-
The second member approaches a limit if /t 1, and increases = 1 it In the particular case when 1. indefinitely with n if is easy to show in a similar manner that the integral increases beyond all limit. Hence the series >
/u,
"t"
2 (log 2)*
<
/j.
o\u
r f\
w(logn)
if /t 511, and diverges if More generally the series whose general term
converges
I
_\U.4
/i
I
I
3 (log 3)*
>
/u.
is
1
n log
7i
2
log
n log 8
1
?i
n(log
log""
p ri)*
In this expression log 2 n 1, and diverges if /x ^ 1. converges if /t denotes log log n, log 8 n denotes log log log n, etc. It is understood, of course, that the integer n is given only values so large that >
p The missing terms in log n are positive. The considered are then to be supplied by zeros. theorem may be proved easily in a manner similar to the demon = strations given above. 1, the function If, for instance, /t 2
8
log n, log n, log n,
,
the series
1
x log x log 2 x the derivative of (log approaches a finite limit
is
1
a:)
if
~
(log
i
/(l
/A),
and only
if
p
xY and
//.
>
this latter function 1.
INFINITE SERIES
338 s
Cauchy
[vill,
theorem admits of applications of another
that the function (x)
sort.
162
Let us suppose let us con
conditions imposed above, and
satisfies the
sum
sider the
+
<j>(n)
+
4>(n
1)
+
+
(n
+
p)
,
where n andp are two integers which are to be allowed to become infinite. If the series whose general term is is convergent, the preceding sum approaches zero as a limit, since it is the difference between the two sums S n + + i and S n p each of which approaches the sum of the series. But if this series is divergent, no conclusion can be drawn. Keturning to the geometrical interpretation given
,
we
above,
find the double inequality
*
f
0(x)dx<0(n)
+
t/n
+
l)
+
+
+<(n
p)<(n).+
f
P
t/n
approaches zero as n becomes
Since (ri)
sum
the
0(n
infinite, it is
evident that the limit of
same as that of the definite integral /n +p 0(x)dx, depends upon the manner in which n and p become infinite. "
in question is the
and this For example, the
limit of the
sum 1
1
n
+
n
1
n
1
+p
+P
=
is the same as that of the definite integral [1/x] dx p/n). It is log(l clear that this integral approaches a limit if and only if the ratio p/n approaches a limit. If a is the limit of this ratio, the preceding sum approaches log(l + a) f"
as
its limit,
as
we have already sum
seen in
+
49.
Finally, the limit of the
Vn is
the
same
Vn +
Vn + p
1
as that of the definite integral
I+P j
f \J n
=
2
Vx
(
Vn + p
Vn).
In order that this expression should approach a limit, it is necessary that the p/Vn should approach a limit a. Then the preceding expression may be written in the form
ratio
2Vn + p + Vn 1
and
it is
evident that the limit of this expression
162. Logarithmic criteria.
Taking the
11 j^
2M
is
a.
series
1 n^
Cauchy deduced a new test for convergence analogous to that which involves -\/u n
as a comparison series,
which
is
entirely
.
vm,
CONSTANT TERMS
162]
339
is always If after a certain term the expression log(l/w n )/lognthan which is greater unity, the series greater than a fixed number If after a certain term log(l/wn)/logn is always less converges.
than unity, the
series diverges.
If log(l/w n )/log n approaches a limit
and
the series converges if I 1, 1 remains in doubt. I >
which
as
n
increases indefinitely, The case in diverges if I
Ki.
=
In order to prove the
part of the theorem,
first
we
will
remark
that the inequality log is
k log n
>
un
equivalent to the inequality
>
un since k
1,
>
Likewise,
nk
un
or
<
-Tk
n
;
the series surely converges. if
log
<
Un
log w,
1/n, whence the series surely diverges. This test enables us to determine whether a given series con a certain verges or diverges whenever the terms of the series, after than the corresponding terms of one, are each less, respectively,
we
shall
have un
>
the series
where A
is
a constant factor and
Un
we
shall
have log
?/
+
log n
//.
p.
1.
For,
if
A <
n
log
<
-S^n
>
A
or
1
,
lo .
log
>
/A
log
A
;
log n
and the right-hand side approaches the limit /A as n increases If K denotes a number between unity and p., we indefinitely. shall have, after a certain term,
log
n
INFINITE SERIES
340
[VIII,
163
Similarly, taking the series
__ ~r n log w(log 2
n(\og n)^
ri)*
we
obtain an infinite suite of tests for con be obtained mechanically from the preceding
as comparison series,
vergence which may by replacing the expression \og(l/un )/logn by log[l/(/m n )]/log 2 then by
lognu
log
n
?i,
n
8 log n
and so forth, in the statement of the preceding tests.* These tests apply in more and more general cases. Indeed, it is easy to show that if the convergence or divergence of a series can be established by means of any one of them, the same will be true of any of those
which follow. with these
It
that no matter how far we proceed no one of them will enable us to determine
may happen
trial tests,
whether the series converges or diverges. Du Bois-Reymond f and Pringsheim have in fact actually given examples of both convergent and divergent series for which none of these logarithmic tests deter mines whether the series converge or diverge. This result is of great theoretical importance, but convergent series of this type evidently
converge very slowly, and it scarcely appears possible that they should ever have any practical application whatever in problems which involve numerical calculation.
Raabe s or Duhamel s test. Retaining the same comparison but series, comparing the ratios of two consecutive terms instead of comparing the terms themselves, we are led to new tests which 163.
to be sure, less general than the preceding, but which are often easier to apply in practice. For example, consider the series are,
of positive terms (7)
?/
+m+
wa H
h
un
-i
,
* See Bertrand, Traitt de Calcul differential et integral, Vol. de Liouville, 1st series, Vol. VII, p. 35. t J
I,
p. 238;
Journal
Ueber Convergenz von Reihen (Crelle s Journal, Vol. LXXVI, p. 85, 1873). Allgemeine Theorie der Divergenz (Mathematische Annalen, Vol. XXXV, .
.
.
.
.
.
1890).
In an example of a certain convergent series due to du Bois-Reymond it would be necessary, according to the author, to take a number of terms equal to the volume of the earth expressed in cubic millimeters in order to obtain merely half the sum of the series.
VIII,
in
CONSTANT TERMS
163]
which the
ratio
u n + /u a approaches unity, remaining constantly l
Then we may write
than unity.
less
341
!+ where a n approaches zero as n becomes this ratio with [n/(n first
+1)]
M
infinite.
The comparison
of
leads to the following rule, discovered
by Raabe* and then by Duhamel.f
If after a certain term the product nan is always greater than a fixed number which is greater than unity, the series converges. If after a certain term the
same product
is
always
than unity, the
less
series diverges.
The second part of the theorem follows immediately. nan 1 after a certain term, it follows that
For, since
<
1
ratio un + /u n terms of the harmonic
and the
+
n+l
ac H
greater than the ratio of two consecutive Hence the series diverges. In order to prove the first part, let us suppose that after a certain term we always have nan >k>l. Let p. be a number which lies l
between 1 and
k,
1
<
is
series.
p.
<
k.
Then the
series surely converges if
term the ratio u n + /u n is of two consecutive terms of the
after
a certain
[_n/(n
+ 1)]
term
is
1
n~*.
l
The necessary condition that
than the ratio
less
whose general should be true
series
this
that
is
(8)
or,
in
developing (1
+ 1/n)
1
s
by Taylor
theorem limited to the term
2
1/n
,
l+-W + 7T
-i<
where Xn always remains infinite.
less
l+a
n
t
,
than a fixed number as n becomes
Simplifying this inequality, we
*
n
<
nan
may
write
it
.
Zeitschrift fur Mathematik und Physik, Vol. X, 1832. Journal de Liouville, Vol. IV, 1838.
in the
form
INFINITE SERIES
342
The
103
[VIII,
left-hand side of this inequality approaches as its limit as n infinite. Hence, after a sufficiently large value of n, the /u.
becomes
left-hand side will be less than It follows that
the series
na n which proves the inequality ,
(8).
is
convergent. approaches a limit
If the product na n the preceding rule.
The
may apply
A
I
n becomes
as
series is
infinite,
convergent
if
we
1>1,
= 1,
except when nan approaches unity remaining constantly less than unity in that case
and divergent
if
I
1.
<
doubt exists
if
I
:
the series diverges. If the product na n approaches unity as its limit, we may compare the ratio w + i/^n with the ratio of two consecutive terms of the series
which converges if terms of the given series
/*>!>
and diverges
may
u
"
I
where
ft,
product
if
The
M^l-
ratio of
two consecutive
be written in the form
+
1
ft, +n
-
n
approaches zero as n becomes infinite. If after a certain term the logn is always greater than a fixed number which is greater than unity, converges. If after a certain term the same product is always less than
ft,
the series
unity, the series diverges. In order to prove the first part of the theorem, let us suppose that
Let
/A
be a number between
a certain term
1
and
k.
Then the
series will surely
ft,
log n
converge
>
if
k
>
1.
after
we have u +l un "
(9\
which may be written
in the
<
~n+1
n
logn
[ Llog(n
+
T l)J
form
logn or,
applying Taylor
1
+
-
+
n
s
theorem to the right-hand
\
where X B always remains
nl
log n
(
less
Simplifying this inequality,
it
side,
than a fixed number as n becomes
becomes
V
infinite.
VIII,
CONSTANT TERMS
lt>;]
343
(n + 1) log (1 + 1/n) approaches unity as n becomes be written, by Taylor s theorem, in the form
The product
may
infinite, for it
(10)
where
e
approaches zero.
approaches
fj.
as
The right-hand side of the above inequality therefore and the truth of the inequality is established for suffi since the left-hand side is greater than k, which is itself
its limit,
ciently large values of n, greater than /^. The second part of the theorem
may be proved by comparing
+ i/w,, with the ratio of two consecutive terms term is l/(nlogn). For the inequality
of the series
ttn
+
1
lg n +
__ >
n
un
which
is
to be proved,
1
may
+ -n
+
1
log(n
1)
be written in the form
n
n/
\
<(*
log n
1_
+ !) log/1 + V
The right-hand
the ratio
whose general
--).
*/
side approaches unity through values
which are greater than
The truth of the inequality is there unity, as is seen from the equation (10). fore established for sufficiently large values of n, for the left-hand side cannot exceed unity. From the above proposition it may be shown, as a corollary, that if the prod uct (8 n log n approaches a limit I as n becomes infinite, the series converges if I 1, and diverges if The case in which I = 1 remains in doubt, unless /3 n logn >
l
always less than unity. In that case the series surely diverges. If p n log n approaches unity through values which are greater than unity, we may write, in like manner,
is
n
n
log n
where approaches zero as n becomes infinite. It would then be possible to prove theorems exactly analogous to the above by considering the product >
2
7,,log n,
and so
Corollary.
forth.
If in
a series of positive terms the ratio of any term to the pre
ceding can be written in the form . l
r ---
un
where
/j.
is
n
n +n i
a positive number, r a constant, and
value remains less than a fixed verges If r
H 1
n
is
greater than unity,
number
as
H
n
n increases
and diverges
a quantity
whose absolute
indefinitely, the series con
in all other cases.
INFINITE SERIES
344 For
we
if
shall
we
164
[VIII,
set
have
-
r
na n = r -
l-
+
n
and hence lim na n if
r
<
1.
The only
=
r.
It
follows that the series converges if r 1, and diverges In order is that in which r = 1. >
case which remains in doubt
to decide this case, let us set
n
From
we
this
n
find
+1
n
log n
__ log
n
-**
,
log n
=
and the right-hand side approaches zero as n becomes infinite, no matter how Hence the series diverges. small the number /* may be. Suppose, for example, that u n + \/u n is a rational function of n which ap proaches unity as n increases indefinitely
:
~ 2 ----\
Then, performing the division indicated and stopping with the term
may
-
write
Un +
l
-
un
where 0(n)
is
_L i -r
1
~
CTl
in
1/n2
,
we
^(^
bl -L -r
>
nz .,
n
a rational function of n which approaches a limit as n becomes
the preceding theorem, the necessary and sufficient condition that the series should converge is that infinite.
By
bi is due to Gauss, general tests for convergence.
This theorem
>
ai
+1
who proved
164. Absolute convergence.
We
it
shall
whose terms may be either positive term all the terms have the same
.
directly.*
now
It
was one
of the first
proceed to study series If after a certain
or negative.
sign, the discussion reduces to
Hence we may restrict ourselves to series previous case. which contain an infinite number of positive terms and an infinite the
*
(Collected Works, Vol. Ill, p. 138.) Disquisitiones generates circa seriem infinitam
1+
a.B = l.y
+
,
VIII,
CONSTANT TERMS
164]
number
We
of negative terms.
lowing fundamental theorem
Any
shall prove first of all the fol
:
whatever
is convergent if the series formed of the abso the terms of of the given series converges.
series
lute values
345
Let
UQ
(11)
+
M! H-----h
u H----
be a series of positive and negative terms, and
l\+
(12)
L\
+
.--+
/
let
+
be the series of the absolute values of the terms of the given series, where Un un If the series (12) converges, the series (11) like .
\
wise converges.
This For we have
157.
is
a consequence of the general
theorem of
and the right-hand side may be made less than any preassigned num ber by choosing n sufficiently large, for any subsequent choice of p. Hence the same is true for the left-hand side, and the series (11) surely converges.
The theorem may
also be proved as follows
un
= (u n + Un ) - Un
and then consider the auxiliary (13)
(u
+U + -)
(u i
series
+U)+ l
:
Let us write
,
whose general term
...+ (u n
+ C7 ) + n
.
.
.
is
un
+ Un
,
.
Let Sn Sn and SJ denote the sums of the first n terms of the series Then we shall have (11), (12), and (13), respectively. ,
,
The series (12) converges by hypothesis. Hence the series (13) also converges, since none of its terms is negative and its general term cannot exceed 2Un It follows that each of the sums Sn and .
SJ, and hence also the sum Sn approaches a limit as n increases ,
indefinitely.
Hence the given
series (11) converges.
It is evident
that the given series may be thought of as arising from the subtrac tion of two convergent series of positive terms.
Any
series is said to be absolutely convergent if the series of the
absolute values of the terms
may
be
In such a series the order of changed in any way whatever without altering the its
terms converges.
INFINITE SERIES
34G
sum of the
Let us
series.
first
[VIII,
KM
consider a convergent series of posi
tive terms,
a
(14)
whose sum
is S,
and
+aH-----\-a
n
----
-\
,
let b
(15)
+b +
.-.
l
+ bn +
..-
be a series whose terms are the same as those of the
first
series
arranged in a different order, i.e. each term of the series (14) is to be found somewhere in the series (15), and each term of the series (15) occurs in the series (14).
Let Sm be the sum of the first m terms of the series (15). Since these terms occur in the series (14), it is evident that n may be chosen so .large that the first m terms of the series (15) are to be found among the first n terms of the series (14). Hence we shall have all
Sm
<
Sn
<
S,
series (15) converges and that its sum a similar manner it is clear that S 5 S
which shows that the not exceed
S.
In
.
S does Hence
=
S. The same argument shows that if one of the above series and (14) (15) diverges, the other does also. The terms of a convergent series of positive terms may also be grouped together in any manner, that is, we may form a series each of whose terms is equal to the sum of a certain mimber of terms of
S
Let us first the given series without altering the sum of the series.* and let consecutive terms are that grouped together, suppose
A,
(16)
be the
new
+ A + Ai + l
>"+A
m
+
....
series obtained, where, for example,
Then the sum Sm of the first m terms of the series (16) is equal to m. the sum 5 V of the first N terms of the given series, where N As m becomes infinite, N also becomes infinite, and hence Sm also >
approaches the limit
S.
Combining the two preceding operations, it becomes clear that any convergent series of positive terms may be replaced by another series each of whose terms is the sum of a certain number of terms of the given series taken in any order whatever, without altering the sum of * It is often said that parentheses may be inserted in a convergent series of positive TRANS. terms in any manner whatever without altering the sum of the series.
Vlll,
CONSTANT TERMS
165]
347
It is only necessary that each term of the given series the series. should occur in one and in only one of the groups which form the terms of the second series. Any absolutely convergent series may be regarded as the differ
ence of two convergent series of positive terms hence the preceding It is evident that an operations are permissible in any such series. ;
absolutely convergent series of numerical calculation as
may if it
be treated from the point of view were a sum of a finite number of
terms. 165.
A series whose terms do not all
Conditionally convergent series.
have the same sign may be convergent without being absolutely con This fact is brought out clearly by the following theorem vergent. on alternating series, which we shall merely state, assuming that it is
already familiar to the student.*
A
series whose terms are alternately positive and negative converges absolute value of each term is less than that of the preceding, the if and if, in addition, the absolute value of the terms of the series
diminishes indefinitely as the number of terms increases indefinitely.
For example, the
series
i-| + |-* + + (-l)-; + We saw in 49 that its sum is log 2.
converges. of the absolute values of the terms of this series
harmonic
series,
A
which diverges.
series
is
The
series
precisely the
which converges but
is called a conditionally conver of Cauchy, Lejeune-Dirichlet, and necessity of distinguishing between
which does not converge absolutely
The investigations gent Riemann have shown clearly the series.
absolutely convergent series and conditionally convergent series. For instance, in a conditionally convergent series it is not always
allowable to change the order of the terms nor to group the terms
These operations together in parentheses in an arbitrary manner. the sum of such or alter a series, may change a convergent may For example,
series into a divergent series, or vice versa.
let
us
again consider the convergent series
i* It there.
is
1
pointed out in
TRANS.
1
+ 2,3
-
1(K>
1 -
4
^
r
1
2n+l
that this theorem
is
_!_ +2
2n
a special case of the theorem proved
INFINITE SERIES
348 whose sum
is
1
1
m becomes infinite.
order,
166
evidently equal to the limit of the expression
+1
!,
as
[VIII,
2n
+
2,
Let us write the terms of this series in another
putting two negative terms
~2~4 + 3~ 6~~8
after each positive term, as follows
~
~~ "*"
"*"
2n
+1
:
+4
4
4r* -f 2
"*"
easy to show from a consideration of the sums S3n SStl+l and S3n+2 that the new series converges. Its sum is the limit of the
It is
,
,
expression
yV 2n i+ =o \
as
m
becomes
2w
infinite.
+1
4ra
evident that the
it is
From
+2
_!_ +2
1
+4
the identity
4n
sum
_
4w
4:7i
+
4
2 \2n
+1
of the second series
2n is
+ 2,
half the
sum
of
the given series. In general, given a series which is convergent but not absolutely convergent, possible to arrange the terms in such a way that the new series converges
it is
toward any preassigned number A whatever. Let Sp denote the sum of the first p positive terms of the series, and S g the sum of the absolute values of the first q negative terms, taken in such a way that the p positive terms and the q negative terms constitute the first p + q terms of the series. Then the sum of Sq As the two numbers p and q increase the first p + q terms is evidently Sp .
each of the sums Sp and Sq must increase indefinitely, for otherwise the series would diverge, or else converge absolutely. On the other hand, since the series is supposed to converge, the general term must approach zero. We may now form a new series whose sum is A in the following manner Let us take positive terms from the given series in the order in which they occur indefinitely,
:
in it until their
sum exceeds A.
Let us then add to these,
in the
order in which
they occur in the given series, negative terms until the total sum is less than A. Again, beginning with the positive terms where we left off, let us add positive terms until the total sum is greater than A. We should then return to the It is clear that the sum of the first n terms of the negative terms, and so on. new series thus obtained is alternately greater than and then less than A, and
that
it
differs
from
166. Abel s test.
A
by a quantity which approaches zero as
The following
test,
due
to Abel, enables us to establish the
convergence of certain series for which the preceding tests based upon the lemma stated and proved in 75. Let MO
+
ui
+
h
its limit.
+
fail.
The proof
is
VIII,
CONSTANT TERMS
166]
349
be a series which converges or which is indeterminate (that is, for which the sum of the first n terms is always less than a fixed number A in absolute value). let
Again,
fQ
1
)
en
)
,
numbers which approach
be a monotonically decreasing sequence of positive Then the series zero as n becomes infinite.
+
foU
(17)
+
eiMi
+
en
+
Wn
.
.
.
converges under the hypotheses made above. For by the hypotheses made above it follows that
for
any value
of
n and |u,
Hence, by the lemma just referred
p.
+ U n+p e n+p
+
+ ie,, + i
i
<
\
Sine? en+ i approaches zero as n becomes infinite,
be
write
2Aen + i.
n may be chosen
than any preassigned positive number for
less
we may
so large that
sum
the absolute value of the
will
to,
+
+
1
-
all
values of p.
The
157. by the general theorem of reduces to the series + un +
series (17) therefore converges When the series u MI 1
+1-
-!-,
+1
1
whose terms are alternately + 1 and 1, the theorem of this article reduces to 165 with regard to alternating series. the theorem stated in
As an example under
+
sin 6
which is
is
+
sin 2
is
if
sin
^
0,
sum
the
+
sin 3 6
+
For
convergent or indeterminate.
zero, while
nometry,
the general theorem consider the series
if
sin
n
+
,
=
0, every term of the series n terms, by a formula of Trigo
sin 6
of the first
equal to the expression nft
.
sin
2
.9 sin -
/n
.
sin
+1
(
2
\
2
which
is less
than |
I/sin (6/2) sin 6 I
converges for
all
values of
i
6.
\
in absolute value.
sin 2 T
It
sin I
may
It
n6 -j-
-f-
be shown in a similar manner that the
series
12
cos 6
converges for
all
values of
cos 2 6
except
follows that the series
n6
cos
n 2krt.
h
INFINITE SERIES
350
[VIII,
Corollary. Restricting ourselves to convergent series, Let general theorem. MO + MI H----- u n +
we may
state a
167
more
f-
be a convergent series, and
let
be any monotonically increasing or decreasing sequence of positive numbers which approach a limit k different from zero as n increases indefinitely. Then the series e
(18)
+ eii
o
H-----h
H----
also converges.
For definiteness
us suppose that the
let
e
s
always increase.
Then we may
write e
=
k
a
ei
,
=
k
a-i
,
en
,
=
an
k
,
,
where the numbers a a\ an form a sequence of decreasing positive numbers which approach zero as n becomes infinite. It follows that the two ,
,
,
,
series
ku
+
+
kui
both converge, and therefore the series
II.
+
ku n
,
(18) also converges.
SERIES OF COMPLEX TERMS
167. Definitions.
+
we
In this section
MULTIPLE SERIES
shall deal with certain gen
eralizations of the idea of an infinite series.
Let
u
(19)
+
u,
+
i, z
+
+
un
+
be a series whose terms are imaginary quantities:
Such a series is said to be convergent if the two series formed of the real parts of the successive terms and of the coefficients of the imaginary parts, respectively, both converge:
+
+
+ &. + &1+6.+- -+^ + a
(20)
(21)
a,
a 2 -f
.
+
an
---
= =
S
,
^".
Let S and be the sums of the series (20) and (21), respectively. Then the quantity S = S + is called the sum of the series (19). It is evident that S is, as before, the limit of the sum Sn of the first n terms of the given series as n becomes infinite. It is evident S"
is"
that a series of complex terms two series of real terms.
is
essentially only a combination of
VIII,
COMPLEX TERMS
168]
When
MULTIPLE SERIES
351
the series of absolute values of the terms of the series (19)
converges, each of the series (20) and (21) evidently converges abso l and IVI = lutely, for \a n In this case the series (19) is said to be absolutely convergent. The sum of such a series is not altered by a change in the order of the <
vX +
terms, nor by grouping the terms together in
Conversely, if each of the series (20) the series (22) converges absolutely, for
any way.
and (21) converges
y a* + b\
;>
|
an
|
+
absolutely, |
bn
. \
of Corresponding to every test for the convergence of a series absolute convergence of positive terms there exists a test for the
Thus, if the absolute value a cer of the ratio of two consecutive terms of a series \un + /un \, after con series the than tain term, is less than a fixed number less unity, 1 = k u Then, since n+l /un u, For, let 7, verges absolutely.
any
series whatever, real or imaginary.
l
.
|
we
after a certain term,
shall
U
which shows that the
rl
_
,
series of absolute values
+ Un +
l
If \u n+l /un
the series converges if
<
\
have also
U + U + converges.
<
|
\
I
approaches a limit
[
<
1,
and diverges if
I
I
>
as 1.
n becomes infinite, The first half is
In the second case the general term u n does not approach zero, and consequently the series (20) and (21) cannot The case I = 1 remains in doubt. both be convergent. self-evident.
More
generally,
if to
be the greatest limit of Vt7n as n becomes
infinite, the
For in the latter case the and diverges if The case in the general term does not approach zero (see 161).
series (19) converges if
w
modulus of which w = 1 remains
doubt
in
w>l.
the series
may
be absolutely convergent, simply
convergent, or divergent.
168. Multiplication of series.
Let
(23)
u
+
Ul
+
uz
+
+
"+-,
(24)
r
+
Vl
+
v,
+
+
vn
+
Let us multiply terms of the first be any two series whatever. series by terms of the second in all possible ways, and then group
INFINITE SERIES
352
the products u Vj for which the sum i+j of the sub the same we obtain in this way a new series
all
together scripts
168
[VIII,
is
t
;
U (25)
+ (H
V
\
+( u
(
If each of the
VI <>v
n
series
series (25) converges,
+ UlV ) + ( M + u vn _ l
*
2
+u
*
vi
+W
2
W
)
^
hw B Vo)H
-{
and (24)
(23)
and
l
its
sum
is absolutely convergent, the the product of the sums of the
is
two given
series. This theorem, which is due to Cauchy, was gener by Mertens,* who showed that it still holds if only one of the (23) and (24) is absolutely convergent and the other is merely
alized series
convergent. Let us suppose for definiteness that the series (23) converges absolutely, and let wn be the general term of the series (25):
Wn = The proposition
+
0^n
MlW-l
H
1-
Un V
.
we can show
will be proved if
that each of the
differences
+W ^o + w
^0
l
H
l
-\
W Sn
1"
+ w 2n+l
~ (U + % - (u + u
l
approaches zero as n becomes in each case, it
S
we
U
+
(v n + l
w n+ i(v
1-
-\
+
infinite.
) n
(l
+ i)0o
+
s, it
h V 2n )
-\
H
-f
h Vn )
Vl H
+ *i
+
H
Since the proof
shall consider the first difference only.
according to the u
=
-{
,
vn + 1 )
the same
is
Arranging
becomes
+
HI (W n + 1
v_i)+
+
u n + a (v
-\
+ V 2n _,) H
h M.,-1
h v n _ 2)-\
t
.+
\-u 2n v
i
.
Since the series (23) converges absolutely, the sum U + Ul H f- Un than a fixed positive number A for all values of n. Like wise, since the series (24) converges, the absolute value of the sum is less
vo
+ Vi +
+
vn
is
less
than a fixed positive number B.
corresponding to any preassigned positive number exists such that
e
a
Moreover,
number
m
A +B c
for any value of p whatever, provided that n m. Having so chosen n that all these inequalities are satisfied, an upper limit of the quan u l} u z U1} U2 tity 8 is given by replacing u ., u 2n by U L\ nJ >
1
,
,
* Crelle s Journal, Vol.
,
LXXIX.
,
,
VIII,
COMPLEX TERMS
169]
MULTIPLE SERIES
v n + 2 -\-----\- v n + p by (./(A respectively, v n + l vn _ l , v vt of the expressions v
+
+
+
+
+
353
+ B), and finally each v by B. + vn _ 2
,
,
This gives 8 1
u
<
+
+ Ul
1
"
+
Un
l
or
A whence,
+B
finally,
A
8 1
< 1
e.
+B Hence the difference 8 actually does approach
zero as n becomes infinite.
Consider a rectangular network which is lim and to the left, but which extends indefinitely down upward ward and to the right. The network will contain an infinite number of vertical columns, which we shall number from left to right from It will also contain an infinite number of horizontal to + oo to + oo we which shall number from the top downward from rows, Let us now suppose that to each of the rectangles of the network a 169. Double series.
ited
.
.
is assigned and written in the corresponding rec a be the quantity which lies in the ith row and in the Let ik tangle. kih column. Then we shall have an array of the form
certain quantity
(26)
We
shall first suppose that each of the elements of this array is real
and
positive.
Now let an infinite sequence of across this array as follows
:
1)
curves
Any
C lt C 2
,
,
Cn
,
be drawn
one of them forms with the two
which bound the array a closed curve which entirely surrounds the preceding one 2) The distance from any fixed point to any point of the curve C B which is otherwise entirely arbitrary, becomes infinite with n. Let S be the sum of the elements of the array which lie entirely inside the closed curve composed of C,- and straight lines
;
,
t
INFINITE SERIES
354
[VIII,
169
the two straight lines which bound the array. If S n approaches a limit S as n becomes infinite, we shall say that the double series \-~r-
+-r>
(27)
and that its sum is S. In order to justify this definition, necessary to show that the limit 5 is independent of the form of the curves C. Let C{, z Cm be another set of curves which recede indefinitely, and let S{ be the sum of the elements inside the closed curve formed by C- and the two boundaries. If m be assigned any fixed value, n can always be so chosen that the curve Cn lies entirely outside of C m Hence S m Sn and therefore Sm ^ S, for any value of m. Since Sm increases steadily with m, it must approach a limit S 5 as m becomes infinite. In the same it follows that S S Hence S = S. way For example, the curve C. may be chosen as the two lines which form with the boundaries of the array a square whose side increases indefinitely with i, or as a straight line equally inclined to the two converges, it is
C"
,
,
,
<
.
,
<
<
.
boundaries. 00
+ fan) +
The corresponding sums 11
+
If either of these
Ol)H
-----
Ka + a
are, respectively, the
+
H-----h
n\
sums approaches a limit
as
-!, n
following
H-----H
n becomes
0n)
infinite,
:
>
the
other will also, and the two limits are equal. The array may also be added by rows or by columns.
For, sup pose that the double series (27) converges, and let its sum be 5. It is evident that the sum of any finite number of elements of the series cannot exceed 5. It follows that each of the series formed of the
elements in a single row (28)
% +
converges, for the
*!
+
+
sum
<*.
+
i
,
of the first n
+1
=
0, 1, 2,
terms a ;o
cannot exceed S and increases steadily with n. the series formed of the elements in the ith row. (29)
o-
+
o-,
+
+
-
o-,
+
,
an
+
+
a, n
sum of Then the new series
Let
o-,
be the
+
surely converges. For, let us consider the sum of the terms of the This sum cannot exceed S, and array 2 tt for which i^p, k^r. increases steadily with r for any fixed value of p; hence it
approaches a limit as r becomes (30)
n
+
o-,
infinite, -f-
+
and that limit
tr
is
equal to
VIII,
COMPLEX TERMS
1(50]
MULTIPLE SERIES
355
+ o^ + + vp cannot any fixed value of p. It follows that exceed 5 and increases steadily with p. Consequently the series (29) Conversely, if converges, and its sum 2 is less than or equal to S. each of the series (28) converges, and the series (29) converges to a for
sum
2, it is
sum
evident that the
of any finite
Hence S
of the array (26) cannot exceed 2.
2
=
5=
number of elements and consequently
2,
s.
The argument
just given for the series
formed from the elements
rows evidently holds equally well for the series formed from the elements in individual columns. The sum of a convergent double series whose elements are all positive may be evaluated by rows, by columns, or by means of curves of any form which recede in individual
indefinitely.
In particular, if the series converges when added by rows, when added by columns, and the sum will be the
will surely converge number of same. it
A
terms
may
example is less,
:
theorems proved for simple series of positive be extended to double series of positive elements. For
if each of the elements of a double series of positive elements than the corresponding elements of a knoivn con
respectively,
vergent double series, the first series is also convergent; and so forth. double series of positive terms which is not convergent is said
A
to be divergent.
The sum
of the elements of the corresponding all limit
array which lie inside any closed curve increases beyond as the curve recedes indefinitely in every direction.
Let us
now
consider an array whose elements are not all positive. it is unnecessary to consider the cases in which
It is evident that
the elements are negative, or in which only a finite number of elements are either positive or negative, since each of these cases reduces immediately to the preceding case. We shall therefore sup pose that there are an infinite number of positive elements and an all
Let a lk be the of negative elements in the array. If the array 7\ of positive elements, general term of this array T. each of which is the absolute value a ik of the corresponding element infinite
number
|
T
\
said to be absolutely convergent. Such an array has all of the essential properties of a convergent array of in T, converges, the array
is
positive elements.
In order to prove
and
T",
us consider two auxiliary arrays T is formed from the array T The array
this, let
defined as follows.
T
by replacing each negative element by a
zero, retaining the positive
is obtained from Likewise, the array a zero and chang element the array by replacing each positive of the Each arrays T and ing the sign of each negative element.
elements as they stand.
T"
T by
T"
INFINITE SERIES
356
[VIII,
converges whenever the array 7\ converges, for each element of
169
T
,
for example, is less than the corresponding element of 7^. The sum of the terms of the series T which lie inside any closed curve is
equal to the difference between the sum of the terms of lie inside the same curve and the sum of the terms of lie
inside
T T"
which which
Since the two latter sums each approach limits as
it.
the curve recedes indefinitely in all directions, the first sum also approaches a limit, and that limit is independent of the form of the boundary curve. This limit is called the sum of the array T. The argument given above for arrays of positive elements shows that the same sum will be obtained by evaluating the array T by rows or by columns. It is now clear that an array whose elements are indiscriminately positive and negative, if it converges absolutely, may be treated as if it were a convergent array of positive terms.
But
it is
essential that the series 7\ of positive terms be
shown
to
be convergent. If the array TI diverges, at least one of the arrays T and If diverges. only one of them, T for example, diverges, the other being convergent, the sum of the elements of the array T which lie inside a closed curve C becomes infinite as the curve recedes indefinitely in all directions, irrespective of the T"
T"
form of the curve. If both arrays and diverge, the above reasoning shows only one thing, that the sum of the elements of the array T inside a closed curve C is equal to the difference between two sums, each of which 7"
T"
increases indefinitely as the curve C recedes indefinitely in all directions. It may happen that the sum of the elements of T inside C approach different limits according to the form of the curves C and the manner in which they recede, that
to say, according to the relative rate at which the number of and the number of negative terms in the sum are made to increase. The sum may even become infinite or approach no limit whatever for certain methods of recession. As a particular case, the sum obtained on evaluating by rows may be entirely different from that obtained on evaluating by columns if is
positive terms
the array
is
not absolutely convergent.
The following example 1
/1\
2\2/
1
/2\
1
SW
(31)
3V3/
I
1
/l\"
2\2/
/2\
3\3/
due
/2\
3\3/
l^V-^-V
2V2/
is
1/
2
1
1
Arndt.*
/3
4\4/
V-1/ 3 V
3V3/ n
to
/2\"
3V3/
4\4/ 1
Let us consider the array
A
p
P
I
P+
I/^liV--
1
P\ P
/
P
p\ p
/
p+
/3\"
4\4/
* Grunert s Archiv, Vol. XI, p. 319.
X
n
X
VIII,
COMPLEX TERMS
169]
MULTIPLE SERIES
357
of positire and an infinite number of negative formed from the elements in a single row or from The sum of the series formed from the those in a single column converges. terms in the nth row is evidently
which contains an
Each
elements.
infinite
number
of the series
2\2 Hence, evaluating the array (31) by rows, the result obtained sum of the convergent series
+ 2a
which (p
is
l)th column, that
converges, and
its
"
+
2Ml
+
equal to the
"
the other hand, the series formed from the elements in the
On
1/2.
+ 28
is
sum
is,
is
jj-1 p
-1
p
p+
Hence, evaluating the array (31) sum of the convergent series
1
by
1
1
p+
p(p +1)
1
p
columns, the result obtained
is
equal to the
3
which
is
1/2.
This example shows clearly that a double series should not be used in a calculation unless it is absolutely convergent.
We
shall also
meet with double
series
whose elements are complex two other
quantities. If the elements of the array (26) are complex, and may be formed where each element of arrays T"
7"
real part of the corresponding element of T is the coefficient of i in the corresponding
T
is
and each element of element of
T.
the T"
If the
array 7\ of absolute values of the elements of T, each of whose elements is the absolute value of the corresponding element of T,
converges absolutely, and converges, each of the arrays T and The sum of the given array T is said to be absolutely convergent. T"
the elements of the array which lie inside a variable closed curve approaches a limit as the curve recedes indefinitely in all directions.
independent of the form of the variable curve, and it The sum of any absolutely of the given array. be evaluated also by rows or by columns. convergent array may
This limit is
is
called the
sum
INFINITE SERIES
358
[VIII,
170
An absolutely convergent double series may be replaced by a simple formed from the same elements. It will be sufficient to show that the rectangles of the network (26) can be numbered in such a way that each rec tangle has a definite number, without exception, different from that of any other In other words, we need merely show that the sequence of natural rectangle. numbers 170.
series
0,
(32)
1,
2,
n,
-..,
-.-,
and the assemblage
of all pairs of positive integers (i, fc), where i^O, k>0, can such a way that one and only one number of the sequence (32) will correspond to any given pair (i, k), and conversely, no number n corresponds to more than one of the pairs (t, k). Let us write the pairs (i, k) in order as
be paired
follows
off in
:
(1,0),
(0,0),
(0,1),
(2,0),
(1,1),
-..,
(0,2),
which i + k = n are written down after those for which i + k n have all been written down, the order in which those of any one set are written being the same as that of the values of i for the various pairs beginning with (n, 0) and going to (0, n). It is evident that any pair (i, k) will be preceded by only a, finite number of other pairs. Hence each pair will have a distinct number when the sequence just written down is counted off according to the natural numbers. where, in general, all those pairs for <
Suppose that the elements of the absolutely convergent double series SSaa- are down in the order just determined. Then we shall have an ordinary series
written (33)
doo
+
io
+
oi
+
2o
+
ii
+
floa
+
+ ao +
a-i,i
+
whose terms coincide with the elements of the given double series. This simple and its sum is equal to the sum of the given double series. It is clear that the method we have employed is not the only pos sible method of transforming the given double series into a simple series, since series evidently converges absolutely,
the order of the terms of the series (33) can be altered at pleasure.
Conversely,
any absolutely convergent simple series can be transformed into a double series in an infinite variety of ways, and that process constitutes a powerful instrument in the proof of certain identities.* It is evident that the concept of double series is not essentially different from that of simple series. In studying absolutely convergent series we found that the order of the terms could be altered at will, and that any finite number of
terms could be replaced by their sum without altering the sum of the series. An attempt to generalize this property leads very naturally to the introduction of double series.
The notion of double series may be generalized. we may consider a series of elements a mn with two oo to + oo subscripts ra and n, each of which may vary from The elements of such a series may be arranged in the rectangles of 171.
In the
Multiple series.
first
place
.
a rectangular network which extends indefinitely in *Tanuery, Introduction a
hi theorie
desfauctions d une
all directions
variable, p. G7.
;
VIII,
it
COMPLEX TERMS
172]
evident that
is
type we have
it
MULTIPLE SERIES
359
be divided into four double series of the
may
just studied.
A
more important generalization is the following. Let us consider a series of elements of the type ,,. .., mp where the subscripts from to -f oo or from values on GO take m tK. 1} any 2 , p may ,
m + oo, but may be
,
,
by certain inequalities. Although no such convenient geometrical form as that used above is available to
when
the
number
restricted
of subscripts exceeds three, a slight consideration
shows that the theorems proved for double series admit of immediate Let us first sup generalization to multiple series of any order p. Let S l are real and positive. the elements all that pose MI ,,,..., mp be the sum of a certain number of elements of the given series, S2 ,
and a certain number of terms previously neglected, sum of S a the sum of S2 and further terms, and so on, the successive sums Sn S 1} S2 being formed in such a way that any particular -, element of the given series occurs in all the sums past a certain one.
the
<S\
,
,
If
Sn approaches a limit S as n becomes
infinite,
said to be convergent, and S is called its sum. double series, this limit is independent of the is
the given series in the case of
As
way
in
which the
sums are formed.
successive
If the elements of the given multiple series have different signs complex quantities, the series will still surely converge if the
or are
series of absolute values of the
terms of the given series converges.
Cauchy s theorem. The following theorem, a generalization of Cauchy s theorem ( 161), enables us to determine in many cases whether a given multiple series is conver 172. Generalization of
which
is
gent or divergent.
Let/(.x, y) be a function of the
two variables x
and y which is positive for all points (x, y) outside a certain closed curve T, and which steadily diminishes in value as the point (x, y) Let us consider the value of the double recedes from the origin.* over the ring-shaped region between dx extended dij integralJJ"/^, y} T and a variable curve C outside T, which we shall allow to recede and let us compare it with the double indefinitely in all directions series 2/(wi, n), where the subscripts m and n may assume any posi ;
tive or negative integral values for
which the point (m, n)
Then the double series a limit, and conversely. approaches
side the fixed curve T. integral
* All that x\>X2
ond
hypotheses.
is
necessary for the present proof
y\>y^
outside T.
TRANS.
It is
is
lies
out
converges if the double
that/to,
l/i)>/(a; 2)
easy to adapt the proof to
still
7/ 2 ) whenever more general
INFINITE SERIES
360
= 0, x =
[VIH,173
x= and y = 0, y = 2, 1, y = 2, F divide the region between and C into squares or portions of squares. Selecting from the double series the term which corresponds to that corner of each of these squares which is farthest from the origin, it
The lines x
is
evident that the
1,
sum 2/(w,
n) of these terms will be less than the
value of the double integral dx dy extended over the region ///(#, y) between F and C. If the double integral approaches a limit as C recedes indefinitely in all directions, it follows that the sum of any
number number
of terms of the series whatever
is always less than a fixed hence the series converges. Similarly, if the double series converges, the value of the double integral taken over any finite hence the integral region is always less than a fixed number ;
;
approaches a
The theorem may be extended
limit.
to multiple
any order p, with suitable hypotheses in that case the of integral comparison is a multiple integral of order p. As an example consider the double series whose general term is 2 2 M where the subscripts m and n may assume all integral l/(?n + n ) values from oo to + o except the values m = n 0. This series for For the double integral 1, and diverges for /x^l. converges p. series of
;
,
>
dx dy extended over the region of the plane outside any circle whose center is the origin has a definite value if ^ 1 and becomes >
infinite if
(133). More generally the multiple p.
series
whose general term
is
1
+ m\ + -.. + miy m = m = = mp =
(m\
where the verges
III.
if
set of values
2n
>
l
SERIES OF VARIABLE TERMS
excluded, con
UNIFORM CONVERGENCE
173. Definition of uniform convergence.
(35)
is
z
j9.*
u
(x)
+ KJ (*)+... + u
A n
series of the
(x)
+
form
-,
whose terms are continuous functions of a variable x in an inter val (a, i), and which converges for every value of x belonging to that interval, does not necessarily represent a continuous function,
*More general theorems
are to be found in Jordan
s
C ours d Analyse, Vol.
I,
p. 163.
VIII,
VARIABLE TERMS
173]
361
we might be tempted to believe. In order to prove the fact we 4 need only consider the series studied in
as
:
1
+x
2
(1
+
a;
2
2
(1
)
+
a;
2 )"
which satisfies the above conditions, but whose sum is discontinuous for x Since a large number of the functions which occur in 0. mathematics are defined by series, it has been found necessary to study the properties of functions given in the form of a series. The first question which arises is precisely that of determining whether
sum
or not the
of a given series
is
a continuous function of the
Although no general solution of
variable.
this
problem
is
known,
study has led to the development of the very important notion
its
of uniform, convergence. series of the type (35), each of
A
which
whose terms
is
a function of x
defined in an interval (a, 6), is said to be uniformly con vergent in that interval if it converges for every value of x between
a and
is
and
if, corresponding to any arbitrarily preassigned positive a positive integer N, independent of x, can be found such that the absolute value of the remainder R n of the given series
b,
number
c,
R n = U n + () + l
is
less
which
The
than
-----H MH + S (*) H
^ n+p (x) ----\
n^N
for every value of and for every value of x the interval (a, &). latter condition is essential in this definition. For any pre e
lies in
assigned value of x for which the series converges it is apparent from the very definition of convergence that, corresponding to any can be found which will satisfy positive number e, a number
N
the condition in question. it
is
But, in order that the series should con necessary further that the same number
N
verge uniformly, should satisfy this condition, no matter what value of x be selected in the interval (a, b). The following examples show that such is not
always the
The
series
val (0, 1).
case.
Thus
in the series considered just
above we have
in question is not uniformly convergent in the inter For, in order that it should be, it would be necessary
(though not sufficient) that a number 1
N exist,
such that
INFINITE SERIES
362 for all values of
x
[VIII,
in the interval (0, 1), or,
what amounts
17.5
to the
same thing, that
N
Whatever be the values of
and
there always exist, however,
e,
positive values of x which do not satisfy this inequality, since the right-hand side is greater than unity.
Again, consider the series denned by the equations
= The sum of the
first
n terms of this
-
S
=
S n _ lf
series is evidently
The
approaches zero as n increases indefinitely.
Sn (#), which
series is therefore
x In order nxe~" *. convergent, and the remainder R n (cc) is equal to that the series should be uniformly convergent in the interval (0, 1), it
would be necessary and
trarily preassigned positive that for all values of n >
corresponding to any arbi exist such a positive integer
sufficient that,
number
N
e,
N
if x be replaced by 1/w, the left-hand side of this inequality is l/n which is 1. Since e greater than 1/e whenever n equal to e~ may be chosen less than 1/e, it follows that the given series is not
But,
>
,
uniformly convergent.
The importance
of uniformly
convergent series rests upon the
following property:
The sum of a series whose terms are continuous functions of a an interval (a, i) and which converges uniformly in that interval, is itself a continuous function of x in the same interval. variable x in
Let X Q be a value of x between a and b, and let x -f h be a value neighborhood of X Q which also lies between a and b. Let n
in the
be chosen so large that the remainder
fl() = u n + is less
l
(x)
+
u n+z (x) H----
than e/3 in absolute value for all values of x in the interval e is an arbitrarily preassigned positive number. Let/(cc)
(a, b),
where
be the
sum
where
Then we may write
of the given convergent series.
denotes the
sum
of the
first
n
-f 1
terms,
+
().
<}>(x)
K*) =
M uO*0
+
M,
00 H
VIII,
VARIABLE TERMS
17.5]
363
Subtracting the two equalities
+
f(x
we
=
It)
(x
+
u
A)
+R
tt
(x
+ A),
find
+
f(x
h)
-/(*)
The number
was
=
+
[>(x
-
A)
<(*)]
+ /^(^u +
A)
- TJ.^o).
we have
so chosen that
O
O
On the other hand, since each of the terms of the series is a continu ous function of x, is itself a continuous function of x. Hence a positive number 77 may be found such that <(o:)
whenever h
is less
than
|
whenever =x
for x
Note.
\h
is
than
less
It follows that
rj.
we
shall have, a fortiori,
This shows that f(x)
rj.
is
continuous
.
would seem at
first very difficult to determine whether uniformly convergent in a given interval. The following theorem enables us to show in many cases that a
It
or not a given series
is
given series converges uniformly.
Let MO (a:)
(36)
a
be
series each
interval (a,
b),
+
+
of whose terms
and
is
-
.
-
+
(*)
+
a continuous function of x in an
let
Jf.
(37)
MI (x)
+
Jtfi
+ ... + *; + ...
a convergent series whose terms are positive constants. Then, tf un ^ n for a H values of x in the interval (a, b) and for all be
M
I
|
values of n, the first series (36) converges uniformly in the interval considered.
For
it is
evident that
we
shall
have
INFINITE SERIES
364
for all values of x between a
the remainder
Rn
and
we
of n greater than N,
shall also
whenever n is greater than N, for For example, the series
N be
If
b.
of the second series
[VIII,
is
less
174
chosen so large that e for all values
than
have
values of x in the interval
all
(a, b).
M + Mi sin x + Mjj sin 2x ----- M smnx ---have the same meaning as above, converges where M M M ,
z
l}
-\
a
\-
-\
,
,
uniformly in any interval whatever. 174. Integration and differentiation of series.
Any series of continuous functions which converges uniformly in an interval (a, b) may be integrated term by term, provided the limits of integration are finite and lie in the interval (a, b*). Let x and x v be any two values of x which lie between a and b, e for all values and let N be a positive integer such that R B (ar)| N. Let f(x) be the sum of of x in the interval (a, b) whenever n <
|
>
the series
and
let
us set
rx
/*i.
Dn
\
Jxt
f(x) dx
rx
i
u dx
I
Jr
The absolute value of D n Hence Dn approaches zero
rx
Uidx
-----
Jx
tA is less
u n dx
I
than
c
x^
rx
*
\
i
I
=
I
whenever
x
R n dx.
Jx
1
as n increases indefinitely, and
n^
we have
the equation r*i I
J*t
r*i Jft
rx r* -----hi u H (x)dx Ui(x)dx-\ (x)dx+l ^x J* 1
*
u
f(x}dx=\
-\
----
.
*
Considering X Q as fixed and Xj as variable, we obtain a series
I
J*t
u Q (x)dx-\----
+lu
n
(x )dx-\
----
J*o
which converges uniformly in the interval (a, continuous function whose derivative is f(x).
b)
and represents a
VIII,
VARIABLE TERMS
174]
365
Conversely, any convergent series may be differentiated term by term the if resulting series converges uniformly.*
For, let
=
f(x)
M
(a;)
+
Ul (x)
+
+
u n (x) -\----
be a series which converges in the interval (a,
ft).
Let us suppose
that the series whose terms are the derivatives of the terms of the
given series, respectively, converges uniformly in the same interval, let denote the sum of the new series
and
<(z)
Integrating this series term by term between two limits x and each of which lies between a and b, we find
= [u
Jx /
(a:)
-
M O (* O )]
+
[MJ
(a;)
x,
-
*SJCK
or
This shows that
is (x)
Examples.
1)
The
the derivative of /(x)
.
integral
x
dx
cannot be expressed by means of a finite number of elementary Let us write it as follows
functions.
:
x
/e
x
The
dx
= C dx J
x.
x
C ex x J
(-/
1
dx
may be developed For we have
last integral
values of
I
= log x
-f
in a series
C ex x J
1
I
dx.
which holds for
all
and this series converges uniformly in the interval from R to + R, no matter how large R be taken, since the absolute value of any * It
is
function.
assumed in the proof also that each term of the new The theorem is true, however, in general. TRANS.
series is
a continuous
INFINITE SERIES
366 term of the
[VIII,
174
than the corresponding term of the con
series is less
vergent series
It follows that the series obtained
F
by term-by-term integration
^ =l+ I + 2Y72 + OC^
OC
3C
n
+ nl.2...n +
converges for any value of x and represents a function whose deriva tive is (tf
!)/.
The perimeter
2) is e is
equal,
an
of
1 12,
by
whose major axis
ellipse
is
2a and whose eccentricity
to the definite integral
=
4c f /o
The product
sum
e2 sin 2
lies
between
and
e2
<j>
(<
Hence the
1).
radical
is
equal to the
of the series given by the binomial theorem
Vl
--
= 1 -- e2 sin 2
e 2 sin 2
e4
2
.
4
2n
6
.
sin*^>
The series on the right converges uniformly, for the absolute value of each of its terms is less than the corresponding term of the convergent series obtained by
=
setting sin
since,
by
Hence the
1.
116,
C\
.,.
.
sin^"0a*
Jo
we
shall
f
--
series
be integrated term by term; and
may
1.8.6...(2n-l) 2
4
.
it
2n
6
.
1
2
have
Vl-e
2 sinV
Jo
= -Sl2 (
- e2 4
-
e*
-
eft
----
256
64
2
3.6...(2n-3)-| 2.4.6--.2n J
V
If the eccentricity e is small,
a very good approximation to the exact value of the obtained by computing a few terms. Similarly, we may develop the integral
integral
is
f Vl-
e 2 sin 2
0drf>
Jo in a series for
any value
of the
Finally, the development of leads to the formula
upper limit
Legendre
s
.
complete integral of the
first
kind
Vlll,
VARIABLE TERMS
174]
367
The definition of uniform convergence may be extended to whose terms are functions of several independent variables. example,
series
For
let
u
(x, y)
+ % (x, y)
h
H
u n (x,
y}-\
be a series whose terms are functions of two independent variables x and let us suppose that this series converges whenever the ij,
and
R
bounded by a closed contour C. lies in a region (x, y} series is said to be uniformly convergent in the region R if,
point
The
corresponding to every positive number e, an integer such that the absolute value of the remainder R n
whenever n
N can be found is
less
than
e
equal to or greater than N, for every point (x, y) It can be shown as above that the sum of inside the contour C. is
such a series
y
is
a continuous function of the two variables x and
provided the terms of the series are
in this region,
all
continu
ous in R.
The theorem on term-by-term
integration also
may
If each of the terms of the series is continuous in
denotes the
I
I
sum
of the series,
f( x y}dxdy >
=11
we
shall
+
I
ul
I
un (x,y}dxdy-\
where each of the double integrals
is
if
f(x, y)
have
u9 (x, y) dx dy
+ 11
be generalized.
R and
(x,
y)dxdy
-\
,
extended over the whole inte
any contour inside of the region R. Again, let us consider a double series whose elements are functions
rior of
of one or
more variables and which converges absolutely for all sets domain D. Let the
of values of those variables inside of a certain
elements of the series be arranged in the ordinary rectangular array, let R c denote the sum of the double series outside any closed
and
curve
C drawn
in the plane of the array. uniformly in the
series is said to converge
Then the given double domain D if correspond
ing to any preassigned number c, a closed curve K, not dependent on the values of the variables, can be drawn such that R c e for <
|
any curve
C whatever
lying outside of of the variables inside the domain D.
K and
It is evident that the preceding definitions
extended without
for
any
set of values
and theorems may be any order whose
difficulty to a multiple series of elements are functions of any number of variables.
INFINITE SERIES
368 Note. integrate
If it
a series does not converge uniformly, it term by term. For example, let us set
SH (x) = nxe-*?, The
S
(x)
whose general term approaches zero as n becomes series
f(x)
=
=
= is
u n (x) converges, and
(z)
+
not always allowable to
its
n
.
sum
is
=
1, 2,
zero, since
.
Sn (x)
Hence we may write
infinite.
1*1
is
= S n - Sn ^
u a (x)
0,
175
[VIII,
MS (x)
+
+
u, (x)
+
,
whence J f(x) dx = 0. On the other hand, if we integrate the series term by term between the limits zero and unity, we obtain a new series for which the sum of the first n terms is
which approaches 1/2 as
its
n becomes
limit as
infinite.
175. Application to differentiation under the integral sign. The proof of the formula for differentiation under the integral sign given in
97
is
based essentially upon the supposition that the limits x If X is infinite, the formula does not always hold.
and X are finite. Let us consider,
for example, the integral
F(a)
= f
+
"
s
max
I
X
i/O
This integral does not depend on y = ax it becomes
dx
a, for if
,
a
>
we make
.
the substitu
tion
y If we tried to apply we should find
the ordinary formula for differentiation to F(a~),
F (o-) = This
I
cos
ax dx
Jo
surely incorrect, for the left-hand side is zero, while the right-hand side has no definite value. Sufficient conditions may be found for the application of the ordinary formula for differentiation, even when one of the limits is
by connecting the subject with the study of series. consider the integral
is infinite,
us
first
Let
p+ f(x)dx,
which we shall suppose to have a determinate value ( 90). Let a u a2 an be an infinite increasing sequence of numbers, all ,
,
,
Vni,
VARIABLE TERMS
175]
greater than a
,
where a n becomes
infinite
369
with
ra
2
/"
we
If
n.
set
+i
Ja n
Ja^ the series
converges and its sum is nn is equal to fa f(x) dx. It
f(x)
we
we
= cosx,
shall
f(x) dx, for the
sum Sn of
be noticed that the converse
should
for example,
If,
j*"
is
the
first
n terms
not always true.
set
a
0,
!
=
TT,
an
-,
=
mr,
-,
have
Un = Hence the
series
cos x dx
I
U
=
.
nit
converges, whereas
the integral
fo cosxdx
ap
becomes infinite. Now let f(x, a) be a function of the two variables x and a which is continuous whenever x is equal to or greater than a and a lies If the integral J f(x, a) dx approaches a in an interval (a i)-
proaches no limit whatever as
I
a
>
limit as
becomes
I
infinite, for
function of a,
any
of a, that limit
value"
is
a
r +x (a)=
f(x,a)dx, i/a.
which may be replaced, as we have just shown, by the sum of a convergent series whose terms are continuous functions
of a
:
U
(a)
=
I
\f(x,
a) dx
,
U, (a)
= /
/(ar,
^t
A,
a)
rfx
,
continuous whenever the series converges uni m formly. By analogy we shall say that the integral f* f(xt a) dx to any converges uniformly in the interval (a aj if, corresponding a can of number a independent preassigned positive quantity e, + for e whenever I be found such that f "f(x, a)dx N, any value
This function F(a)
is
,
N
>
<
|
t
of a which lies in the interval (a * See
TRANS.
W.
F.
,
a^*
OSGOOD, Annals of Mathematics, 2d
If the integral converges series, Vol. Ill (1902), p. 129.
INFINITE SERIES
370
For
if
f(x,
cr)
uniformly, the series will also.
we
[vm,i75
a n be taken greater than N,
shall have \R.\
r+*
=
I
Jan
dx
hence the function F(a) is continuous in this case throughout the interval (a a^. Let us now suppose that the derivative df/da is a continuous ,
when x ^ a and a
function of x and a
a
<
<
a^ , that the integral
da has a
value for every value of a in the interval (a
finite
that the integral converges uniformly in that interval. in question may be replaced by the sum of the series
dx
=F
(a-)
+
F!
(tr)
+
,
The
a^, and integral
+ Fn (a)
where
The new
series converges uniformly, and its terms are equal to the corresponding terms of the preceding series. Hence, by the theorem proved above for the differentiation of series, we may write
In other words, the formula for differentiation under the integral sign provided that the integral on the right converges uniformly. The formula for integration under the integral sign ( 123) also may be extended to the case in which one of the limits becomes still holds,
Let f(x, a) be a continuous function of the two variables a a a a If the integral // V(*, ) dx is uni in the interval formly convergent (a a^, we shall have infinite.
x and
a, for x
>
<
<
t
,
.
,
r+* /
(A)
/>
+
I
A|
To prove
r^ f, dx da f(x,a}da=\ ^ J J us
this, let
first select
a
f(x,a)dx.
number
I
>
a
;
then
have
C (B)
^u
ra
l
dx
/ a
\
J*f
ra
*
f(x,
a)da= J
I
r
i
da Ja
l
f(x, a)dx.
we
shall
VIII,
As
VARIABLE TERMS
176]
371
increases indefinitely the right-hand side of
I
this
equation
approaches the double integral + ao
.
f(x, a)dx,
between these two double integrals
for the difference
is
equal to
r +x
ai
da
I
/(cc,
<x)dx.
Jl
Suppose
N chosen
+ :0
/
a)dx
/(x,
so large that the absolute value of the integral than c whenever / is greater than N, for any
is less
Then the absolute value of the value of a in the interval (a a^. a and therefore it difference in question will be less than c a 1 Hence the left-hand zero as I increases indefinitely. will ,
,
\
approach
side of the equation (B) also approaches a limit as the symbol nite, and this limit is represented by
I
becomes
infi
/!
X+0
dx
f(x, a) da.
I
J*t
This gives the formula (A) which was to be proved.* 176. Examples.
1)
Let us return to the integral of
F(a}-
I
-
er"*
The
is positive.
:
dx,
x
Jo where a
91
+
/
integral / I
+
-
er ax sin x dx
,
Jo * The formula for differentiation may be deduced easily from the formula (A). For, a en for a suppose that the two functions f(x, a) and fa (x, a) are continuous * + that the two integrals F(a) = fa */(*, ) dx and *() = fa *S ( x ^ dx have x a <
<
,
>
;
;>
finite
values
;
and that the
the formula (A),
if
a lies
latter
fduC J
"/<*t)d*=
J
"o
"o
where
converges uniformly in the interval (a
in the interval (a
<*!)
>
f
Jmt
dx
may f
du
= ( Ja
+
+ */(*, o
a)dx- C
Ja o
From
C fu (x,u)du,
a has heen replaced by u under the integral sign. be written in the form
"*(!*)
Jao
i).
Jao
for distinctness
formula
,
we have
f(x,
whence, taking the derivative of each side with respect to a, we find
But
this
INFINITE SERIES
372
[VIII,
176
obtained by differentiating under the integral sign with respect to a, converges all values of a greater than an For arbitrary positive number k.
uniformly for we have
/
~ +00
+<=0
e- ax smxdx
I
Ji
and hence the absolute value values of
a
1
e~ ax dx
=
greater than It follows that
of the integral
k, if
*-
\
I
N, where
>
(X)
.
on the
N
is
left will be less than e for all chosen so large that keky 1/e >
sin
Q
I
er al
a
Ji
ctj&
,
Jo
The
indefinite integral
was calculated
F (a) =
~e-<*
x
+ a sinx)~| + Jo + a2
1
(cosx
I"
1
whence we
119 and gives
in
1
+ a2
find
=C-
F(a)
arc tan
a
,
C may be determined by noting that the definite integral F(a) as a becomes infinite. Hence C = x/2, and we finally find the
and the constant approaches zero formula
+
r
e~ ax ,.
J This formula
is
sin Sll
x
,
dx
=
1
arc tan
.
,
established only for positive values of n-, but we saw in is the sum of an alternating series whose remainder n
R
the left-hand side
91 that is
always than 1/n. Hence the series converges uniformly, and the integral is a con tinuous function of a, even for a = 0. As a approaches zero we shall have in less
the limit +ao
f
(39)
2
Jo 2) If in the
formula
rv-
134
we
set
x
= yVa,
where +
(40)
r t/o
2
a
is
positive,
we
find
V
and it is easy to show that all the integrals derived from this one by successive differentiations with respect to the parameter a converge uniformly, provided that a is always greater than a certain positive constant A;. From the preceding formula we may deduce the values of a whole series of integrals :
22
(41)
EXERCISES
VIII, Exs.]
We
number
these an infinite
By combining
373
of other integrals
/^
-
-4-
be evaluated.
oo
t-**W**
I
2
!
Jo
and we
All the integrals on the right have been evaluated above,
+
2
1
\n = 1 ^0032^ -^--^ /-
may
have, for example,
(2/3)
+ /_iy, 7
(
2 fl 2
Vn
a-i
^*
"
1.2.3.--2n
+
find
...
1.8.6..
2
.(8n-l)8
2"
or, simplifying, /
+
*>
e~ a v*cos2pydy
I
(42)
^o
1
=
~
2
_ 5!
Iff
%/-
\a
e
.
EXERCISES 1.
;
1
.
2
Derive the formula
[Z"
n
=1+
(logZ)><]
Si lOgZ
+
uz"
where Sp denotes the at a time.
sum
^1.2
(logz)
of the products of the first
2
+
"
h 1
.
n
2
(logz)"
n natural numbers taken p r
,,
n
[MURPHY.] [Start with the formula
(log)
1.2 and 2.
differentiate
n times with
1
.
2
n
+ ..1
respect to z.]
Calculate the value of the definite integral
j by means 3.
of the formula for differentiation
under the integral
Derive the formula *
/
^ r + e -+ &dx =
=1
Jo [First
show
that dl/da
=-
21.]
,
sign.
INFINITE SERIES
374 4.
Derive the formula
J_ e
by making use of the preceding 5.
From
the relation
derive the formula
a_
*?
da
exercise.
i
= = VTre- 2,.*
Va
rviil,
CHAPTER IX POWER SERIES
TRIGONOMETRIC SERIES
In this chapter we shall study two particularly important classes power series and trigonometric series. Although we shall speak of real variables only, the arguments used in the study of
of series
power
without change to the case where the
series are applicable
variables are complex quantities, by simply substituting the expres sion modulus or absolute value (of a complex variable) for the expres sion absolute value (of a real variable).*
POWER
I.
SERIES OF A SINGLE VARIABLE
177. Interval of convergence.
+ AiX + A
A
(1)
where the
A
coefficients
Let us 2
+
X*
A 1} A 2
,
first
consider a series of the form
AH X
-f
+
-,
are all positive, and where
,
the independent variable A is assigned only positive values. evident that each of the terms increases with A Hence, .
It is if
the
any particular value of A, say X l} it converges a fortiori for any value of A less than X l Conversely, if the series diverges for the value A2 it surely diverges for any value of A We shall distinguish the following cases. greater than A2 series converges for
.
,
.
1)
Such
The
series (1) may converge for any value of is the case, for example, for the series
Y
1 f "
I+
A
A
whatever.
2
+
lT2
A"
>
"
+ l72T^ +
""
2) The series (1) may diverge for any value of A except The following series, for example, has this property
A=
:
1
+ A + 1 2A
2
.
+1.2.3-.-
H
nX n
H
.
3) Finally, let us suppose that the series converges for certain values of A and diverges for other values. Let x be a value of A
A
for
which
it
converges, and
let
* See Vol. II,
A
2
be a value for which
2GG-275.
375
TKAKS.
it
diverges.
SPECIAL SERIES
376
From
the remark
made
series converges if
A
above,
X
follows that
it
and
j,
[IX,
it
l
is less
if
diverges
than
AT>A 2
.
X
X2 The The only But all .
2 uncertainty is about the values of A between X l and the values of A for which the series converges are less than
we
hence they have an upper limit, which values of of
A
X
the values of
X
2
,
and
Since all the
shall call R.
which the
for
which
for
.
177
series diverges are greater than any value converges, the number R is also the lower limit of for which the series diverges. Hence the series (1)
it
X
X
diverges for all values of greater than R, and converges for all values less than R. It may either converge or diverge when R. of
X=
X
For example, the
series
In this case R =1. converges if A 1, and diverges if A ^ 1. This third case may be said to include the other two by suppos <
ing that
R may
be zero or
may become
Let us now consider a power
a
(2)
where the
-f
a1 x
+a
z
infinite.
series, i.e.
x
a series of the form
+ an x +
+
2
a
,
and the variable x may have any From now on we shall set A = |a,-|, X = \x\.
coefficients a,
whatever.
{
series (1) is the series of absolute values of the
real values
Then the
terms of the series
Let R be the number defined above for the series
(1).
(2).
Then the
any value of x between It remains of x whose This follows immediately from a funda
series (2) evidently converges absolutely for
- R and + R, by the very definition of the number R. to be shown that the series (2) diverges for any value absolute value exceeds R. mental theorem due to Abel * :
If the
for any particular value x it converges values of x whose absolute value is less than \x \.
series (2) converges
absolutely for
any
,
In order to prove this theorem, let us suppose that the series (2) be a positive number greater than converges for x = x and let the absolute value of any term of the series for that value of x.
M
,
Then we
shall have, for
any value of
A and we
may
lt
n,
n
x
<M,
write /
\
_
/
/X <M
*
Recherche sur la sene
,
1
H
m
in
x
(m
1)
-\
1
.
2t
IX,
POWER
177]
SERIES
377
whenever
It follows that the series (1) converges
X<|cc
|,
which
proves the theorem. x the series (1) In other words, if the series (2) converges for x is less than x of absolute values converges whenever Hence ,
X
.
|
R was
cannot exceed R, for
|cc
1
supposed to be the upper limit of
X
for which the series (1) converges. up, given a power series (2) whose coefficients may have either sign, there exists a positive number R which has the follow
the values of
To sum
The series (2) converges absolutely for any value of x ing properties R and -f R, and diverges for any value of x whose absolute between :
The
value exceeds R.
which
R
origin
if
is
R
interval
(
R,
+
is
R)
This interval extends from
convergence.
called the interval of cc in the case in
oo to
+
conceived to have become infinite, and reduces to the = 0. The latter case will be neglected in what follows.
The preceding demonstration gives us no information about what R. The series (2) may be absolutely happens when x = R or x = For example,
convergent, simply convergent, or divergent. for each of the three series
for the ratio of
any term to the preceding approaches x as
in each case.
The
series diverges for
x
first
series diverges for
= 1, and converges for = 1.
x
aj
=
=
1.
1.
R=
1
its limit
The second The third con
verges absolutely for x
The statement
Note. for
it is
theorem may be made more general, argument that the absolute value of any
of Abel
sufficient for the
s
term of the series
+a
l
X
be less than a fixed number.
+
----1- a X n
Whenever
H---this condition is satisfied,
the series (2) converges absolutely for any value of x whose absolute value is less than \x .
The number
R
is
which
is
the greatest limit of the Sequence
in
160,
For
if
we
connected
in
a very simple
consider the analogous sequence
way with
the
number
to
defined
SPECIAL SERIES
378
[IX,
evident that the greatest limit of the terms of the new sequence is 1/w 1/w, and diverges if sequence (1) therefore converges if
The
it is
X
178. Continuity of a
series
X
<
>
;
178
hence
power series. Let f(x) be the sum of a power R to + R, in the interval from
which converges f(x)
(3)
=
a
We
shall first let R be a positive number less than R. that the series (3) converges uniformly in the interval from than R R to For, if the absolute value of x is less
and
+
.
remainder
Rn
Rn =
of the series (3)
is
a n+l x n + l
show
R ,
the
+ an+p xn + P +
+
less in absolute value
than the remainder
1
4.+I*** +-*.+;**** But the series (1) converges for of the corresponding series (1). R. R , since R may be found Consequently a number
X=
N
<
such that the latter remainder will be number c whenever n ^ N. Hence
less
Rn
positive
provided that
\x\
<
R
than any preassigned <
c
whenever n
>
N
\
.
is a continuous f(x} of the given series R. and R For, let x function of x for all values of x between is evident It R. than number whose absolute value is less be
It follows that the
sum
+
any
number R may be found which is less than R and greater than \x \. Then the series converges uniformly in the interval sum f(x) of the (_ R + R as we have just seen, and hence the 1
that a
,
~),
series is continuous for the value
x
,
since x belongs to the interval
in question.
R of the and The function f(x} remains continuous,
This proof does not apply to the end points interval of convergence.
+R
however, provided that the series converges for those values. x = R, its Indeed, Abel showed that if the series (3) converges for series sum for x = R is the limit which the sum /(#) of the approaches as x approaches R through values less than J?.f
Let S be the sum of the convergent series
S
=
+
al
R
+a
2
R* -\----
+
an
R"
H----
,
* This theorem was proved by Cauchy in his Cours d Analyse. It was rediscovered by Hadamard in his thesis. to the case of t As stated above, these theorems can be immediately generalized series of
imaginary terms. In this case, however, care TRANS. See Vol. II, 266.
the generalization.
is
necessary in formulating
IX,
m]
and
let
SERIES
379
n be a positive integer such that any one of the sums
than a preassigned positive number e. If we set x = R0, and to 1, a; will increase from to R, and we increase from
is less
then
POWER
let
shall have
/(z) =/(0J?)
n be chosen
If
=
+a
a
=
2
2
2
+
+a
n $R"
R(l
-
4. ....
write
+ # (1 + + + --+a n+l R + + an+p R n+ ? + ai
2
0)
n
(4)-
+a
we may
as above,
S -f(x)
0R
l
2
2
)
l
a n R n (l
0")
-.
and the absolute value of the sum of the series in the second line can B + 2 H+P On the other hand, the numbers O n + not exceed e. -, Abel s lemma in form a decreasing sequence. Hence, by 75, proved l
,
,
we
shall
have n + l Rn + \a n + l 6
l
-i
+ a n+p O n+p R n+p
----
It follows that the absolute value of the
<
sum
n 6 +lt
<
e.
of the series in the
Finally, the first line of the right-hand which side of the equation (4) is a polynomial of degree n in vanishes when 0=1. Therefore another positive number rj may be third line cannot exceed
e.
found such that the absolute value of this polynomial is less than c and unity. Hence for all such values whenever lies between 1 r\
we
of
shall
have \S~f(x)\<3e.
Hence f(x) e is an arbitrarily preassigned positive number. approaches S as its limit as x approaches R. In a similar manner it may be shown that if the series (3) con But
=
R, the sum of the series for x verges for x the limit which /(x) approaches as x approaches R. greater than to the preceding.
An
application.
if
Indeed,
we
replace
S =UQ + S = v +
(6)
be two convergent (7)
u
o
+
R is equal to R through values
x, this
case reduces
This theorem enables us to complete the results of
UI vl
+ +
series, neither of (ot>i
4-
Uiu
)
U Z ----\
t>
2
u n H----
+
H-----h
v
,
+
which converges absolutely.
H----
168
Let
regarding the multiplication of series. (5)
x by
=
+
(
uo
H-----
1-
no) H----
The
series
SPECIAL SERIES
380 converge or diverge.
may the
sums
power
If
two given
of the
it
S
sum S
its
converges,
series, i.e.
=
<S<S
[IX,
.
is
179
equal to the product of
For, let us consider the three
series
f(x)
= =
+
v
tf>(x)
+
+
vix
v n x n H----
,
-\
-----h
Each of these series converges, by hypothesis, when z = 1. Hence each of them 1 and + 1. For any such converges absolutely for any value of x between value of x Cauchy s theorem regarding the multiplication of series applies and gives us the equation /(*)0(z)
(8)
=
*(z).
By Abel s theorem, as x approaches unity the three functions /(x), #(x), \ft(x) approach S, S and 2, respectively. Since the two sides of the equation (8) meanwhile remain equal, we shall have, in the limit, S = SS The theorem remains true for series whose terms are imaginary, and the proof ,
.
same
follows precisely the
lines.
If a
179. Successive derivatives of a power series.
=a +
f(x)
which converges
av x
+a
in the interval
2
x 2 -\-----h
*"
power
series
H----
R, -f R) be differentiated term
(
by term, the resulting power series
In order to prove this, it will be converges in the same interval. show that the series of absolute values of the terms of
sufficient to
the
new
series,
A where A
t
=
v
+ 2A
and
|a,.|
9
X=
X-\ \x\,
h
nA n Xn ~
converges for
l
-\
,
and diverges for
X
X>R.
For the
first
ber between
X
R, and let R be a part let us suppose that X X the R R. Then and R, auxiliary series <
num
<
<
1
h
4-
(
Y 4-
4-
I}*
4
converges, for the ratio of any term to the preceding approaches X/R which is less than unity. Multiplying the successive terms ,
of this series, respectively,
by the factors
IX,
POWER
179]
each of which obtain a
new
is less
SERIES
A1
X
t
is
R
than a certain fixed number, since
we
series
which also evidently converges. The proof of the second part is similar
where
381
+ 2A X + 2
to the above.
+ nA n Xr +
l
If the series
>
greater than R, were convergent, the series
A X l
l
+
2A t X*
+ .-- + nA n X n
+
l
+
and consequently the series 2^ n would con is less than the each of its terms since corresponding term of verge, Then R would not be the upper limit of the the preceding series. values of X for which the series (1) converges.
would converge
also,
X"
The sum /j (a:) of the series (9) is therefore a continuous function x inside the same interval. Since this series con R -f- R ), where R R f1 (x) verges uniformly in any interval ( of the variable
<
,
)
174. the derivative of f(x) throughout such an interval, by Since R may be chosen as near R as we please, we may assert that
is
the function f(x) possesses a derivative for any value of x between R and -f R, and that that derivative is represented by the series obtained by differentiating the given series term by term * :
/(*)
(10)
=
!
+
+ na n x*- + ....
+
2a2 x
1
Repeating the above reasoning for the series (10), we see that f(x) has a second derivative,
= 2a + 6a 2
/"(a)
3
x
+
.
.
.
+ w (n - 1) a n x n ~ + 2
-,
and so forth. The function f(x) possesses an unlimited sequence of derivatives for any value of x inside the interval + R), and
(/,
these derivatives are represented by the series obtained by differen tiating the given series successively term by term :
(11) If
we
or, in
*
= =
f<*\x)
set
x
1.2---na n
+
2.3---n(n
in these formulae,
we
+ l~)a
n
---+ l x-i
.
find
general,
Although the corresponding theorem
proof follows
somewhat
different lines.
is
true for series of imaginary terms, the
See Vol.
II,
266.
TRANS.
SPECIAL SERIES
382
The development of /(#) thus obtained ment given by Maclaurin s formula
is
179
[IX,
identical with the develop
:
/(*)
=/(0)
+ f/
(<>)
+
+ iTf^/^O) +
+ f^/"(<>)
-.
an are equal, except for certain The coefficients a 0) a l9 numerical factors, to the values of the function f(x) and its succes sive derivatives for x = 0. It follows that no function can have two ,
distinct developments in if
Similarly,
,
series.
power
a power series be integrated term by term, a new obtained which has an arbitrary constant term and
power series is which converges
in the
same interval
as the given series, the given If we integrate again, first two terms are arbitrary and so
series being the derivative of the
we
obtain a third series whose
new
series.
;
forth.
Examples. 1) The geometrical progression X
1
X s -\-----
+ X*
whose ratio is x, converges + 1, and its sum is 1/(1 + x). and x, where x the limits of log (1
+
x*)
found in
49
2)
(l)
n
X n -\----
for every value of x
<
,
between
= 1,
for the series on the right con
1
=1- x +* - x + 2
^-^
4
6
and
+1
+
(-
...
we may x2
1 )"
"
write
+
Integrating this series term by term between the limits Ice
I
Since the
<
1,
new TT
and
= 1.
For any value of x between
where
1
Integrating it term by term between 1, we obtain again the development
:
This formula holds also for x verges when x
\-
we
find
series converges for
111
x
= 1,
it
follows that 1
and
x,
row EH
17 ]
IX,
.
3) Let F(x) be the
m where
m
is
sum
m(m
1)
[m
1
+
(m
p +1) 1.2.--OJ-1)
which
is
a:
<
(in,
Then we
1.
(m
(m -1).
a;)
have
p +1) J
-1
1.2..
(1+
shall
|
and then
terms
collect the
in
Using the identity
powers of x. 1)
!)
<
Let us multiply each side by
(m
i(m
+
+
*
383
of the convergent series
any number whatever and !
like
SERIES
(m
-(m
!)
_
/>)
m(?rc
we
easily verified,
find the
p -f-1)
(m
1)
1.2--.J9
1.2-.-.P
formula
P
,
,
or
From
this result
we
find, successively,
F (x) _ m ~F(x)~l+x log [/
()]
= m log (1 +
a-)
+
log
or
To determine the constant C we need merely notice that F(0) = l. Hence C = 1. This gives the development of (1 + a*) found in 50 "
:
Y_1. 4) Replacing we find
m(m-l)...(m-.p+l)
.
a;
by
x 2 and
/,
1.32.4
H-----
=1+,1
-
-
by
1/2
, 1
in the last
,
formula above,
1.3. 5- ..(27tv ^ 2.4.6 2n
1 and +1. This formula holds for any value of x between x where and both sides between the limits x, grating
<
|
obtain the following development for the arcsine arc
x sm x = -
3
1
a-
7:
-^
,
-f
1.3a; 5
+^
-=-
1
H----,
T
.
3 5 .
,
-
-
(2n
:
z -1) x
g 2.4.6---2n .
"
+l
Inte 1,
we
SPECIAL SERIES
384
[IX,
180
180. Extension of Taylor s series.
Let/(x) be the sum of a power which converges in the interval ( R, + R~), a- a point inside that interval, and x + h another point of the same interval such that |a; + h\ R. The series whose sum is f(x + A), series
<
|
|
o
may be
+
+
ai(
A)
+a
2
(x
+
2
A)
H
\-
+
an (x
A)"
-]
,
replaced by the double series obtained by developing each and writing the terms in the same power
+
of the powers of (x A) of h upon the same line :
-f-
2a 2 x h
n
(12)
n(n
1)
1.2
This double series converges absolutely. For if each of its terms be replaced by its absolute value, a new double series of positive terms is obtained :
(13)
If
we add
we
the elements in any one column,
obtain a series
+
which converges, since we have supposed that x h R. Hence the array (12) may be summed by rows or by columns. Taking the sums of the columns, we obtain f(x Taking the sums A). |
\
<
1
\
+
of the rows, the resulting series 2 A, and the coefficients of A, h
is
are
,
tively.
(14)
arranged according to powers of
f (x
),f"(x
)/2l,
,
respec
Hence we may write A) =/(.r
)
+ .L
.
71
we assume that A R x This formula surely holds inside the interval from x R +\x to x + R but it may happen that the series on the right \x converges in a larger interval. As an example consider the function if
.
<
|
[
|
\
\
,
IX,
(1
+
POWER
180]
m
m
SERIES
386
not a positive integer. The development x for all x of holds values of 1 and between powers x which lies in of that be a value interval. Then x we Let
x~)
where
,
is
according to
+ 1. may
write (i
+ x} m =
(i
+ x + x - XY = (i +
(i
+
m *)
,
where z
We may
now develop
new development
(1
=X X l+X
+ z) m
will hold
according to powers of
whenever
If x 1 and 1 + 2x between former interval than the larger
\z\
<
is positive,
.
1, i.e.
the
and
2,
this
for all values of
new
x
interval will be
Hence the new formula
(1, +!)
enables us to calculate the values of the function for values of the variable
which lie outside the original interval. Further investiga remark leads to an extremely important notion, that
tion of this
of analytic extension.
We
shall consider this subject in the second
volume. Note.
It is evident that the theorems
proved for series arranged may be extended immedi
according to positive powers of a variable x
a, or, ately to series arranged according to positive powers of x more generally still, to series arranged according to positive powers whatever. We need only consider of any continuous function them as composite functions, being the auxiliary function. Thus a series arranged according to positive powers of I/a; con verges for all values of x which exceed a certain positive constant in absolute value, and it represents a continuous function of x for all such values of the variable. The function Va/2 a, for example, may 2 be written in the form The expression (1 a/ar )* x(\ a/a: )*. 2 may be developed according to powers of I/a; for all values of x a in which exceed absolute value. This gives the formula <(x)
(x)
2
V
1
.
2
.
3
-
(2P
-
2.4.6---2p
3)
a"
a-
2
-
1
a whenever x which constitutes a valid development of Va; 2 Va. When x Va, the same series converges and represents the func >
<
tion
Va: 2
a.
This formula
may
be used advantageously to obtain
a development for the square root of an integer whenever the first perfect square which exceeds that integer is known.
SPECIAL SERIES
386
181
[IX,
Dominant functions. The theorems proved above establish a close analogy between polynomials and power series. Let ( r, + r) be the least of the intervals of convergence of several given power When |cc|
by the ordinary rules for polynomials. In general, any integral poly nomial in /1 (x),/2 (a;), ,/() may be developed in a convergent
power series in the same interval. For purposes of generalization we shall now define certain expres Let f(x~) be a power sions which will be useful in what follows. series
=
f(x)
and
+
a
aj_x
+a
2
x2
h an
-)
H
x"
,
be another power series with positive coefficients
let (x)
=
+
a
a^x
+
a 2 x2 H
f-
an x n
-\
which converges in a suitable interval. Then the function said to dominate * the function f(x) if each of the coefficients a n
is
<(a;)
is
greater than the absolute value of the corresponding coefficient of
/(*): 1 \<
0\
Poincare has proposed the notation f(x) to express the relation
which
<
<(*)
between the two functions f(x)
exists
and (x~).
The
lowing fact, Let P(a a\,
which
,
of f(x) ties
aQ
,
it <(#),
dominant functions is based upon the fol an immediate consequence of the definition. be a polynomial in the first n -f- 1 coefficients
utility of these
)
,
whose
is
coefficients are all real
and
positive.
!, -, a n be replaced by the corresponding is clear that we shall have
|P(Oo, a 1}
-,
a n ~)\
For instance, if the function the series which represents
(x)
[<(:r)]
In general,
2
,
a 1}
,
If the quanti coefficients of
an ).
dominates the function /(#), dominate [/(ic)] 2 and so
will
,
w iU
dominate [/()]". Similarly, if and are dominant functions for / and /t respectively, the product will dominate the product jff\ and so forth. on.
[<(#)]"
,
^>!
<<j
;
*This expression pour
la fonction
majorantes."
will be used as a translation of the phrase est Likewise, "dominant functions will be used for "
"
/(a;)."
TRANS.
majorante "
fonctions
IX,
POWER SERIES
181]
387
+
Given a power series/(x) which converges in an interval ( R, R), the problem of determining a dominant function is of course indeter But it is convenient in what follows to make the domi minate. nant function as simple as possible. Let r be any number less than
R and arbitrarily near R. the absolute value of shall call
its
Since the given series converges for x
terms will have an upper
Then we may
M.
write, for
any value of
limit,
= r,
which we
n,
or
Hence the
series
M
Mx n
M + M-xr
x
r"
r n
n
whose general term is M(x /r ), dominates the given function /(x). This is the dominant function most frequently used. If the series /(x) contains no constant term, the function
M may
be taken as a dominant function.
It is evident that r
that
may
be assigned any value less than R, and with r. But can never be less than
M
M decreases, in general,
not zero, a number p less than R can always be found such that the function -4 /(1 x/p) dominates the function /(x). For, let the series
A
.
If
A
is
M + M -r + M ^r +
+ M^ +
2
M
where than rA
/M
whence
|a n p
>
A
be a and n ,
first
>
n <
A
.
1,
dominant function.
we
shall
On the
|
dominates the function f(x). ently.
than or
r"
If p be a
number
less
have
other hand, |ac
We
shall
=A
.
Hence the
series
|
make
use of this fact pres
More generally still, any number whatever which equal to A may be used in place of M.
is
greater
SPECIAL SERIES
388
may be shown
It
is
in a similar
a dominant function, where The knowledge
Note.
/t
[IX,
manner that
is
if
a
=
0,
182
the function
any positive number whatever.
of a geometrical progression
which dominates the func
tion f(x) also enables us to estimate the error made in replacing the function f(x) by the sum of the first n + 1 terms of the series. If the series M/(l x/r)
dominates /(z),
of the given series
of the
dominant
evident that the remainder
it is
is
less in absolute value
than the corresponding remainder
It follows that the error in question will
series.
be less than
1
(T jfW 182. Substitution of one series in another.
*
(15)
=
=/(>/)
o
+
iy 4-
+
Let
y+
an
be a series arranged according to powers of a variable y which con R. Again let \y\<.
verges whenever /1
A\
-JL
be another 2
series,
^
TitA
\
I
which converges
t
A
w.
i
in the interval
(
r,
+
r).
If
8
in the series (15) be replaced by their y ) y ) developments in series arranged according to powers of x from (16), a double series y>
+ I
,.
7,
.,
(17)
is
obtained.
l
O
7,
7,
~.
a 2 (b\
We
shall
now
investigate the conditions under which
this double series converges absolutely. In the first place, it is necessary that the series written in the first row,
IX,
POWER
182]
should converge absolutely, is also sufficient.
condition
i.e.
SERIES
that
For
\b
389
should be less than R.*
This
\
if it is satisfied,
the function <(>(x)
dominated by an expression of the form m/(l x/p), where number than and where r. We greater any positive p \b Let R be another therefore suppose that m is less than R.
will be
m
is
<
\
may
number which lies between m and R. Then the function dominated by an expression of the form
positive
f(y)
is
R
R
y
2
R If y be replaced by wi/(l x/p) in this last series, and the powers of y be developed according to increasing powers of x by the binomial
theorem, a
new double
series
M ^ mx M j^ +
(18)
-
+ nMt-
is obtained, each of whose coefficients is positive and greater than the absolute value of the corresponding coefficients in the array (17), since each of the coefficients in (17) is formed from the coefficients
a
,
a*!,
a2
,
,
b 0) bi, b 2
,
-by means of additions and multiplications
The double
series (17) therefore converges absolutely pro only. vided the double series (18) converges absolutely. If x be replaced its absolute value in the series (18), a necessary condition for abso lute convergence is that each of the series formed of the terms in any
by
If this condition be one column should converge, i.e. that \x\ p. column of in the is equal to the sum the terms satisfied, (n + l)th <
Then a further necessary condition
is
that
we should have
or
(19)
in
* The case in which the TRANS. what follows.
series (15) converges for
y
= R (see
177) will
be neglected
SPECIAL SERIES
390
[IX,
182
Since this latter condition includes the former, \x\ p, it follows it is a necessary and sufficient condition for the absolute con <
that
vergence of the double series (18). The double series (17) will therefore converge absolutely for values of x which satisfy the It is to be noticed that the series inequality (19). converges <j>(x)
and that the corresponding value of y For the inequalities absolute value.
for all these values of x, less
than
R
in
i. t
m.
.
x,
(
1
is,
On
f[$(x)~\-
m
\x\
!
necessitate the inequality (17) by columns, we find
that
is
$(x)\<.R
.
Taking the sum of the
series
the other hand, adding by rows, we obtain a powers of x. Hence we may write
series arranged according to
where the
coefficients c
CQ
=
,
c i} c 2
-f- fti
w
are given by the formulae
,
+
ft 2
t>o
~r
~r
<^
n
"o
i >
(21)
which are easily
verified.
The formula
(20) has been established only for values of x which the inequality (19), but the latter merely gives an under satisfy limit of the size of the interval in which the formula holds. It may
be valid in a
much
solution requires a shall return to
We
it later.
1) Since the number R which occurs in (19) may as we please, the formula (20) holds whenever x
Special cases.
be taken as near
This raises a question whose larger interval. of of a complex variable. functions knowledge
R
the inequality \x p(l m/R). Hence, if the series (15) for value of converges y whatever, R may be thought of as infinite, any be r taken as near as we please, and the formula (20) applies p may satisfies
whenever
<
|a;|
<
r,
that
is,
in the
same interval
in
which the
series
In particular, if the series (16) converges for all (16) converges. values of x, and (15) converges for all values of y, the formula (20) is
valid for all values of
x.
IX,
POWER
182]
2) tion
When
391
SERIES
the constant term b of the series (16) is zero, the func dominated by an expression of the form
is <(z)
x
l- p
An r and where m is any positive number whatever. where p case shows that the in the that used to similar general argument formula (20) holds in this case whenever x satisfies the inequality <
(22) v
z|<
R
where
This \b
<
is
is
\
evidently
Examples.
1)
arises
satisfied,
+ x).
in
and the
Cauchy gave a method
y
,
The inequality depend upon
practice.
coefficients c n
for obtaining the binomial
theorem from
Setting
=
write
whence, substituting the
If the
often
case
the development of log(l
we may
m
R as we please. The corresponding interval than that given by the inequality (19). larger
special
R
R +
as near to
is
of validity
P r
first
expansion in the second,
right-hand side be arranged according to powers of x, it is evident that will be a polynomial of degree n in ju, which we shall call
the coefficient of
x"
n 1, and must This polynomial must vanish when ^ = 0, 1, 2, reduce to unity when ^ = n. These facts completely determine Pn in the form
Pn (n)-
,
.M
~""" p _M(M-l)---(M-n+l) P "
2) Setting z
where
=
(1
+
1
z) /*,
where x
lies
between
-
1
and
+
1,
we may
write
SPECIAL SERIES
392 The
first
\x\<
the this
1.
first
183
is valid for all values of y, and the second is valid whenever Hence the formula obtained by substituting the second expansion in holds for any value of x between 1 and + 1. The first two terms of
expansion
formula are
T It
[IX,
follows that
(1
+
1
x)
/*
e
approaches
zero through positive values.
f(
of a
series
power
the interval
(
r,
less
than
e as
z approaches
__ Let us
}
consider the reciprocal
first
_ b 2 x*
\
l+b lX +
+
...
which begins with unity and which converges -f-
in
Setting
?).
y
we may
.
through values
183. Division of power series.
Ax
__
=
bv
x
+
b 2 x 2 H----
,
write /(*)
=1 ~
= r+~y
y
+
y2
-
y
*
+
whence, substituting the first development in the second, we obtain an expansion for f(x) in power series, (25)
f(x)
=1-
+ (b\ - 6.) * + 2
b lX
which holds inside a certain
-,
In a similar manner a devel
interval.
opment may be obtained for the reciprocal whose constant term is different from zero.
of
any power
series
Let us now try to develop the quotient of two convergent power series
$(x) If b
is
b
ti
+
not zero, this quotient
= ( a +a
x
+a
2
biX
may
x*
+
+
b 3 x*
----
-\
be written in the form
--
)
X
Then by the case just treated the left-hand side of this equation is the product of two convergent power series. Hence it may be written in the
form of a power
series
which converges near the origin
:
-f
Clearing of fractions and equating the coefficients of like powers we find the formulae
of x,
IX,
POWER
184]
=
an
(27)
l cH
from which the sively.
+ biC
+
+---
H_l
coefficients c
SERIES b H c9
c lt
,
393 0, 1, 2,
may be
cn
,
=
n
,
-,
calculated succes
same as
It will be noticed that these coefficients are the
we should
obtain by performing the division indicated by the ordinary rule for the division of polynomials arranged according to increasing powers of x. .
those
If b
that
= 0, the result is different. = x fa (x), where is a
Let us suppose for generality and \l/\(x) is a
k
A;
ij/(x~)
power
series
whose constant term
positive integer
Then we may write
not zero.
is
and by the above we shall have also
It follows that
\
the given quotient
=
/
"77
*tI
*
\
ck
,~
*
<
expressible in the form
is
-j- Cj.
+ j a; -p
,
where the right-hand side is the sum of a rational fraction which becomes infinite for x = and a power series which converges near the origin. Note.
In order to calculate the successive powers of a power
venient to proceed as follows. (a
+
a\x
+
+
Assuming the
a n xn
+
"
)
=
c
it is
series,
con
identity
+
ct x
+
+
c n xn
+
,
us take the logarithmic derivative of each side and then clear of fractions. This leads to the new identity
let
+
na n xn - 1
(29) a,(
The
coefficients
+
-)(c
+
Cix
n 4- c n x
+
+
)
x"
the various powers of x are easily calculated. Equat we find a sequence of formulae from which may be found successively if c be known. It is evident that
of
ing coefficients of like powers, CQ, Ci,
,
cn
,
184. Development of according to powers of
2xz
1/Vl z.
-|-
Setting y
z2
=
.
Let z 2,
2xz
us develop
we
or 1
(30)
Vl^2xzTP
=
1
+
2xz
~ 2
28
+
- (2xz 8
1/Vl - 2xz when
shall have,
-
z 2) 2
+
+
\y\<
z2 1,
SPECIAL SERIES
394
[ix,i86
Collecting the terms which are divisible by the same power of expansion of the form
-- = p
l
(31)
VI -
+
?xz
+ p lZ + pzZ 2 +
...
z2
+ pnZ n +
z,
we
obtain an
...,
where 2
*
Pn is a polynomial of the nth degree in x. These poly be determined successively by means of a recurrent formula. Dif ferentiating the equation (31) with respect to z, we find and where, nomials
in general,
may
(1
-
2xz
by the equation
or,
(x
-
Equating the
(
is
Vl n is
z",
(1
-
+
2xz
z 2 )(Pi
+ 2P 2 z
H
)
.
obtain the desired recurrent formula
and moreover P and the formula
=X
between three consecutive Legendre = X\ P2 = 2 Hence Pn = n
(31)
P!
,
X
,
X
.
be written
may
~ 2xz
+
z2
the Legendre polynomial of the nth order -V -&-n
id
= 2
We
we
=
)
identical with the relation
88),
for all values of n,
X
Pn * n +
h
coefficients of
This equation
where
(31),
+ PIZ +
z)(P
polynomials
+ z 2 )*
.
4
.
shall find later the interval in
II.
POWER
6
2n dxn
which
this
i\_n
\\&*
1)1
formula holds.
SERIES IN SEVERAL VARIABLES
185. General principles.
The
properties of power series of a single
may be extended easily to power series in several independ ent variables. Let us first consider a double series 2 a mn x m y n where variable
t
m
,
and n vary from zero to + oo and where the coeffi cients a mn may have either sign. If no element of this series exceeds a certain positive constant in absolute value for a set of values x x y = 2/0) the series converges absolutely for all values of x and which y satisfy the inequalities \x\ \x \, \y\ \y \. the integers
o>
<
<
For, suppose that the inequality I
amn x Vo \<M
or |
a mu
M <
,
I
I
xo m \yo\ ,
\
.
,
IX,
DOUBLE POWER SERIES
185]
395
m
is satisfied for all sets of values of and n. Then the absolute value of the general element of the double series 2a mn .r m is less than the m n But corresponding element of the double series 2M\x/x \y/y
y
.
\
the latter series converges whenever
sum
is
|#|<|a:
|,
\
|y|<|y
an d
its
|>
M 2/o
we
by taking the sums of the elements by columns and then adding these sums. Let r and p be two positive numbers for which the double series 2 a mn \rm pn converges, and let R denote the rectangle formed by the four straight lines x = r, x = = p, y For every point r, y p. inside this rectangle or upon one of its sides no element of the double series as
see
m n exceeds the corresponding element of the series ^\amn \r p in abso lute value. Hence the series (33) converges absolutely and uni
formly inside of R, and it therefore defines a continuous function of the two variables x and y inside that region. It may be shown, as for series in a single variable, that the double series obtained by any number of term-by-term differen tiations converges absolutely and uniformly inside the rectangle
bounded by the lines x = r = r + c, y = p e y = p + e c, x where c and e are any positive numbers less than r and p, respec These series represent the various partial derivatives of tively. m - n is F(x,y). For example, the sum of the series 2 ma mn x y equal to cF/dx. For if the elements of the two series be arranged accord ,
,
l
t
ing to increasing powers of x, each element of the second series is equal to the derivative of the corresponding element of the first. m n Likewise, the partial derivative d m+n F/dx dy is equal to the sum of a double series whose constant factor is a l 2 1 2 n. ran .
Hence the
m
.
.
amn are equal to the values of the correspond ing derivatives of the function F(x, y) at the point x = y 0, except for certain numerical factors, and the formula (33) may be written in the form coefficients
/8 m + "F\
F(x, \
y)=
>yj
SPECIAL SERIES
396
186
[IX,
It follows, incidentally, that no function of two variables can have two distinct developments in power series. If the elements of the double series be collected according to their degrees in x and y, a simple series is obtained :
=
F(x, y)
(35)
<
+
fa
+
2
+
+
+
<
,
where is a homogeneous polynomial of the nth degree y which may be written, symbolically,
in
x and
The preceding development therefore s series (
Taylor Let (a-
coincides with that given by
51).
be a point inside the rectangle R, and (x + be a neighboring point such that x + h + \k\ r, \y for any point inside the rectangle formed by the lines ,
y
~)
<
\
|
*
=
ar
[r-|a:
y
|],
=
\
\
1
y
h,
<
[p
- |y
p.
y
+
k)
Then
|],
be developed in a power series arranged xv and y y according to positive powers of x the function F(x, y)
may
:
F\
ZdT 1.2
For
if
m f)i/n / x=x ,!
1. ;*..*"*
each element of the double series
be replaced by its development in powers of h and k, the new multi Arrang ple series will converge absolutely under the hypotheses. ing the elements of this new series according to powers of h and k,
we obtain the formula The reader will be
(36).
able to
show without
difficulty that all the
preceding arguments and theorems hold without essential altera tion for power series in any number of variables whatever. in n functions. Given a power series f(x, y, z, ) shall say that another series in n variables y, z, ) is dominates the first series if each coefficient of positive y,z, ) and greater than the absolute value of the corresponding coefficient 186.
Dominant
variables,
we
(,
<j>(x,
of f(x, y,
z,
).
The argument
in
185 depends essentially upon
IX,
DOUBLE POWER SERIES
186]
the use of a dominant function. verges for x
=
y
r,
d>(x,
= p,
if
the series 2,\a mn x
con "y"\
the function
M
=
y}
For
397
x\
/
/x\ 3/2 -
=
m /vY* -
(
) \ r/
y\
I
M
is greater than any coefficient in the where The function dominates the series 2,a mn x m yn
>
(
\P/
series 2,\amn r
m p"\,
.
is
another dominant function.
For the
coefficient of
x m yn in
1/^(3*,
y)
equal to the coefficient of the corresponding term in the expan sion of M(x/r -+- y/p} m+n and therefore it is at least equal to the m n in coefficient of x is
,
y
(x,
y).
Similarly, a triple series
where r, r which converges absolutely for x = r, y = r z = is dominated an are three positive numbers, by expression of the form ,
,
r",
r"
M
y>
l_Ul_\/l_-
and also by any one of the expressions
M
M x\[~. -
/?/
!-(-, r/\_ )
If f(x, y, z) contains no constant term,
sions diminished
by
may
182)
(
\r
z\
~1
)
J
r"/
any one of the preceding expres
M may be selected as a dominant function.
The theorem regarding the another
+
substitution of one
power
series in
be extended to power series in several variables.
If each of the variables in a convergent power series in p variables yp b & replaced by a convergent power series in q variables y\i y-ii x \j x ii xq which has no constant term, the result of the substitu tion may be written in the form of a power series arranged according )
*>
to
powers of x
l
,
x2
,
,
xq provided that the absolute value of each ,
of these variables is less than a certain constant.
SPECIAL SERIES
398
[IX,
186
Since the proof of the theorem
number
of variables,
is essentially the same for any shall restrict ourselves for definiteness to
we
Let
the following particular case. (37)
be a power series which converges whenever y -bm
(y=-t>i*--b 2
(38) f
Cj
00
^2
~~r~
x^ **^
"
*
*
~T~
j*
t
<
r
*
and
z
<
r
,
and
let
.., ~J~
be two series without constant terms both of which converge if the absolute value of x does not exceed p. If y and z in the series (37) be replaced by their developments from (38), the term in y m becomes a new power series in x, and the double series (37) becomes a triple z"
series,
each of whose coefficients
may be calculated from the coeffi means of additions and multiplications by It remains to be shown that this triple series converges abso when the absolute value of x does not exceed a certain con from which it would then follow that the series could be
cients a mn
only.
lutely stant,
,
bn
,
and
cn
arranged according to increasing powers of the function f(y, z)
In the
x.
first place,
dominated by the function
is
(39)
and both of the
series (38) are
dominated by an expression of the form
N (40) ,
(x\ N= ^T^ .V (-) \P/ L*i
x
n ,
>
1
71=
1
P
where
M and N are two positive numbers.
If
y and z
in the double
series (39) be replaced by the function (40) and each of the products m n y z be developed in powers of x, each of the coefficients of the result
ing triple series will be positive and greater than the absolute value of the corresponding coefficient in the triple series found above. It will therefore be sufficient to
show that
this
new
verges for sufficiently small positive values of x. the terms which arise from the expansion of any series (39) is
MAT
-"
Vp/
triple series con
Now
the
sum
of
term ym z* of the
IX,
REAL ANALYTIC FUNCTIONS
187]
which is the general term of the two series x \m
series obtained
i
y
Mry _p_ L*\rl 1 _ x
399
by multiplying the
x P
_
term by term, except for the constant factor M. Both of the latter series converge if x satisfies both of the inequalities
-
x
x
<
<
It follows that all the series considered will converge absolutely, and therefore that the original triple series may be arranged accord ing to positive powers of x, whenever the absolute value of x is less than the smaller of the two numbers pr/(r + JV) and pr /(r + N).
Note.
The theorem remains valid when the
series (38) contain
r and c r For provided that \b the expansion (37) may be replaced by a series arranged according to powers of y b and z c by 185, which reduces the discus
constant terms b
and
c
<
,
.
\<
,
sion to the case just treated.
III.
IMPLICIT FUNCTIONS
ANALYTIC CURVES AND SURFACES 187. Implicit functions of a single variable. The existence of implicit functions has already been established (Chapter II, 20 et ff.) under certain conditions regarding continuity. When the left-hand sides
of the given equations are power series, more thorough investigation is possible, as we shall proceed to show.
=
an equation whose left-hand side can be developed power series arranged according to increasing powers x x and of y y Q where the constant term is zero and the coeffi is cient of y Then the equation has one and y different from zero. root which as one x approaches XQ and that root only approaches n can be developed in a power series arranged according to powers of
Let F(x, y~) in a convergent
be
,
,
>/
x
-
x
.
For simplicity let us suppose that x = y = 0, which amounts to moving the origin of coordinates. Transposing the term of the first degree in y, we may write the given equation in the form (41)
y
= f(x, y} = a^x +
a 20 a; 2
+
a n xy
+ a^y* H
,
SPECIAL SERIES
400
[IX,
187
where the terms not written down are of degrees greater than the second. We shall first show that this equation can be formally sat isfied by replacing y by a series of the form
if
= Cl x +
y
(42)
c2
+
x*
+
-.-
cn x
n
+
---
the rules for operation on convergent series be applied to the series For, making the substitution and comparing the coeffi right.
on the
we
cients of x,
ci
find the equations
=
a
c2
io>
=
a so
-f
anCj
+
a 02 cf,
;
and, in general, cn can be expressed in terms of the preceding c s k and the coefficients a ik where i n, by means of additions and
+
,
Thus we may write
multiplications only. cn
(43)
where Pn integer.
is
<
= Pn (a
10
,
a
an
w>
,
,
a0n ),
a polynomial each of whose coefficients is a positive of the operations performed will be estab
The validity we can show
that the series (42) determined in this way shall do this by all for sufficiently small values of x. converges means of a device which is frequently used. Its conception is due lished
if
We
to
Cauchy, and Let
it
is
based essentially upon the idea of dominant
functions.
be a function which dominates the f unction /(x, y), where # 00 = bol = Let us then and where b mn is positive and at least equal to a mn .
\
consider the auxiliary equation (41
Y=
)
Y=C,x+
)
The values
C
=
2b mn x m Y n
=b
l
lo
C2
,
+
C 15 C 2
=
b 20
+
...
+c
n
z"
form
+ --..
can be determined as above,
,
b n C l -{-b 02 Cl,
-,
in general
(43
=P
C.
)
\c n
< \
positive
Cn
and
,
n
(b w ,b 20
,
,*)
from a comparison of the formulae (43) and (43 ) since each of the coefficients of the polynomial Pn is Hence the series (42) surely converges \a mn \^b mn
It is evident
that
C 2 x*
of the coefficients
and are and
Y)
to find a solution of this equation of the
and try (42
4>(x,
.
EX,
REAL ANALYTIC FUNCTIONS
187]
401
whenever the series (42 ) converges. Now we may dominant function Y) the function
select for the
<j>(x,
"
where M,
r,
equation (41
M ~ M 7Y Then the
and p are three positive numbers. )
auxiliary
becomes, after clearing of fractions,
p
+
M
p
+
M
-L
_
* r
This equation has a root which vanishes for x
=
0,
namely
:
F= The quantity under the
radical
may
be written in the form
where P
Hence the
root
Y may
be written
] It follows that this root
verges in the interval cide with that which
(
Y may
be developed in a series which con and this development must coin
a, -f a),
we should obtain by direct substitution, that the series (42) converges, a fortiori, in Accordingly ). the interval ( This is, however, merely a lower limit of a, + a). the true interval of convergence of the series (42), which may be is,
with (42
very
much
larger.
It is evident
from the manner
in
which the
coefficients c n
were
determined that the sum of the series (42) satisfies the equation (41). = 0, and Let us write the equation F(x, y) in the form y f(x, y) = be substi let y P(x) be the root just found. Then if P(x) + tuted for y in F(x, y), and the result be arranged according to powers of x and z, each term must be divisible by z, since the whole expression vanishes then F[x, P(x) -f *]
when
z
= sQ(#,
= )
,
We shall have for any value of x. where Q(x, z) is a power series in x
402 and
SPECIAL SERIES z.
z be replaced by
if
Finally,
[IX,
P(x} in Q(x,
y
we
z),
188
obtain
the identity
where the constant term of Q t must be unity, since the of y on the left-hand side
(44)
is
unity.
y) = [y
F(x,
coefficient
Hence we may write
P(x)] (1
+
ax
+
fa
+
...).
This decomposition of F(x, y) into a product of two factors is due to Weierstrass. It exhibits the root y = P(x), and also shows that there is no other root of the equation F(x, = which vanishes y) with x, since the second factor does not approach zero with x and y.
The preceding method
Note.
for determining the coefficients cn is
46. But it is now evident essentially the same as that given in that the series obtained by carrying on the process indefinitely is
convergent. 188.
The general theorem. Let us now consider a system of p equa
tions in
p
-f q variables.
=
0,
,
-,y,)=
...........
0,
(*u **
yi>y 2 ,
%)==
o,
i(i,*a x2
(x 1}
2
-
,
* ,x9
yi,y a yi,y 2
;
i
>*;
F1} F
j
J
yP )
series near that point.
suppose that the Jacobian
D(Fl} F2
2
,
,
,
yk = 0, We shall further
Fp vanishes when x
where each of the functions and is developable in power
Fp )/D(y
-,
not vanish for the set of values considered.
1
,
Under
y2
,
i
,
yp ) does
these conditions
and only one system of solutions of the equations
there exists one
(45)
of the form
where fa, fa, when x l = a; 2 =
<
,
y,
are
=
xq
power
series in
x l} x 2
,
,
= 0.
a*,
tvhich vanish
In order to simplify the notation, we shall restrict ourselves to the case of two equations between two dependent variables u and v and three independent variables x, y, and z :
/^CN (4o)
(F = au l
<
I
F = 2
a u
-f bv
+
Since the determinant ab
equations (46)
may
Vv
+ ex +dy + ez +cx+dy+ez
----
-\
^----
= 0, = 0.
ba is not zero, by hypothesis, the two be replaced by two equations of the form
IX,
REAL ANALYTIC FUNCTIONS
188]
403
mnpqr
(v
=
^b mnpqr
where the left-hand sides contain no constant terms and no terms It is easy to show, as above, that of the first degree in u and v. these equations
be satisfied formally by replacing u and v by
may
power
series in x,
(48)
u
and z
//,
=
2c,. t
:
v
,a;V*S
= 2c
x yk z {
ikl
1 ,
where the coefficients c ikl and c\ kl may be calculated from a mnpqr and In order to b mn r by means of additions and multiplications only. show that these series converge, we need merely compare them with the analogous expansions obtained by solving the two auxiliary (
equations
and p are positive numbers whose meaning has been to a single explained above. These two auxiliary equations reduce
where M,
r,
equation of the second degree
x
+
+ 4M _
x
if
+z
,r.
2P
+
|
4Af
2p
+
1
2
p
~
4(p
___ ~
+ 2M)
+z
_
r
which has a single root which vanishes for x
__
y
= y = z = 0,
namely
:
2
p
4( P
-f
2M) r
where a
= r [p/(p + 4M)]
This root
may
2 .
be developed in a convergent power series when to x, y, and z are all less than or equal
ever the absolute values of
Hence the series (48) converges under the same conditions. v t be the solutions of (47) which are developable in x and If we set u = ?/ t + u v = v^ -f v in (47) and arrange the series. result according to powers of x, y, z, u v each of the terms must be divisible by u or by v Hence, returning to the original varia bles x, y, z, u, v, the given equations may be written in the form
a/3.
Let
,
,
,
.
M
-/ +(-iH =0,
SPECIAL SERIES
404
[IX,
189
^
are power series in x, y, z, u, and v. where /, In this ft form the solutions u = u l} v = v-i are exhibited. It is evident also that no other solutions of (47 ) exist which vanish for x = y = z = 0. For any other set of solutions must cause ffa to vanish, and a comparison of (47) with (47 ) shows that the constant term is unity in both / and l} whereas the constant term is zero in = cannot be met by and hence the condition both/! ffa u v which and functions vanish x when replacing by y = z = 0. ,
,
<&/\
<j>
<;
Let us consider the equation
189. Lagrange s formula.
=
y
(49)
where
is
a function which
= (a)
+
-
(y
x$(y)
,
developable in a power series in y
is
<j>(y)
4>(y)
+
a
a)
+
(a)
(y
~
a)2 0"(a)
1
+
a,
,
l
.
which converges whenever y a does not exceed a certain number. By the 187, this equation has one and only one root which general theorem of approaches a as x approaches zero, and this root is represented for sufficiently small values of x by a convergent power series y In general,
if
is
f(y)
=
a
+
aix
a2 x2
-f
a function which
+
.
developable according to positive
is
a, an expansion of f(y) according to powers of x may be obtained powers of y by replacing y by the development just found,
f(y) =f(a)
(50)
and
this
+A
expansion holds for
The purpose
of
Lagrange
t
+ A 2 x* +
+
A,,x
+
.
.
,
all values of x between certain limits.
s
formula
AI, terms of
x
A2
to determine the coefficients
is
-,
,
An
,
be noticed that this problem does not differ essentially from the general problem. The coefficient A n is equal to the nth derivative of f(y) for y = 0, except for a constant factor n!, where y is defined by (49); and this derivative can be calculated by the usual rules. The calculation appears to be very complicated, but it may be substantially shortened by applying the fol lowing remarks of Laplace (cf. Ex. 8, Chapter II). The partial derivatives of in
a.
It will
the function y defined by (49), with respect to the variables x by the formulae
whence we
and
a, are
find immediately d
i*.i\
u
(51)
ox
where u =f(y).
On
the other hand,
da
da it is
! acJ
=
easy to show that the formula
W-
-!><*)
dx I
c>a
given
IX,
is
REAL ANALYTIC FUNCTIONS
189]
identically satisfied,
where F(y)
is
405
an arbitrary function of
y.
For either
side becomes
We
on performing the indicated differentiations.
for
shall
now
prove the formula
any value of n. It holds, by (51), for n = 1. In order to prove it in gen assume that it holds for a certain number n. Then we shall have
eral, let us
dx + l
But we
also have,
from
(51)
da n
and
(51
~l
),
du
l-
cxj ca L ca [*]-[>* L
caJ
whence the preceding formula reduces cn +
1
dx
to the
form
Bn
u
which shows that the formula in question holds for all values of n. Now if we set x = 0, y reduces to a, u to /(a), and the nth derivative of u with respect to x is given by the formula
da"-
Hence the development
of f(y)
/(2/)=/(a)
1
by Taylor
+ z0(a)/
(a)
s series
becomes
.
(52)
It gives an expression for the is the noted formula due to Lagrange. shall find later the root y which approaches zero as x approaches zero.
This
We
limits
between which
Note.
It follows
this
formula
is
applicable.
from the general theorem that the root
y,
considered as a
be represented as a double series arranged according This series can be obtained by replacing each of the to powers of x and a. Hence the series (52) may coefficients A n by its development in powers of a. function of x and a,
may
be differentiated term by term with respect to
Examples. (53)
1)
a.
The equation y
=
a
+ ?(y*~l)
SPECIAL SERIES
40G
has one root which is equal to a when x following development for that root
190
[IX,
=
0.
Lagrange
V
27
da
s
formula gives the
:
2
.
(54)
_J_M" d- (q2-l)n
,
1.2...nV27
On
the other hand, the equation (53)
=
y
-
may
-
do--
be solved directly, and
Vl
+
2ax
x2
its
roots are
.
The root which is equal to a when x = is that given by taking the sign ~~. Differentiating both sides of (54) with respect to a, we obtain a formula which differs from the formula (32) of 184 only in notation. 2)
Kepler
s
equation for the eccentric anomaly u,*
=
u
(55)
a
+
e sin
u
,
which occurs in Astronomy, has a root u which is equal to a f or e = 0. formula gives the development of this root near e = in the form (56)
=
u
e
a
1.2 da
Laplace was the
Lagrange
s
"
da-i
1.2..-n
show, by a profound process of reasoning, that this series converges whenever e is less than the limit 0.662743 first to
190. Inversion.
Let us consider a series of the form
y
(57)
where
=
aix
+
a z x2
+
+
a n xn
+
,
from zero and where the interval of convergence is( r, + r). If y be taken as the independent variable and x be thought of as a function of y, by the general theorem of 187 the equation (57) has one and only one root which approaches zero with y, and this root can be developed in a power series in y ai is different
:
x
(58)
=
The
coefficients bi, b 2
(57)
by
biy
-f
b2 y
2
+
b^y
3
+
+
bn
y"
+
.
63, may be determined successively by replacing x in expansion and then equating the coefficients of like powers of y. The values thus found are ,
this
bi
=
&2
,
=
The value Lagrange
s
of the coefficient bn of
formula.
1
-
3 ,
.
af
the general term
may
be obtained from
For, setting ^(X)
the equation (57)
=
bs
~
a?
<*!
may
=
tti
+
a2 X
+
+
OnX n
~l
+
,
be written in the form 1
*See p. 356.
p. 248,
TRANS.
Ex. 19; and ZIWET, Elements of T?teoretical Mechanics, 2d
ed.,
IX,
REAL ANALYTIC FUNCTIONS
l!ii]
and the development is given by Lagrange
of the root of this equation
formula
s 1
where the subscript
+
which approaches zero with y
form
in the
d-i
?/"
+
407
/
1
\" "
indicates that
we
=
are to set x
after performing the
indicated differentiations.
The problem 191.
tion of
just treated has
sometimes been called the reversion of series.
we
Analytic functions. In the future any number of variables x, y, z,
shall say that a func
is
analytic
developed, for values of the variables near the point x in a power series arranged according to increasing
x
~x
o>
V
2/o
j
z
~z
" o>
values of the differences x
if it
y
,
,
can be z
,
,
powers of
which converges for sufficiently small x The values which ar y z .
,
,
,
,
may may be restricted by certain conditions, but we shall not go into the matter further here. The developments of the pres ent chapter make clear that such functions are, so to speak, inter take on
Given one or more analytic functions, the operations of integration and differentiation, the algebraic operations of multipli cation, division, substitution, etc., lead to new analytic functions. related.
Likewise, the solution of equations whose left-hand member is ana Since the very simplest functions,
lytic leads to analytic functions.
such as polynomials, the exponential function, the trigonometric functions, etc., are analytic, it is easy to see why the first functions studied by mathematicians were analytic. These functions are still predominant in the theory of functions of a complex variable and in
the study of differential equations.
Nevertheless, despite the funda it must not be forgotten
mental importance of analytic functions,
that they actually constitute merely a very particular group the whole assemblage of continuous functions.* 192. Plane curves.
Let us consider an arc
AB
among
of a plane curve.
We
AB
shall say that the curve is analytic along the arc if the coordinates of any point which lies in the neighborhood of any fixed point of that arc can be developed in power series arranged
M
M
according to powers of a parameter (59)
T"
y=
t
-rw-^o + aiC
to)+*2(t
=y +b f(f)
f
which converge for * In the second
whose derivatives
l
(t-
)
+
b z (t
t
,
y + ... + a n (t - t y + + b n (t to
o
sufficiently small values of
t
t
volume an example of a non-analytic function exist throughout an interval (a, b).
.
will be given, all of
SPECIAL SERIES
408
192
[IX,
A point 3/o will be called an ordinary point if in the neighbor hood of that point one of the differences y x can be y x as a of the series in other. convergent power represented powers for can be in a series in If, example, y y developed power x0) x ,
y
(60)
y
(
=
x
^(x
>
)
+
ca
x
(x
2
)
H-----1- c n (x
z
n
H---- ,
)
h and -a- + h, the point (x y is an ordinary point. It is easy to replace the equation (60) by two equations of the form (59), for we need only set for all values of x between x n
,
~)
(61)
from zero, which is the case in general, the equa tion (60) may be solved for x x in a power series in y y which is valid whenever y is In this case each of y sufficiently small. the differences x x , y y can be represented as a convergent If c v is different
power
series in
zero, that
x
axis.
is
In that
oped in a
powers of the
other.
This ceases to be true
to say, if the tangent to the curve case, as
we
shall see presently,
x
series arranged according to fractional
It is evident also that at a point
is
if c t is
parallel to the
x
may
be devel
powers of y
where the tangent
is
y
.
parallel to
the y axis x x can be developed in power series in y yoy but cannot be x t/o y developed in power series in x If the coordinates (a-, ?/) of a point on the curve are given by the .
M
that point is an ordinary point if equations (59) near a point , at least one of the coefficients a l} bi is different from zero.* If a l is
not zero, for example, the
in
powers of x
of y
t-t
y
in
x
first
equation can be solved for
t
t
and the second equation becomes an expansion of x x when this solution is substituted for powers ,
.
The appearance
of a curve at an ordinary point is either the cus tomary appearance or else that of a point of inflection. Any point which is not an ordinary point is called a singular point. If all
the points of an arc of an analytic curve are ordinary points, the arc is said to be regular. * This condition is sufficient, but not necessary. However, the equations of any curve, near an ordinary point may always be written in such a way that 04 and b l do not both vanish, by a suitable choice of the parameter. For this is actually
M
accomplished in equations
(61).
,
See also second footnote,
p. 409.
TRANS.
IX,
KEAL ANALYTIC FUNCTIONS
193]
and
If each of the coefficients a is
different
the form (x
from zero, the xu )* = (t
hand member
is
first )
2
[
b v is zero,
409
but a 2 for example, ,
of equations (59) ----f a 8 (t ) -\
may
]*,
be written in
where the
developable according to powers of
t
t
t t is developable in powers of (x if t a^)*, and second equation of (59) be replaced by that development, a development for y z )* y in powers of (x
right-
Hence
.
in
t
we
the
obtain
:
In this case the point (z y is usually a cusp of the first kind.* The argument just given is general. If the development of x x in powers of t t begins with a term of degree n, y y can be developed according to powers of (x aThe appearance ). of a curve given by the equation (59) near a point (x y ) is of ,
~)
,
one of four types
a point with none of these peculiarities, a point of inflection, a cusp of the first kind, or a cusp of the second kind.* :
A
Skew curves. skew curve is said to be analytic along an arc the coordinates x, y, z of a variable point can be developed in power series arranged according to powers of a parameter t t 193.
M
AB if
- n ---+ y ) + t,} + -.., (* SS K9 +Cl (t-tj + -~+ Cn (t - ) n + f
x
J
(62)
=x + =y +
ai(t
-
b, (t
-----h a n *) H (t
tu
-i
)
,
n
b n (t
*
t
in the
M x
is
xoi y
so can be developed in powers of the third. z
y<
according to
M
-,
M
A
of the arc. neighborhood of any fixed point point said to be an ordinary point if two of the three differences
power
series arranged
It can be shown, as in the preceding paragraph, that the point will surely be an ordinary point if not all three of the coefficients
a l ,b l
,
cl
vanish.
point must
Hence the value
of the parameter
t
for a singular
satisfy the equations f
^=
dy_
dt
dt
Q U
dz_ dt
* For a cusp of the first kind the tangent lies between the two branches. For a cusp of the second kind both branches lie on the same side of the tangent. The point is an ordinary point, of course, if the coefficients of the fractional powers TRANS. happen to be all zeros. t These conditions are not sufficient to make the point 3f which corresponds to a value t of the parameter, a singular point when a point of the curve near Jf Such is corresponds to several values of t which approach t as approaches the case, for example, at the origin on the curve defined by the equations x = t 3 ,
M
M
M
.
,
SPECIAL SERIES
410 Let x
[IX,
M
z be the coordinates of a point equations are given in the form
whose
y
,
,
F(x,y,z)=0,
(63)
where the functions Fand
The point
M
F
t
F,(x,
194
on a skew curve T
0,
are power series in x
x
,
y
y
,
z
z
.
will surely be an ordinary point if not all three of
the functional determinants
D(F, F,}
(F,
D(x, y)
FQ
D(F,
D(y, z)
FQ
D(z, x)
vanish simultaneously at the point x = x y = ?/ z = z For if the determinant D(F, Fi)/D(x, y), for example, does not vanish at the equations (63) can be solved, by X Q and y 188, for x yQ ,
M
as
.
,
,
power
series in z
z
.
194. Surfaces. A surface S will be said to be analytic throughout a certain region if the coordinates x, y, z of any variable point can be expressed as double power series in terms of two variable t and u HO parameters t
M
x = y y = = U
ex J
(64)
a 10 (t b 10 (t c
f
-
io(t
*
+ ) + +
oi( M
)
b 01 (u
- OH -w)
c oi ( u
?
o)
,
-\
,
H
1
M
of that region, where any 1 and the three series converge for sufficiently small values of t will u u of the surface be said to be an point ordinary in the neighborhood of
.
point
if
fixed point
M
A
one of the three differences x
x
,
y
y
D(x, y}
D(z, x)^ D(t, u)
D(t, u)
can be
Every point
M
D(y, g)
z
z
,
expressed as a power series in terms of the other two. for which not all three of the determinants
D(t, u)
vanish simultaneously is surely an ordinary point. If, for exam ple, the first of these determinants does not vanish, the last two of the equations (64) can be solved for t equation becomes an expansion of x
t
and u
u
,
and the
first
x in terms of y
y and u these values. upon replacing by Let the surface S be given by means of an unsolved equation F(x, y, z) = 0, and let x y z be the coordinates of a point of the surface. If the function F(x, y, z) is a power series in x x y zo) an(i ^ n t a ll three of the partial derivatives z z
z
t
,
o>
and u
t
y<
dF/dx 8F/8y 8F/dz vanish simultaneously, the point 3/ an ordinary point, by 188. ,
M
,
,
is
surely
TRIGONOMETRIC SERIES
IX,195J
411
The definition of an ordinary point on a curve or on a sur be an independent of the choice of axes. For, let 3/ (x y ) Then the coordinates of any neigh ordinary point on a surface S. Note.
face
is
,
,
boring point can* be written in the form (64), where not all three of the determinants D(y, z)/D(t, u), D(z, x)/D(t, u}, D(x, y)/D(t, it} u u t Let us now select any new vanish simultaneously for t
=
=
,
.
axes whatever and let
A =
+ fay + yiz + 8 y -f y z + S x + fay + y z + 8
ai x
Y= a^x + Z=
a3
{3 2
be the transformation which carries
1}
z
2
3
3
,
the
x, y, z into
new
coordinates
X, Y, Z, where the determinant A = D(X, Y, Z)/D(x, y, z) ent from zero. Replacing x, y, z by their developments we obtain three analogous developments for X, Y, Z (64),
is
differ
in series ;
and we
cannot have 1>(X,
Y}
=
for
t
=
,
u
=
ii
,
J^Z) =~ D(Z, X} =
Z>(
D(t, u)
J)(t, if)
D(t, u)
since the transformation can be written in the
X = A X + B Y+ y = A X + B Y+ z
form
C\Z + D ly C Z + J9 = A X + B Y+C Z + D 1
1
2
2
2
2
,
3
3
3
t
,
and the three functional determinants involving vanish simultaneously unless the three involving
.Y,
x, y,
Y, z,
Z
cannot
also vanish
simultaneously.
TRIGONOMETRIC SERIES
IV.
195. Calculation of the coefficients.
MISCELLANEOUS SERIES The
series
which we
shall study
section are entirely different from those studied above. Trigonometric series appear to have been first studied by D. Ber in
this
noulli, in connection
with the problem of the stretched string. The we are about to give,
process for determining the coefficients, which is due to Euler.
Let first
/(a:) be
a function defined in the interval
suppose that a and
tively,
which
is
b
have the values
TT
We shall + TT, respec
(a, &).
and
always allowable, since the substitution x
=
(a -f 1}
2-7TX
7T
r-*o
a
* See footnote, p. 408.
TRANS.
SPECIAL SERIES
412
Then
reduces any case to the preceding. (65) /(#)
= -^ -f (i cos x + b
holds for
all
(tn
:
sin x)
\-
-\
values of x between
TT
and
195
[IX,
the equation
if
(am cos mx
+
TT,
+ & m sin mx)
where the
-\
coefficients
ao) a a bm are unknown constants, the following device shall first write down enables us to determine those constants. i>
)
bi>
>---
m>
We
for reference the following formulae, which were established above, for positive integral values of m and n :
L
L f \J
sin
mx dx =
cos
m
;
= 0,
rfa;
if
;
r
cos wcc cos
nx dx
71
"
cos(m ^
ri)x *
+ cos (m + ri)x rfx =
if
m
= TT,
if
m 3=
_
x
.
7
5
/cos
-mxdx =
r+
^
,
=
n
;
0;
"
sin
I
\J
dx
I
V/-7T
7T
(66)
m^
mx
sin
nx dx
cos
(m
7T
+*
f
n) x
r+ I
I
r+
7r .
sm 2 ma;rfx=
sin
cos
(m
-f
n)x ^
o 2
/
C/-7T
"
"
cos 2mrc
1
I
^
.
,
^x
=TT,
ifw^O;
mx cos nx dx 77
sin
X
(m
+ n) x +
sin (TO
n)x
__
2
TT and -f Integrating both sides of (65) between the limits the right-hand side being integrated term by term, we find /"
-\-
I
/^
7T
/(x) dx
J-TT
=
I
"t"
TT,
T
dx
=
7ra
,
/-
which gives the value of a Performing the same operations upon after the equation (65) having multiplied both sides either by cos mx .
IX,
TRIGONOMETRIC SERIES
195]
413
or by sin mx, the only term on the right whose integral between TT and + TT is different from zero is the one in cos 2 mx or in sin 2 mx.
Hence we +
/
find the formulae
"
mx dx =
f(x) cos
(*
7nz m ,
The values
respectively.
*
f(x) sin
\J
TT
follows
+
I
mx dx =
7rb m ,
TT
of the coefficients
be assembled as
may
:
=-
a
**
am
/(a) da,
I
t/
=-
n
/(a) cos
I
*i^/
(67) bm
r +)r
i
=
f(a~) sin
I "
T7 t/-n-
ma da,
jr
mar
c?nr.
The preceding calculation is merely formal, and therefore tenta For we have assumed that the function f(x) can be developed
tive.
in the form (65), and that that development converges uniformly between the limits TT and -f TT. Since there is nothing to prove, a priori, that these assumptions are justifiable, it is essential that we investigate whether the series thus obtained converges or not.
Eeplacing the coefficients a and
b{
{
simplifying, the
S m+l =
f
1
sum
+7r
I
fl
/()
-^
of the first
+ cos(a
a-)
by
(m +
from (67) and seen to be
their values
1) terms
+ cos2(rt
is
1 -----
x}-\
\-COSm(a
x) \da.
But by a well-known trigonometric formula we have
-sin
2
+ cos a -f cos 2a +
+ cos m a =
2m -+1
a
-
>
a
2sm.
whence .
sin
or, setting
(68)
a
=x S
2m-+ 1 (a
a-)
-f-
^
The whole question is reduced to that of sum as the integer m increases indefinitely.
finding the limit of this In order to study this
question, we shall assume that the function f(x) satisfies the fol lowing conditions :
SPECIAL SERIES
414
The function f(x)
1)
+
and
TT,
except for
a,
between c
and
we we
c
+
shall call f(c shall call f(c
c
approaches zero, f(c + 0). Likewise, f(c 0) as
c
we
-f-
c)
is
It
where f(x) and /(TT
is
is
TT
value
e)
approaches a limit which approaches a limit which If the function f(x)
zero.
approaches shall have /(c)
f(c f(c + 0) 0), f(x) to take the arithmetic mean of these values =
for /(c).
its
in the following
continuous for x = c,
is
of values of x, for which
TT.
is
As
h.
number
196
manner. Let c be a number For value of c a number h can be found + any continuous between c h and c and also between
and
such that f(x)
between
shall be in general continuous
finite
may change suddenly TT
[IX,
+ 0) =/(c 0). If = c, and we shall agree [/(c + 0) + f(c 0)]/2 f(c
discontinuous for x
evident that this definition of /(c) holds also at points We shall further suppose that /( TT + e)
continuous.
+
which we
shall call /( TT 0) and zero values. through positive approaches /(TT The curve whose equation is y f(x) must be similar to that of have Fig. 11 on page 160, if there are any discontinuities. that the function in is the from seen interval integrable already /(x)
approach
e)
limits,
0), respectively, as
e
We
to
TT
+
TT,
and
it is
evident that the same
is
true for the product
of f(x) by any function which is continuous in the 2) It shall be possible to divide the interval ( finite
number
of subintervals in such a
way
same TT,
thaty(cr)
interval.
+ TT)
is
into a
a monoton-
ically increasing or a monotonically decreasing function in each of
the subintervals.
For brevity we shall say that the function f(x) satisfies Dirichlet s conditions in the interval ( It is clear that a function TT, TT).
+
which finite
is
number
Dirichlet
196.
for
<S
TO
continuous in the interval
s
TT,
(
maxima and minima
in
+
TT)
and which has a
that interval, satisfies
conditions.
h
f (x) [sin nx/sin x] dx. The expression obtained leads us to seek the limit of the definite integral
The +1
of
integral
/
C
sin
nx
f(x) since dx Jo
n becomes infinite. The first rigorous discussion of this ques was given by Lejeune-Dirichlet.* The method which we shall employ is essentially the same as that given by Bonnet. t as
tion
* Crelle s Journal, Vol. IV, 1829. t Mtmoires des savants etrangers publics par
1
Academic de Belgique, Vol. XXIII.
IX,
TRIGONOMETRIC SERIES
196]
Let us
first
415
consider the integral r>h
./
(69)
I
=
.
I
nx
sin
.
,
dx >
c/O
where h is a positive number less than TT, and which satisfies Dirichlet s conditions in the interval
is
a function
<j>(x)
a constant C, we may write
it is
is
easy to find the limit of sink I
sin
J=C\ Jo and the limit of J
as
If (x)
For, setting y
= nx,
7
2-dy,
y
n becomes
Next suppose that
176. infinite is CTr/2, by (39), a positive monotonically decreasing
is (x)
function in the interval all
y
/.
(0, A).
The integrand changes sign Hence J may be written
(0, h).
for
values of x of the form kir/n.
j= Uo - Ul + u - u + 2
+ (- I) m
+ (- 1)*%- +
3
-
where
C
sn nx
"
I
Jk*
0",,
<
<
1,
.
dx
supposed to lie between mir/n and Each of the integrals uk is less than the preceding. (ra -f T)7r/n. For, if we set nx = &TT + y in u k we find
and where the upper limit h
is
,
y
+
kw\
n
/
sin y
y
+
*
A:TT
it is evident, by the hypotheses regarding <(#), that this inte Hence we shall have gral decreases as the subscript k increases.
and
the equations ./
= UQ
which show that / a positive number
lies
less
\Ml
between than
ii
,
?/
/
But
and n
that
Jo
^4/
y^ s
^"i)
is
u^
)
.
It follows that
to say, less
/
is
than the integral
smnx -dx. x
this integral is itself less than the integral
Jo
where A denotes the value of the
y
definite integral //[(sin y)/y~\ dy.
416
SPECIAL SERIES
The same argument shows that the .
^
,
[IX,
196
definite integral
sm nx
.
Xh *(x)- x -dx, where
shown
any positive number less than h, approaches zero as n If c lies between (i l)?r/n and i-rr/n, it can be
c is
becomes
infinite.
as above that the absolute value of
C
"
nx
sin
.
X
Jc
,
than
is less
C
+
dx
J
snnx
Jiz n
and hence, a fortiori,
less
than
_ c
\
n
Hence the integral approaches zero as n becomes infinite.* This method gives us no information if c = 0. In order
to dis
cover the limit of the integral J, let c be a number between and h, such that is continuous from to c, and let us set (x)
=
(x)
(c)
+
Then
\f/(x").
interval (0, c) from value zero when x
=
J=
nx
/""sin
I <(c)
X
JO
and then subtract
is
i^(a-)
the value
dx
C
+
I
we
i(-f 0),
(7r/2)
(70)
-
nx
sin
$(x)
Jo
X
/e JO
in the
dx
+
when x
in
=
the
to the
form
-- dx
C
nx
sin
7
Jc
find
.
I I
<(c)
/
write
r
-
0)
<(+
we
If
c.
and decreases
positive
f(x)
sinnx -dx+ -^
.
,->
I <#>(x)
Jc
smnir x
In order to prove that ./ approaches the limit (Tr/2) 0), it will be sufficient to show that a number m exists such that the absolute ^>(+
* This result of the
mean
may
be obtained even more simply by the use of the second theorem Since the function is a decreasing function,
for integrals (75).
(r)
that formula gives .
.
()
sin
nx
,
ax
=
0(c)
r
-J-J
sin
wx dx
=
and the right-hand member evidently approaches
-1
M
-j* zero.
(cos
nc
-
cosn|)
,
IX,
TRIGONOMETRIC SERIES
196]
value of each of the terms on the right positive
made
number e/4 when n
is
417 than a preassigned
is less
greater than m.
the remark
By
above, the absolute value of the integral sin
nx
/
ax
Jo is less
than
A$(+
=A
0)
[<(+
0)
<(e)].
Since <(z)
approaches
x approaches zero, c may be taken so near to zero that A [>(+ 0) - (c)] and (7r/2)[>(+ 0) - (c)] are both less than e/4. The number c having been chosen in this way, the other two terms on the right-hand side of equation (70) both approach zero as n becomes infinite. Hence n may be chosen so large that the abso 0) as
<(+
lute value of either of
them
(71)
We
is less
than
e/4.
It follows that
lim/=|*(+0). A (=QO shall
now proceed
to
remove the various
restrictions
which
x ) ^ n the preceding argument. If is a monotonically decreasing function, but is not always positive, the function \j/(x) + C is a positive monotonically decreasing func tion from to h if the constant C be suitably chosen. Then the
have been placed upon
(
(x)
(x)
formula (71) applies to *
sinnx
(*
dx
(*)
I
JO
Moreover we may write
\j/(x).
= C
k
C h sinnx
sinnx
-dx-C
Jo
dx, JO
and the right-hand side approaches the limit (w/2) \j/(-{that
is,
<#(+
0)
<(#)
Ch I
Jo
Hence the
(?r/2) C,
0).
is a monotonically increasing function from to a monotonically decreasing function, and we shall have
If is
(?r/2)
^(*)
sin nx - dx = - C
sin
I
*
h,
(x~)
nx - dx.
t7o
in this case also. (7r/2)<(+ 0) that is function which satisfies DirichFinally, suppose any let s conditions in the interval (0, Then the interval (0, A) A).
integral approaches <(#)
may
be divided into a finite
number
each of which
of subintervals
(0, a), (a, b),
a monotonically increasing or decreasing function. The integral from to a approaches the limit of Each the other which are of the type integrals, (7T/2) 0). (b, c),
-,
(7,
A), in
is
(x~)
<(-)-
sinnx
=/ Ja
dx,
418
SPECIAL SERIES
approaches
For
zero.
if
is (x~)
[IX,
a monotonically increasing function, can be formed \j/(x)
for instance, from a to b, an auxiliary function in an infinite variety of ways, which increases to
b, is
continuous from
Then each
to a,
monotonically from from a to b.
and coincides with
(x)
of the integrals
C
sinnx
approaches
$(+
sinnx <**,
/ t/O
which
107
0) as
n becomes
infinite.
Hence
their difference,
is
It follows that the formula precisely H, approaches zero. which satisfies Dirichlet s condi (71) holds for any function tions in the interval (0, h). (x)
Let us
now
f=f(x)dX)
(72)
where f(x) to A.
consider the integral
is
(X/K7T,
a positive monotonically increasing function from may be written
This integral
x n ff x \ = C vf( )~ 1
Jo
and the function tf>(x)
L
to h.
lim 7
(73)
-,
x
sill
= f(x) x/sin x
increasing function from that
s
sm x J
^
is
dx,
a positive monotonically
Since /(+ 0)
= /(+
=
<(+
0), it follows
0).
This formula therefore holds
if f(x) is a positive monotonically function from to h. It can be shown by successive increasing as that the restrictions upon f(x) can all be removed, steps, above,
and that the formula holds Dirichlet
s
for
any function f(x) which
conditions in the interval (0,
satisfies
li).
197. Fourier series. A trigonometric series whose coefficients are given by the formulae (67) is usually called a Fourier series. Indeed it was Fourier who first stated the theorem that any function arbi
trarily defined in an interval of length 2?r may be represented by a series of that type. By an arbitrary function Fourier understood
a function which could be represented graphically by several cur vilinear arcs of curves which are usually regarded as distinct curves. shall render this rather vague notion precise by restricting our discussion to functions which Dirichlet s conditions.
We
satisfy
IX,
TRIGONOMETRIC SERIES
197]
419
In order to show that a function of this kind can be represented we must find the by a Fourier series in the interval ( TT, + TT), Let us divide limit of the integral (68) as m becomes infinite.
two integrals whose limits of integration are and let us and (TT (TT + x)/2 and 0, respectively, r)/2, and = of these z in the second make the substitution y integrals. Then the formula (68) becomes
this integral into
sn *
C
1 TT
When
x
lies
between
lie
Jo
between and TT.
TT
ox
s,
Jf(x v
I
and
sin
2z)
4-
TT,
(TT
Hence by the
(2m
sm
+
1) *
,
dz
a;)/2 and (TT last article the
.
+ #)/2
both
right-hand
side of the preceding formula approaches
as
m becomes infinite.
It follows that the series (65) converges
and
TT and + TT. sum is/(x) for every value of x between Let us now suppose that x is equal to one of the limits of the Then S m + may be written in the form TT for example. interval,
that
its
l
= -1
C*., J/( 7rJ
The
first
Setting y
TT
I
+ ,
X
2y)
sin(2m4-1)// j -dy i
;
smy
TT integral on the right approaches the limit /( the form 2 in the second integral, it takes TT
+
0)/2.
=
sm* Hence the sum of the trigonometric = TT. It is evident 0)] /2 when x 0) +/( that the sum of the series is the same when x = + TT. we If, instead of laying off x as a length along a straight line,
which approaches /(TT series is [/(TT
lay
it off
0)/2. TT
+
as the length of an arc of a unit circle, counting in the
420
SPECIAL SERIES
[IX,
197
positive direction from the point of intersection of the circle with the positive direction of some initial diameter, the sum of the series
any point whatever will be the arithmetic mean of the two limits approached by the sum of the series as each of the variable points ra and taken on the circumference on opposite sides of ra, m. If the two limits /( TT + approaches are 0) and f(ir at
ra",
0)
different, the point of the circumference
on the negative direction
of the initial line will be a point of discontinuity.
In conclusion, every function which (
TJ
val
may
-f-
is defined in the interval Dirichlet s conditions in that inter satisfies be represented by a Fourier series in the same interval.
and which
"")
More generally, let f(x) T^e a function which is defined in an interval (a, a 2?r) of length 2?r, and which satisfies Dirichlet s conditions in that interval. It is evident that there exists one and
+
only one function F(x) which has the period 2?r and coincides with This function is represented, for /(#) in the interval (a, a + 2?r). all values of x, by the sum of a trigonometric series whose coeffi cients a m and b m are given by the formulas (67):
am
The
=
~
F(r) cos
I
coefficient
a m for example, ,
r*
1
am
mx dx,
F(x) cos
I
*V-tte
bm
may
=
-
F(x) sin
I
mx dx
.
be written in the form
i mx dx + -
r-
2 *-
F(x) cos
I
vj_ n
mx dx,
where a
is supposed to lie between 2hir TT and 2hjr + TT. Since has the period 2?r and coincides with F(x} f(x) in the interval (a, a + 2?r), this value may be rewritten in the form
(
=~
I
r
+
=
/
/ (.r) cos
mx dx
2jr
f(x) cos
/
mx dx
.
ft
we should bm
(75)
mx dx +
Jm*+m
,^a +
t/
^a: + 2T
!r
f(y) cos
I
-
V.
Similarly,
"
Ja
\
(74)
/l2
1
am
find
=~ "ft
I
Ja
f(x) sin
mx dx
.
Whenever a function f(x) is defined in any interval of length 2?r, the preceding formulae enable us to calculate the coefficients of its development in a Fourier series without reducing the given interval to the interval
(
TT,
+
TT).
IX,
TRIGONOMETRIC SERIES
198]
198. Examples. it
x
<
1
r
It
J-ir
1
r JI -n
=
a
am
=
nm
If
m
by
1
c
Tt
Jo
dx
I
!
m
If
7T/4 for
is
it
is
odd, b m
cosmTT
2
cos(
mir)-
>
.
is
4/mir.
Multiplying
all
the coeffi
of the series
+
+
2m + 1
3
x 0, and + ff/4 for and the sum of the series <
<
-,
mit
sum
-i-
1
of discontinuity,
mx dx =
Jo
Sin3x
=
cos
T* sin mx ax A =
,
see that the
v
(76)
r* JQ
-\
Tt
is zero.
,
-\
j sm mx dx
we
=
i
mxdx
.
7T/4,
<
n
It
J -T,
even, b m
is
<
H
cos
= ~I f 71
cients
dx
I
It
Let us find a Fourier series whose sum is 1 for f or x + it. The formulae (67) give the values
1)
+1
and
0,
<
421
x
<
<
The point x = is a point when x = 0, as it should be. when sin x is positive, ?r/4 it.
zero
is
More generally the sum of the series (76) is 7t/4 when sin x is negative, and zero when sin x 0. The curve represented by the equation (76) is composed of an infinite number of segments of length n of the straight lines y = 7T/4 and an infinite num ber of isolated points (y
The
2)
=
0,
x
=
on the x
kit)
coefficients of the Fourier
axis.
development of x
in the interval
from
to
27T are
=
do
1
=
dm
1 -Tt
= -1
om
r-* I
Tt
=
xdx
Jo
r2
2?r,
x cos mxdx
=
r2
rxcos?/ixl 2;r
"
x sin
1
I
-{
Jo
11171
\_
n Jo
c^ -mit Jo
rxsinmx-l 2 *
"
Jo
mx dx =
\
--1
-\
mit
\_
Jo
(**"
|
iwt Jo
sin
mx dx =
cos
2 mx dx = ---
,
m
Hence the formula x
__
it
22
sinx
sin3x
sin 2x
is valid for all values of x between and 2?r. If we set y equal to the series on the right, the resulting equation represents a curve composed of an infinite num ber of segments of straight lines parallel to y x/2 and an infinite number of
isolated points.
Note. to say,
if
function /(x) defined in the interval ( - Tt, + Tt) is even, that is x) =/(x), each of the coefficients b m is zero, since it is evident that
If the
/(
nO I
J-n Similarly,
if
coefficients a m
/(x) is
is
n-n
f(x) sin
mx dx
I
Jo
/(x) sin
mx dx
.
an odd function, that is, if /( x) = a A function /(x) which
zero, including
.
/(x), is
each of the
defined only in
SPECIAL SERIES
422 the interval from
to
it
may
be defined
in the interval
of the equations
/(-*)= /() if
we choose
by a
to
do
or
/(
-
z)
from
to
it
by either
= - f(x)
Hence the given function /(x) may be represented by a series of sines, in the interval from to ic.
so.
199
[IX,
either
series of cosines or
199. Expansion of a continuous function. Weierstrass theorem. function which is defined and continuous in the interval (a,
Let /(z) be a
The following
6).
remarkable theorem was discovered by Weierstrass Given any preassigned posi tive number e, a polynomial P(x) can always be found such that the difference f(x)P(x) is less than e in absolute value for all values of x in the interval (a, 6). Among the many proofs of this theorem, that due to Lebesgue is one of the simplest.* Let us first consider a special function \j/(x) which is continuous in :
the interval
and
and which is defined as follows ^(x) = for Then \f/(x) = 1, where fc is a given constant. x + 1 we shall have
(1, +1)
=
2kx for Moreover for -
V (x)
<
1
x
<
:
<
(x -f
x 0, x |) k. <
|
<
same values
for the
1
<
of x the radical can be developed in a uniformly con It follows that x|, and
vergent series arranged according to powers of (1 - x 2 ). hence also i//(x), may be represented to any desired
degree of approximation in
the interval
+
by a polynomial. Let us now consider any function whatever, /(x), which is continuous in the interval (a, 6), and let us divide that interval into a suite of subintervals ax a2 ai), (a\ a 2 ), (a,,_i a n ), where a = a a n = 6, a,,_i 1,
(
,
(<*oi
in such a
1)
,
,
<
<
<
<
<
that the oscillation of /(x) in any one of the subintervals is less than e/2. Let L be the broken line formed by connecting the points of the curve y =f(x) whose abscissae are a a x a 2 b. The ordinate of any point on L is evidently a continuous function 0(x), and the difference is /(x) less than e/2 in absolute value. For in the interval (a M _!, aM ), for example,
way
,
,
,
,
(x)
we
have
shall
/(x)
where x
-
a M _i
-
= 0(a M
0(x)
=
[/(x)
-/(
(1
-
a^_i). unity, the absolute value of the difference/
The function
can be (x)
6)
+
[/(x)
Since the factor d
split
up
into a
sum
is less
of
is
-/(aM )]
0,
positive
and
than
e(l
n functions
-
6
+
less
e)/2
=
than e/2.
same type as of L. Then
of the
A n be the successive vertices For, let A Q AI, A 2 equal to the continuous function (x) which is represented throughout the interval (a, 6) by the straight line A A l extended, plus a function 0i(x) which is represented by a broken line A whose first side A Q A\ lies on the A{ x axis and whose other sides are readily constructed from the sides of L. Again, the function fa (x) is equal to the sum of two functions 2 and where f 2 is ^ 2 zero between a and cti, and is represented by the extended straight line V (x).
,
,
,
(x)
^
is
A
,
between first
aj
and
while
represented by a broken line A(, Ai whose Finally, we shall obtain the equation where is a continuous function which vanishes + + and o,-_i and which is represented by a segment of a straight line 6,
three vertices
= fi + between a
A\A i A2 An
lie
2 is
-
on the x axis. ,
^
* Bulletin des sciences mathtmatiques,
p. 278, 1898.
IX,
TRIGONOMETRIC SERIES
200]
423
X
= mx + n, where m between ai_i and b. If we then make the substitution and n are suitably chosen numbers, the function \j/i(x) may be defined in the interval ( 1, + 1) by the equation can be represented by a polynomial with any desired degree of Since each of the functions f;(x) can be represented in the interval (a, 6) by a polynomial with an error less than c/2n, it is evident that the sum of these polynomials will differ from /(x) by less than e.
and hence
it
approximation.
It follows from the preceding theorem that any function f(x) which is contin uous in an interval (a, b) may be represented by an infinite series of polynomials e be a sequence which converges uniformly in that interval. For, let i c2 of positive numbers, each of which is less than the preceding, where e n approaches zero as n becomes infinite. By the preceding theorem, corresponding to each of ,
,
,
,
a polynomial P(x) can be found such that the difference /(z) P,(x) Then the series the interval (a, b). e,- in absolute value throughout
the
e s
less
than
^1
(3)
+ P2 (X) t
Pi
(X)]
+ [Pn (X) - P.-i ()] +
+
For which Moreover the series con
converges, and its sum is/(x) for any value of x inside the interval the sum of the first n terms is equal to P B (x), and the difference /(x) is less
than
e,,
approaches zero as n becomes
,
is
infinite.
(a, b). <S
M
,
Sn will be less verges uniformly, since the absolute value of the difference /(x) than any preassigned positive number for all values of n which exceed a certain fixed integer
200.
A
N, when x has any value whatever between a and
continuous function without a derivative.
b.
We shall conclude this chapter
by giving an example due to Weierstrass of a continuous function which does not possess a derivative for any value of the variable whatever. Let 6 be a posi Then the function tive constant less than unity and let a be an odd integer. F(x) defined by the convergent infinite series -r
F(x)
(78)
=
^
V
b"
cos
(a"
nx)
x, since the series converges uniformly in any product ab is less than unity, the same statements hold for the series obtained by term-by-term differentiation. Hence the func We shall tion F(x) possesses a derivative which is itself a continuous function. is
continuous for
all
interval whatever.
now show that
values of
If the
the state of affairs
is
essentially different
if
the product ab exceeds
a certain limit. In the
first
place, setting
m-l
Sm = -
^
b"
^
b"
(cos [a TT(Z
+
h)]
cos (a n TTX) }
,
{cos
+
h)]
cos
,
n
H=U R, n
we may (79)
[a"
TT(X
write ffl*Li!fiL>. h
.+*..
(a"
nx)}
SPECIAL SERIES On
the other hand,
function
na n h |
easy to show, by applying the law of the
it is
cos(a"7rz),
[IX,
that the difference
+
cos[a"7r(z
Hence the absolute value
in absolute value. \
cos(a
h)]
Sm
of
n
mean
200
to the
itx) is less
than
than
is less
m-l an b n
= n
ab-l
n=(l
m
and consequently also less than 7r(ab) /(ab 1), lower limit of the absolute value of B m when h
We
always have
shall
where a m
an integer and m
is
= am +
am x lies
/j
_
ab
assigned a particular value.
>1.
,,
+ 1/2.
1/2 and
between p
Let us try to find a
if
is
If
we
set
?
5S
, 1 a"
where em is equal to 1, it is evident that the sign of h is the same aa that of e m and that the absolute value of h is less than 3/2a m Having chosen h in this way, we shall have an - n a m 7c(x + h) = an - m n(a m + e m ). a"7f(x + h) .
,
Since a
is
odd and
em
=
1,
the product an -
(a m
-f
am +
l
e m ) is
even or odd with
a m + 1, and hence cos[a"7r(z
Moreover we
shall
coa(a
or, since
m am
a"~
n 7tx)
is
= =
follows that
=
A)]
l)
(
m am = 7rz) - m a Tt cos m )
cos(a"-
cos
(a"
even or odd with CQS
It
+
.
have
we may
(fin 7f%\
am 1
(
m cos[a"-
-
*(a m
m n)
"
(a"
+
m)]
,
,
^""*
m^
COS {a n
7T^
write (
_ 1)g, +
* i
"m
n Since every term of the series is positive, it is greater than 6 m since
consequently
or, since
\h\<
3/2a
its ,
sum is greater than the
lies
between
1/2 and
m ,
\
If
a and b satisfy the inequality a6>l
(80)
we
shall
+
,
have 3
whence, by
ab-l
(79),
F(x
+ h)-
2
F(x) >
|B|-|S n
3
x
a6-l
first
+
term, and
1/2.
Hence
EXERCISES
IX, Exs.]
425
m
As becomes infinite the expression on the extreme right increases indefinitely, while the absolute value of h approaches zero. Consequently, no matter how be chosen, an increment h can be found which is less than in abso and for which the absolute value of [F(x + h) F(x)~\/h exceeds any preassigned number whatever. It follows that if a and b satisfy the relation (80), small
e
lute value,
the function F(x) possesses no derivative for
any value of x whatever.
EXERCISES 1.
Apply Lagrange
s
formula to derive a development in powers of x of that ay + x which is equal to a when x = 0.
=
root of the equation y 2
Solve the similar problem for the equation y
2.
a
+
=
xy m + l
Apply the
0.
=
result to the quadratic equation a bx + ex 2 0. Develop in powers of c that root of the quadratic which approaches a/6 as c approaches zero.
Derive the formula
3.
l+x Show
4.
that the formula
Vl + holds whenever x
Show
5.
I
\
is
x
1
+
x
2 \1
greater than
+
2.4
x/
1/2.
that the equation c
+
2 1
x2
2
holds for values of x less than
when x |
>
1
+
4 \1
.
1
.
4
in absolute value.
.
G \1
+
+
x2
What
is
the
sum
of the series
?
|
1.2
Show
7.
and
powers
when sinx
respectively, of sin x
m2 - 1
.
=
mx =
[Make use
2 2 0) l)(m sm s x + (m sm 5 x 1.2.3.4.5 m 2 (m 2 -4) 4 sm x 1.2.3.4 .
.
?n 2 1
.
sin 2 x H
,
J
.
of the differential equation
satisfied
From
n
.
dy 2 is
sinmx and cosmx which reduce
are developable in series according 10
1.2.3 1.2
which
1-2.3
+*
:
L cos
a
that the branches of the function
1,
sm mx = m f sm x
8.
2
Derive the formula
6.
to
x2 /
by u
dy
cosmx and by u = sinmx, where y
=
sinx.J
the preceding formulae deduce developments for the functions
cos (n arc cos x)
,
sin (n arc cos x)
.
CHAPTER X PLANE CURVES The curves and
surfaces treated in Analytic Geometry, properly
speaking, are analytic curves and surfaces. However, the geomet rical concepts which we are about to consider involve only the exist ence of a certain number of successive derivatives. Thus the curve
whose equation is y = f(x) possesses a tangent if the function f(x) has a derivative / (#) it has a radius of curvature if / (#) has a ;
derivative
/"(cc);
and so
forth.
I.
ENVELOPES Given a plane curve C whose
201. Determination of envelopes.
equation (1)
f(x, y, a)
=
involves an arbitrary parameter a, the form and the position of the If each of the positions of the curve C is curve will vary with a.
tangent to a fixed curve E, the curve E is called the envelope of the curves C, and the curves C are said to be enveloped by E. The before is to establish the existence us problem (or non-existence) of
an envelope for a given family of curves C, and to determine that envelope when it does exist.
Assuming that an envelope E
exists, let (x, y] be the point of tanE of with that one of the C which corresponds to a cer curves gency tain value a of the parameter. The quantities x and y are unknown functions of the parameter a, which satisfy the equation (1). In
order to determine these functions, let us express the fact that the tangents to the two curves E and C coincide for all values of a.
Let Sx and 8y be two quantities proportional to the direction cosines of the tangent to the curve C, and let dx/da and dy/da be the derivatives of the unknown functions x = Then a y = $(&) <(),
necessary condition for tangency
is
dx da
dy
_
da
C * 426
X,
ENVELOPES
201]
427
On the other hand, since a in equation (1) has a constant value for the particular curve C considered, we shall have (
+ ^Sy = dy
sx
(3)
dx
0,
"
which determines the tangent to C. Again, the two unknown func = y = ^() satisfy the equation
tions x
<(),
f(x, y, a)
where a
also,
is
now
cfdx
oITda ex
df + cfdt/ aj+a= da oa
combining the equations
cy
(2), (3),
The unknown functions x = y and the equation (1). Hence <(),
an envelope
exists, is to be
and
(4),
= \f/(d) are solutions of this equa
tion case
Hence
the independent variable.
(4)
or,
= 0,
the equation of the envelope, in
found by eliminating the parameter a
=
between the equations f 0, df/da = 0. Let R(x, y) be the equation obtained by eliminating a between (1) and (5), and let us try to determine whether or not this equation
Let C be the particu represents an envelope of the given curves. which corresponds to a value a of the parameter, and let
lar curve
M
y ) be the coordinates of the point (atwo curves ,
/(*, y, a
(6)
The equations x
= (a),
a
a
y
=
(1)
=
*-
0,
=
of intersection of the
<>.
t/a
and (5) have, in general, solutions of the form which reduce to x and y respectively, for
\j/(a),
Hence
.
)
ti
,
for a
aQ
we
dx \da/o
shall
c // Q
have
\da/o
This equation taken in connection with the equation (3) shows that the tangent to the curve C coincides with the tangent to the curve described by the point (x, y*), at least unless df/dx and
df/dy unless the point Q is a singular point for the curve C It follows that the equation R(x, represents either y) the envelope of the curves C or else the locus of singular points on are both zero, that .
these curves,
is,
M
=
PLANE CURVES
428 This result
may be
[X,
202
C
has
If each of the curves
supplemented.
one or more singular points, the locus of such points is surely a part 0. of the curve R(x, y) Suppose, for example, that the point (x, y) Then x and y are functions of a which a is such singular point.
=
satisfy the three equations ^\
f(x,y,a) = Q,
Q
/
^
= 0,
/>
^
= 0,
=
0. It follows that x and y = obtained the eliminating a between by equation R(x, y) satisfy In the general case the 0. and df/Sa the two equations /
and hence also the equation df/da
=
=
curve R(XJ y) one of which
= is
is
composed
two analytically
of
the true envelope, while the other
distinct parts, the locus of
is
the singular points.
Example. Let us consider the family of curves
= /-2/ + 2
/(*,*/,)
The
-)
2
=
-
elimination of a between this equation and the derived equation
y
gives y*
y
(*
jf.
l,
2
= 0, 1.
y
which represents the three straight
The given family
by a translation of the curve
of curves
y
y*
2
+x = 2
lines
y
=
0,
may be generated along the x axis.
This curve has a double point at the origin, and it is tangent to I at the points where it cuts the each of the straight lines y = line the y = is the locus of double points, straight y axis. Hence lines two the whereas y = 1 constitute the real envelope. straight 202.
If the curves
C have an
envelope E, any point of the envelope
the limiting position of the point of intersection of two curves of the family for which the values of the parameter differ by an infini
is
tesimal.
For, let f(x, y, a)
(7)
=
f(x,y,a
0,
+
h)
=
two neighboring curves of the family. The the points of intersection of the two determine equations (7), which be replaced by the equivalent system curves, may evidently be the equations of
f(x,
y,
a)
=
f(x, 0,
y,
a
+
h)-f(x,y, -
a)
_
0,
X,
ENVELOPES
202]
429
=
as h approaches zero, that the second of which reduces to Sf/da two curves of the second the as approaches the first. This prop is,
In Fig. 37, a, for instance, the of the two neighboring curves C and C as C approaches the curve C approaches the point of tangency erty
is
fairly evident geometrically.
N
point of intersection
M
FIG. 37, b
as its limiting position. Likewise, in Fig. 37, b, where the given curves (1) are supposed to have double points, the point of intersec tion of two neighboring curves C and C approaches the point where
C
cuts the envelope as
The remark
just
C
made
approaches C. explains
the locus of singular points
why
found along with the envelope. For, suppose that f(x, y, a) is a ?/ ) chosen at polynomial of degree m in a. For any point Q (x is
M
random
,
in the plane the equation
(8)
/(.r
,
7,0,0
=
m
will have, in general, distinct roots. Through such a point there different curves of the given family. But if the pass, in general,
m
point
M
lies
on the curve
R (or,
y)
= 0,
the equations
are satisfied simultaneously, and the equation (8) has a double root. The equation R(x, y) may therefore be said to represent the locus of those points in the plane for which two of the curves of
=
the given family which pass through one.
The
figures 37, a,
and
37,
b,
have merged into a single clearly the manner in which
it
show
two of the curves through a given point merge into a single one as that point approaches a point of the curve R(x, y) = 0, whether on the true envelope or on a locus of double points.
PLANE CURVES
430 It often
Note.
becomes necessary to
[X,
203
find the envelope of a family
of curves F(x, y, a,
(9)
ft)
=
whose equation involves two variable parameters a and b, which = 0. This case themselves satisfy a relation of the form ft) does not differ essentially from the preceding general case, however, for may be thought of as a function of a defined by the equation (a,
ft
=
the rule obtained above, we should join with the given the equation equation obtained by equating to zero the derivative of its left-hand member with respect to a <
0.
By
:
dFdt __ -n
dp ~0
07
I
ca
But from the
relation
ft)
(a,
=
d(fr
we have db da
d
da
"
7
da
co
db
also
_
whence, eliminating db/da, we obtain the equation
_ da db
db
da
and The parameters a and
required envelope. between these three equations 203. Envelope of a straight line
ft
if
0,
may
determine the be eliminated
desired.
As an example
of a straight line.
=
F=
which, together with the equations
let
us consider the equation
D in normal form zcosa + y
(11)
where the variable parameter
is
with respect to this parameter,
a
0,
f(a)
the angle a. Differentiating the left-hand side find as the second equation
we
xsinct
(12)
sin
+
y cos
a
f(a)
=
0.
These two equations (11) and (12) determine the point of intersection of any one of the family (11) with the envelope E in the form
(x=f(a)cosa -f y = f(a) sin a + f
\
It is
(a) sin (a) cos
a,
a
.
E
which is described by this easy to show that the tangent to the envelope For from the equations (13) we find the line D. (z, y) is precisely
point
whence dy/dx
=
cot
(
dx
I
dy
= - [/() + /"()] sin a da = [/(a) + cos a da
a which ,
/"(a)]
is
,
,
precisely the slope of the line D.
X,
ENVELOPES
203]
Moreover, point upon
if
it,
431
denote the length of the arc of the envelope from any fixed
s
we
have, from (14), ds
=
+
=
dy*
+
[/(a)
/"(a)]
da-
,
whence
=
da
+/
[//(or)
Hence the envelope
known
for /(a) the derivative of a
As an example
let
(<*)].
be a curve which
will
easily rectifiable
if
we merely choose
function.*
=
us set f(a)
sin
I
we
cessively in the equation (11),
is
a
cos a.
Taking y
find (Fig. 38)
OA =
I
AB
=
= I. The required respectively hence curve is therefore the envelope of a straight line of constant length Z, whose extremities ;
The formulae
on the two axes.
lie
always
= suc OB = Zcosa,
and x
sin a,
(13) give in this case
x
=
I
sin 3
and the equation
a
=
y
,
I
cos 3
a-
of the envelope
-A*
,
is
AA?
if*-
1
which represents a hypocycloid with four cusps, of the form indicated in the figure. As a varies from to 7T/2, the point of con tact describes the arc
DC.
s=
p IG
Hence the length 3 1 sin
I
of the arc, counted
a cos a da =
sin 2
Jo
from D,
is
a.
2
M
Let I be the fourth vertex of the rectangle determined by OA and OB, and the foot of the perpendicular let fall from / upon AB. Then, from the tri angles AMI and APM, we find, successively,
AM = AI cos a = Hence
I
AP = sin AB with the envelope.
OP = OA
the line
cos 2
I
3
a
,
AP = AM sin a =
a, and the point Moreover
M
is
I
cos 2
a sin a
.
the point of tangency of
BM = l-AM=lsm*a; hence the length of the arc
*
Each
DM = 3BM/2.
which occur
of the quantities
a
in the
formula for
s,
s
= f (a) + ff(a)
daf,
the angle between the x axis and the perpendicular let fall upon the variable line from the origin is the distance from the from origin to the variable line; and / is, except for sign, the distance the point where the variable line touches its envelope to the foot of the perpen
has a geometrical meaning
:
is
ON
;
M
dicular let fall upon the line from the origin.
Legendre
s
formula.
ON
f(<x)
(<*)
N
The formula
MN
for 5 is often called
PL AXE CURVES
432 204. Envelope of a
where
a, 6,
[X,
Let us consider the family of
circle.
and p are functions
of a variable
parameter
204
circles
The
t.
points where a
circle of this family touches the envelope are the points of intersection of the circle
and the straight
line (x
(16)
This straight line
by the center
is
-
a) a
+
-
(y
+
b) b
and
the variable circle (15),
.
M T to the curve C described
perpendicular to the tangent
(a, 6) of
=
pp
its
distance from the center
p dp/ds, where
is
denotes the length of the arc of the curve C measured from
y
some
fixed point.
Consequently, if the meets the circle in the two
line (16)
N
N
points bisected
and the chord T by the tangent
angles. consists
of
/<j[
/
s
M
It follows
two
NN
,
is
at right
that the envelope
which
parts,
are,
in
general, branches of the same analytic curve. Let us now consider several special cases. 1) If
tact
NN
p
is
constant, the chord of con
reduces to the normal
PP
to
the curve C, and the envelope is com posed of the two parallel curves C\ and
FIG. 39
C{ which are obtained by laying off the constant distance p along the normal, on either side of the curve C. reduces to the tan 2) If p = s + K, we have p dp/ds = p, and the chord gent to the circle at the point Q. The two portions of the envelope are merged into a single curve T, whose normals are tangents to the curve C. The curve C is called the evolute of T, and, conversely, T is called an involute of C (see 206), If dp ds, the straight line (16) no longer cuts the circle, and the envelope is
NN
>
imaginary.
Secondary caustics.
Let us suppose that
the radius of the variable circle tional to the distance
is
propor
from the center to a
fixed point 0. Taking the fixed point O as the origin of coordinates, the equation of the
becomes
circle
where
A; is a constant factor, of the chord of contact is
(x If 8
-
a)
+
a
and
8
(y
-
b)
b
+
and the equation
k*(aa
E +
bb
)
=
0.
FIG. 40
denote the distances from the
center of the circle to the chord of contact and to the parallel to 2 origin, respectively, the preceding equation shows that 5 = k d
through the Let P be a k*MO, and let C be the .
point on the radius
MO
(Fig. 40),
such that
MP =
it
-
X,
CURVATURE
05]
the equation just found shows that the chord of con
Then
locus of the center.
433
P
the perpendicular let fall from upon the tangent to C at the center M. denote that branch of the Let us suppose that k is less than unity, and let as does the point 0. which lies on the same side of the tangent
tact
is
E
MT
envelope
Let
MN
MO
the two angles which the two lines r, respectively, denote make with the normal MI to the curve C. Then we shall have
i
and
.
_ =
Mq
~_ Mp
sin
i
_ Mq _ MQ _
~~
and
1
M~ MP~ k
Now let us imagine that the point is a source of light, and that the curve C which O lies from another medium separates a certain homogeneous medium in whose index of refraction with respect to the first is l/k. After refraction the will be turned into a refracted ray MR, which, by the law of incident
OM
ray
MR
Hence all the refracted rays the extension of the line NM. are normal to the envelope, which is called the secondary caustic of refraction. The true caustic, that is, the envelope of the refracted rays, is the evolute of the refraction,
is
secondary caustic. The second branch it
would correspond
E
of the envelope evidently has
to a negative index of refraction.
no physical meaning If
we
set k
1,
;
the
E
E
becomes the locus reduces to the single point 0, while the portion with respect to the tangents to C. of the points situated symmetrically with This portion of the envelope is also the secondary caustic of reflection for inci envelope
dent rays reflected from C which issue from the fixed point 0. It may be shown in a manner similar to the above that if a circle be described about each point of C with a radius proportional to the distance from its center to a fixed straight the envelope of the family will be a secondary caustic with respect to a system of parallel rays. line,
II.
205. Radius of curvature.
CURVATURE The
first
idea of curvature
is
that the
greater than that of another if it recedes more rapidly from its tangent. In order to render this somewhat vague idea precise, let us first consider the case of a circle. Its
curvature of one curve
is
curvature increases as
its
radius diminishes
;
it
is
therefore quite
curvature the simplest func tion of the radius which increases as the radius diminishes, that Let AB be an arc of a circle of the radius. is, the reciprocal natural to select as the measure of
its
l/R
subtends an angle o at the center. The angle and between the tangents at the extremities of the arc AB is also Hence the measure of the curva the length of the arc is s = This last definition may be extended to ture of the circle is w/s. of radius
R which
R
Let AB be an arc of a plane curve without a at the point of inflection, and w the angle between the tangents extremities of the arc, the directions of the tangents being taken the direction from A in the same sense according to some rule, an arc of any curve.
434
PLANE CURVES
toward B, for instance.
Then the quotient w/arc
A
AB
is called
the
the point B approaches the this quotient in general approaches a limit, which is called the curvature at the point A. The
As
average curvature of the arc AB. point
[X,205
radius of curvature at the point A is defined to be the radius of the circle
which would have the same curvature which the given curve has at the point
A
it is
;
therefore equal to the recipro Let s be the
cal of the curvature.
length of the arc of the given curve
measured from some fixed point, and a the angle between the tangent and
FIG. 41
some
fixed direction,
the x axis, for example.
Then
it is
clear
that the average curvature of the arc AB is equal to the absolute value of the quotient Aa/A.s hence the radius of curvature is given ;
by the formula
R=
As
,.
,
lim
=
Aa
dsart
Let us suppose the equation of the given curve to be solved for y in the form y =f(x). Then we shall have
a
= arc tan y
,
da
=
y"dx y "
_
.
ds
=
and hence (
*=
17 >
y,,
Since the radius of curvature indicates that
we
essentially positive, the sign are to take the absolute value of the expression is
on the right. If a length equal to the radius of curvature be laid from A upon the normal to the given curve on the side toward which the curve is concave, the extremity / is called the center of off
curvature.
The
circle described
about 7 as center with
called the circle of curvature. The coordinates (x center of curvature satisfy the two equations is
R ,
y
as radius )
of the
which express the fact that the point lies on the normal at a dis tance R from A. From these equations we find, on eliminating x l}
X,
CURVATURE
205]
In order to
tell
435
which sign should be taken,
us note that
let
if
y"
is
y must be positive hence the positive positive, as in Fig. 41, y^ is negative, y y is nega should be taken in this case. If ;
sign
y"
tive,
and the positive sign should be taken in
this case also.
The
coordinates of the center of curvature are therefore given by the
formulae 1
(18)
+
y
2
**-*=-
yi-9*=jf->
1 ,
y
+
y
2
fT-
the coordinates of a point (x, y) of the variable curve are we have, by 33, given as functions of a variable parameter t,
When
and the formulae (17) and (18) become jz
+ dy*?
(dx*
_
**
.
j }
dy(dx*
y
+ dif~) &x =
i
dxd y-dyd*x 2
and the radius of curvature is first kind y can be developed according hence to powers of x l/2 in a series which begins with a term in x of radius the therefore is infinite, and y has a finite value, but
At a point of inflection At a cusp of the infinite.
y"
0,
;
y"
curvature
is zero.
When
Note.
the coordinates are expressed as functions of the arc s of the
curve,
x the functions
and f
=
+
dy2
=
da 2
,
(/>"
Solving these equations for
where
e
=
(20)
+
and hence they <
cially elegant
=
f (a)
,
satisfy the relation 2(S)
since dx 2
y
,
<
and
\f/
+ ,
V
/2
(S)=1,
also satisfy the relation
-fy
^i"
we
=
.
find
and the formula for the radius of curvature takes on the espe form
1,
s[*
PLANE CURVES
436
[X,
206
206. Evolutes. The center of curvature at any point is the limit ing position of the point of intersection of the normal at that point with a second normal which approaches the first one as its limiting For the equation of the normal is position.
where
X
and
Y are
the running coordinates.
In order to find the
limiting position of the point of intersection of this normal with another which approaches it, we must solve this equation simulta
neously with the equation obtained by equating the derivative of the left-hand side with respect to the variable parameter x,
i.e.
The value of Y found from this equation is precisely the ordinate of the center of curvature, which proves the proposition. It follows that the locus of the center of curvature is the envelope of the normals of the given curve,
i.e.
its
evolute.
Before entering upon a more precise discussion of the relations between a given curve and its evolute, we shall explain certain con ventions.
Counting the length of the arc of the given curve in a from a fixed point as origin, and denoting by a the
definite sense
angle between the positive direction of the x axis and the direction of the tangent which corresponds to increasing values of the arc, we shall have tan a y and therefore
=
cos
,
a
On the right the sign -f- should be taken, for if x and s increase simultaneously, the angle a is acute, whereas if one of the varia bles x and s increases as the other decreases, the angle is obtuse 81).
(
Likewise,
tangent, cos
(3
=
if (3
dy/ds.
COS
denote the angle between the y axis and the The two formulae may then be written or
= dx as
where the angle a
is
>
Sin
a
~ = dy
-j
as
counted as in Trigonometry.
On
the other hand, if there be no point of inflection upon the given arc, the positive sense on the curve may be chosen in such a way that s and a increase simultaneously, in which case R ds/da
=
along the arc. Then it is easily seen by examining the two possible cases in an actual figure that the direction of the segment all
X,
CURVATURE
206]
437
starting at the point of the curve and going to the center of curva a -f 7r/2 with the x axis. The coordinates ture makes an angle a 1 f the center of curvature are therefore given by the formulae X L y\) (
=
>
=
xl
x
+ R cos
+
a
(
}
\
=x
R
sin a,
"/
= + (\ a
y
\
+ R cos a,
whence we find cfaj efo/!
In the
= =
cos
ads
R
cos
a da
sin
ads
R
sin
a
tZa
a dR
show that dy l /dx l
these formulas
first place,
= + cos a dR = sin
sin
a cR,
cos
a
=
rf7?
.
cot a,
which
proves that the tangent to the evolute is the normal to the given Moreover curve, as we have already seen.
=
ds\
or dsi
=
dx\
+
dy\
=
dR*
,
Let us suppose for definiteness that the radius
dR.
of curvature constantly increases as we proceed along the curve C and let us choose the positive sense of l to Jl/2 (Fig. 42) from
M
,
motion upon the evolute (D) in such a way that the arc s 1 of
(Z))
increases simultane
Then the preceding formula becomes d$i = dR, whence s t = R + C. It R 1} and we follows that the arc /!/2 = R 2 ously with R.
see that the length of any arc of the evolute equal to the difference between the tivo
is
radii of curvature of the curve C which cor respond to the extremities of that arc.
This property enables us to construct the be involute C mechanically if the evolute
M
(Z>)
FIG. 42
If a string be attached to (D) at an arbitrary point A and rolled around (D) to /2 , thence following the given.
tangent to
M
t
,
the point
string, now held struction may be
taut,
is
M
2
will describe the involute
wound
C
as the
This con
further on round (Z>).
stated as follows
:
On
=
each of the tangents
IM of
=
where I + s the evolute lay off a distance IM const., s being I, values various the length of the arc AI of the evolute. Assigning to the constant in question, an infinite number of involutes may be drawn, all of which are obtainable from any one of them by laying off
constant lengths along the normals.
PLANE CURVES
438 All of these properties
may
[X,
be deduced from the general formula
for the differential of the length of a straight line
dl
=
dar 1
207
COS
(Zcr 2
(D!
COS
2
o>.
segment
(
82)
.
segment is tangent to the curve described by one of its extremities and normal to that described by the other, we may set = 7T/2, and the formula becomes dl da^ = 0. If the TT, w.2 straight line is normal to one of the two curves and Z is constant, = 0, and therefore cos 2 = 0. dl = 0, cos The theorem stated above regarding the arc of the evolute depends essentially upon the assumption that the radius of curvature con If stantly increases (or decreases) along the whole arc considered. this condition is not satisfied, the statement of the theorem must In the first place, if the radius of curvature is a maxi be altered. mum or a minimum at any point, dR = at that point, and hence Such a point is a cusp on dxi = dy l = 0. If the
<0l
o>
the evolute.
(Fig. 43),
for example, the radius at the point
If,
of curvature
is
we
a
M
maximum
shall
have
= IM arc 77, = IM -
arc//!
A-!/!,
72 3/a
,
whence arc /! 7/2
=
-
2 737 t
to the difference 207. Cycloid. circle
which
M
72
M
72 Hence the difference 7 77 and not and arcs the two between 2 Hi
2
is
equal
their sum.
cycloid is the path of a point upon the circumference of a without slipping on a fixed straight line. Let us take the
The
rolls
n Fio. 44 fixed line as the
x axis and locate the origin at a point where the point chosen on When the circle has rolled to the point 7 (Fig. 44)
the circle lies in the x axis.
the point on the circumference which
was
at
has come into the position M,
X,
CURVATURE
207]
439
where the cirenlar arc IM is equal to the segment OI. Let us take the angle between the radii CM and CI as the variable parameter. Then the coordinates of the point
x
M are
= 01 - IP =
- E sin 0,
R(f>
-
y
MP = 1C + CQ = R
- R cos 0,
M
P
on the two lines 01 and IT, respec where and Q are the projections of It is easy to show that these formulae hold for any value of the angle 0. tively. In one complete revolution the point whose path is sought describes the arc If the
OBO\.
motion be continued
= y =
x
sin
R(l
and the slope
cos 0)
dx
,
R(<}>
dy
,
of the tangent
is
0/2, the triangle
d <(>)
d-x
,
d*y
,
infinite
number
find
= Rsin0d0 2 = .Rcos0 d0 2
,
,
seen to be sin
dx
MTC =
the preceding formulae
= R(l cos = R sin d0
dy ~ which shows that the tangent
we obtain an we
indefinitely,
From
of arcs congruent to this one.
at
^
=
cot
i
2
cos
1
M
MTC being
M
the straight line T, since the angle Hence the normal at is the
is
M
isosceles.
MI
straight line through the point of tangency I of the fixed straight line with the moving circle. For the length of the arc of the cycloid we find (Zs 2
if
= E2 d02 [sin 2 +
(1
-
cos 0) 2 ]
= 4R 2 sin 2 - d0 2
the arc be counted in the sense in which
the arc from the point
as origin,
=
we
= 2it, we
ds
= 2R sin
4fi(
~ d0
Hence, counting
have
1-cos-Y 2/
OBO
find that the length of one whole section t is 8R. is therefore 4.R, from the origin to the maximum
OMB
B
length of the arc the length of the arc
,
2i
increases with 0.
it
shall
V Setting
or
2
The and
BM (Fig. 44) 4R cos 0/2. From the triangle MTC the 2R cos 0/2 hence arc BM 2MT. length of the segment MT is
is
;
Again, the area up to the ordinate through
A=
f Jo
M
ydx= JoC E 2 (l-2cos0 +
sin
2 A= /3-0-2sm0 + - 20\ 4 ^)# V"
is
cos 2 0)d0
.
.
/
Hence the area bounded by the whole arc OBOi and the base OOi is,
three times the area of the generating circle.
is 37T.K 2 ,
that
(GALILEO.) The formula for the radius of curvature of a plane curve gives for the cycloid
PLANE CURVES
440
[X,
208
On the other hand, from the triangle MCI, MI = 2R sin 0/2. Hence p = 23fl, and the center of curvature may be found by extending the straight line MI This fact enables us to determine the evolute past I by its own length. easily.
For, consider the circle which is symmetrical to the generating circle with Then the point respect to the point I. where the line cuts this second circle is evidently the center of l = MI. But we have curvature, since
M
MI
M
T
arc
M
=
TtR
-
arc
T
M
or
B
M
BOi, the point
OB
IM =
TtR
-
arc
IM = nE -
= OH - OI = IH = T E
M
BB
already described, with respect to
appearance
.
describes a cycloid which is congruent to the first one, the and the maximum at O. As the point describes the arc describes a second arc B Oi which is symmetrical to the arc .
208. Catenary. The catenary is the plane curve to a suitably chosen set of rectangular axes is
Its
01,
M
Hence the point cusp being at
arc
is
indicated by the arc
MAM
whose equation with respect
in the figure (Fig. 45).
FIG. 45
From
(21)
we
find
4
denote the angle which the tangent for y gives If
X e~ a
X
_
ea
-
ea
The radius
of curvature
is
+
e
cos0
,
MPN,
2
=
a
en
+
given by the formula
y"
But, in the triangle
TM makes with
MP =
a
MNcostf,- hence
COS
= e
"
a2
the x axis, the formula
a -
X,
CURVATURE
209]
It follows that the
the normal
MN.
The length
radius of curvature of the catenary is equal to the length of evolute may be found without difficulty from this fact.
The
of the arc
AM of
the catenary
x
px e a
/
=
y sin
tangent
If .
MT, we
+
_y e
,
dx
=
2
Jo or s
441
given by the formula
is
a I * a -
e
(
e
_ *\ 1
2\
.
/
Pm
a perpendicular be dropped from from the triangle PMm,
P
(Fig. 46)
upon the
find,
Mm = MP sin = s
.
Hence the arc 209.
AM
is
Mm.
equal to the distance
The curve described by the point m (Fig. 45) is called the an involute of the catenary and has a cusp at the point A. The
Tractrix.
tractrix.
It is
length of the tangent to the tractrix is the distance mP. But, in the triangle = a hence the length measured along the tangent to y the tractrix from the point of tangency to the x axis is constant and equal to a.
MPm, mP = The
tractrix
is
mP
;
cos
the only curve which has this property.
M
Moreover, in the triangle TP, radius of curvature and the normal is
Mm is
x
mT = a
2.
Hence the product
a constant for the tractrix.
of the
This property
shared, however, by an infinite number of other plane curves. The coordinates (xi, y\) of the point TO are given by the formulae
x\
x/ a or, setting e
=
x
s
=
cos
a
x
(f>
eu
_
e ~a
tan 0/2, the equations of the tractrix are
xi
(22)
=
a cos
6
+
a log (tan-
As the parameter 8 varies from n/2 to n, the point (x l y v ) describes the Amn, approaching the x axis as asymptote. As varies from n/2 to 0, ,
point (xi yi) describes the arc the y axis. The arcs Amn and ,
and
AM
Am n Am n
,
symmetrical to the
first
with respect to
correspond, respectively, to the arcs
210. Intrinsic equation. Let us try to determine the equation of a plane when the radius of curvature R is given as a function of the arc s, = Let a be the angle between the tangent and the x axis then = ds/da, and therefore ds ds (s).
;
da =
A
AM
of the catenary.
curve
E R
arc
the
first
=
.
.
integration gives
a=
tt
/
ds -
|
/
.
*()
PLANE CURVES
442
and two further integrations give x and y x
XQ
+J
cos
I
a ds
,
[X,
in the
form
y
+ f
y
sin
J *o
*o
a ds
210
.
The curves defined by these equations depend upon the three arbitrary con But if we disregard the position of the curve and think 2/o, and a
stants x
,
.
only of its form, we have in reality merely a single curve. sider the curve C defined by the equations
the general formulae
For,
if
we
first
con
be written in the form
may
X cos X n
=x + = 2/0 +
x
y
si
O-Q <*o+
Y cos
cr
the positive sign be taken. These last formulae define simply a transforma If the negative sign be selected, the curve obtained tion to a new set of axes. A plane curve is axis. is symmetrical to the curve C with respect to the if
X
therefore completely determined, in so far as its form is concerned, if its radius The equation B #(s) is of curvature be known as a function of the arc. More generally, if a relation between called the intrinsic equation of the curve.
any two of the quantities B, s, and a be given, the curve is completely deter mined in form, and the expressions for the coordinates of any point upon it be obtained by simple quadratures. For example, if B be known as a function of a, ds = /(a) da, and then dx = cos af(a) da,
may
dy
=
sin
af(a) da
B =/(), we
first
find
,
If J? is a constant, for instance,
whence x and y may be found by quadratures. these formulae give
x
=
and the required curve from the consideration
XQ is
+ B sin a
y
,
=
B cos a
y
,
This result
a circle of radius B.
is
otherwise evident
of the evolute of the required curve, which must reduce to a single point, since the length of its arc is identically zero. As another example let us try to find a plane curve whose radius of curva
ture
is
proportional to the reciprocal of the arc,
=
r
I
B=
a?/s.
The formulas
give
s
s
ds
*
_
i
*
and then
Although these integrals cannot be evaluated in explicit form, it is easy to gain an idea of the appearance of the curve. As s increases from to + cc, x and y each pass through an infinite number of maxima and minima, and they approach Hence the curve has a spiral form and approaches the same finite limit. asymptotically a certain point on the line y
=
x.
CONTACT OSCULATION
X,2ll]
CONTACT OF PLANE CURVES
III.
211. Order of contact.
C and C
Let
be two plane curves which on C let us assign,
To every point
are tangent at some point A.
according to
443
m
m
any arbitrary law whatever, a point
on C
,
the only
m
requirement being that the point should approach A with m. Taking the
a.TcAm
what amounts
or,
to
Am
the same thing, the chord the principal infinitesimal, let us
what law of correspond
investigate
make the
ence will
as first
order of the infin
mm
with respect to Am as large as possible. Let the two curves be referred to a system of rectangular itesimal
p
or oblique cartesian coordinates, the axis of
common tangent AT.
y not being parallel
to
the
Let
(C)
(C")
Y=F(x)
be the equations of the two curves, respectively, and let (o y ) be the coordinates of the point A. Then the coordinates of will be [a- -f h, f(x -f A)], and those of will be [# k, F(x Q &)], ,
m
m +
+
where k is a function of h which defines the correspondence between the two curves and which approaches zero with h.
=
Am
The principal infinitesimal may be replaced by h ap, for the ratio ap/Am approaches a finite limit different from zero as the
m approaches the point A. Let us now suppose that is an infinitesimal of order r -f 1 with respect to h, for a certain
mm
point
method of correspondence. Then mm denote the angle between the axes, we
mm = [F(x + 1
/(*
jfe)
+ *) + (k-
is
of order 2r
h) cos
be an infinitesimal of order not less than r -f
k
=
h
where a and approaches
/3
-f
a h,* +
l
F(x
,
are functions of h
zero.
The second
+
k)
-f(x
OJ
+
h and F(x
hence each of the differences k
1,
+
+
h
+
k) f(x that is,
+ A) =
which approach
of these formulae
a/t^ 1 )
If 6
2 2 A) sin 6
(k
-f(x +
A)
must
finite limits as
may
= fih
+ A)
;
ph +\
*1
A
be written in
the form
F(x
+ 2.
shall have
.
PLANE CURVES If the expression
F(x
+h+
[X,2ii
ah r+l ) be developed
in
powers of
a,
the terms which contain a form an infinitesimal of order not less
than r
Hence the
-f 1.
& is
difference
=
F(x Q
an infinitesimal whose order
+ h)-f(x + is
h)
not less than r
+1
and may exceed
equal to the distance mn between are cut by a parallel the two points in which the curves C and of order the infinitesimal Since the to the y axis through ra. r
+
But
1.
this difference
A
is
C"
mm
is
increased or else unaltered by replacing
m
by
n, it
follows that
the distance between two corresponding points on the two curves is an infinitesimal of the greatest possible order if the two corresponding
points always ble order is r
on a parallel to the y axis. If this greatest possi 1, the two curves are said to have contact of order r
lie
+
at the point A.
This definition gives rise to several remarks. The y axis whatever not parallel to the tangent A T. Hence, in order to find the order of contact, corresponding points on the two Notes.
was any curves
line
may be
defined to be those in which the curves are cut by any fixed line D which is not parallel to the tan
lines parallel to
The preceding argument shows that gent at their common point. the order of the infinitesimal obtained is independent of the direc tion of
7),
a conclusion which
is
easily verified.
Let
mn
and
mm
m
of the curve C which are not be any two lines through a point Then, from the triangle parallel to the common tangent (Fig. 46).
mm
n,
mm
sin
mn
sin
mm
n
mnm and mm n which is zero or TT, since the chord m n approaches the tangent AT. Hence mm /mn approaches a finite limit different from zero, and mm is an infinitesimal of the same The same reasoning shows that mm cannot be of order as mn. order than higher mn, no matter what construction of this kind is used to determine m from m, for the numerator sin mnm always approaches a finite limit different from zero. The principal infinitesimal used above was the arc Am or the chord Am. We should obtain the same results by taking the arc An of the curve C for the principal infinitesimal, since Am and An are infinitesimals of the same order. As the point
m
approaches the point A, the angles
approach limits neither of
X,
CONTACT
212]
OSCULATION
445
two curves C and C have a contact of order r, the points m on C may be assigned to the points m on C in an infinite number of ways which will make mm an infinitesimal of order r + 1, for that purpose it is sufficient to set k = h + aha + where s^r and where a is a function of h which remains finite for h = 0. On the If
1
l
,
other hand,
if s
<
r,
mm
the order of
cannot exceed
5 -f 1.
212. Analytic method. It follows from the preceding section that the order of contact of two curves C and C is given by evaluating the order of the infinitesimal
with respect to h. Since the two curves are tangent at A, It may happen that others of the (x ). F(x^) /() and F (x ) derivatives are equal at the same point, and we shall suppose for
=
=f
the sake of generality that this
F(x )=f(x o)
(23}
is
true of the first
,
\ I F"(* )
-.,
=/"(*),
but that the next derivatives -F (n +
Applying Taylor
s series to
n derivatives
F (* ) =/ (*), F^(x ) =/>(x +
:
),
and / are unequal. (cc ) (a- ) each of the functions F(x) and f(x), we 1)
(n
1)
find
y
or,
=
subtracting,
F~y=
(24)
where
and
1.2.^(
are infinitesimals.
It follows that the order of contact equal to the order n of the highest derivatives of F(x) and f(x) which are equal for x x The conditions (23), which are due to Lagrange, are the necessary c
of two curves
e
is
=
and
sufficient conditions that
order n
+
x
.
x should be a multiple root of
1 of the equation But the roots of this F(z) =/(
PLANE CURVES
446 curves
C and
C"
;
hence
it
contact of order n have n
may
+
212
[X,
be said that two curves which have
1 coincident points of intersection.
y changes sign with h if n is (24) shows that F and that it if does not n is odd. Hence curves which have even, contact of odd order do not cross, but curves which have contact of
The equation
even order do cross at their point of tangency. It is easy to see why Let us consider for definiteness a curve C
this should be true.
which cuts another curve C in three points near the point A. If be deformed continuously in such a way that each of
the curve
C"
the three points of intersection approaches A, the limiting position of C has contact of the second order with (7, and a figure shows that
the two curves cross at the point A.
This argument
is
evidently
general. If the equations of the two curves are not solved with respect to Y and y, which is the case in general, the ordinary rules for the
calculation of the derivatives in question enable us to write down the necessary conditions that the curves should have contact of
order
n.
culties.
The problem
We
shall
is therefore free from any particular examine only a few special cases which
diffi
arise
First let us suppose that the equations of each of the frequently. curves are given in terms of an auxiliary variable
(X = A f(u),
"
(c)
an(^ A (^o) = and that ^( ) = i- e that the curves are tan gent at a point A whose coordinates are f(t ), Iff (to) w not -
<
(Xo)>
(t(,)-
zero, as
we
shall suppose, the
common tangent
is
not parallel
to
the
and we may obtain the points of the two curves which have the same abscissae by setting u = t. On the other hand, x x is of the first order with respect to t t and we are led to evaluate the with respect to t t In order that the two order of i/f() curves have at least contact of order n, it is necessary and sufficient that we should have y
axis,
,
.
<()
(25)
,K*
)
=
<X*o) ,
*
(
=*
(
,
,
*
(*o)
-
*
Co)
,
and the order of contact will not exceed n if the next derivatives (n + 1) and (n + 1) (*o) are unequal. (*o) consider the case where the curve C is represented by the Again, two equations 4>
(26)
*=/(0,
y
=
X,
CONTACT
212]
OSCULATION
447
and the curve C by the single equation F(x, y} = 0. This case may be reduced to the preceding by replacing x in F(x, y) by /() and considering the implicit function y ^(t) defined by the equation
KO] =
F[/(0,
(27)
Then the curve C
1
is
also represented
(28)
y
*=/(*)>
In order that the curves C and
C
o.
by two equations of the form
=
^CO-
should have contact of order n at
A which
corresponds to a value tn of the parameter, it is But the necessary that the conditions (25) should be satisfied. are function successive derivatives of the implicit given by the i/r() a point
equations
?+ 2
[/ (29)
Hence necessary conditions by t
for contact of order
n will be obtained
inserting in these equations the relations
=t
The
,
x ==y(
),
i/^o)
==
resulting conditions
<(^o
may
be expressed as follows
:
Let
e
^wo given curves will have at least contact of order n if and
only if (30)
= F(
The
0,
F(*o)
=
0,
..-,
R")(^)
= 0.
=
are the values of t which cor roots of the equation F() two given curves. Hence to of intersection of the respond points
=
the preceding conditions amount to saying that t t is a multiple root of order n, i.e. that the two curves have n + 1 coincident points of intersection.
PL AXE CURVES
Given a fixed curve C and another curve
213. Osculating curves.
C which depends upon n
it is
way
1
-f-
F(x,
(31)
[X,213
parameters
y, a, b,
c,
,
a, b,
t)
=
c,
,
I,
0,
+
1 parameters in such a possible in general to choose these n that C and C shall have contact of order n at any preassigned
=
For, let C be given by the equations x =/(), y point of C. Then the conditions that the curves C and C should have contact of order
n
where
at the point
=
t
tQ
are given by the equations (30),
where
=
F(0 If
t
be given, these n
+1
equations determine in general the n +1 The curve C obtained in this way is
-, parameters a, b, c, called an osculating curve to the curve C. /.
Let us apply this theory to the simpler classes of curves. The equation of a straight line y = ax -+ b depends upon the two param eters a and b the corresponding osculating straight lines will have ;
contact of the
first order. If y =f(x) is the equation of the curve C, the parameters a and b must satisfy the two equations
A x = ax o)
hence the osculating line
+
f (x ) =
b,
a
;
we should
the ordinary tangent, as
is
expect.
The equation (32)
of a circle (x
-
2
a)
+
(y
- by -
R*
=
depends upon the three parameters a, b, and R hence the corre sponding osculating circles will have contact of the second order. ;
Let y
= f(x)
be the equation of the given curve
and
C
;
we
shall obtain
R by
requiring that the circle should meet this curve in three coincident points. This gives, besides the
the correct values of
a, b,
equation (32), the two equations (33)
x
-a + (y- b}y = 0,
1+
y
2
+
(y
V)y"
=
0.
The values
of a and b found from the equations (33) are precisely the coordinates of the center of curvature ( 205) hence the oscu ;
lating circle coincides
with the
tact is in general of order two, circle
of curvature. Since the con may conclude that in general the
circle
we
of curvature of a plane curve crosses the curve at their point
of tangency.
CONTACT
X,2i:<]
OSCULATION
449
All the above results might have been foreseen a priori. since the coordinates of the center of curvature
For,
depend only on
any two curves which have contact of the second y\ and But the center of curva order have the same center of curvature.
x, y,
y",
ture of the osculating circle is evidently the center of that circle itself; hence the circle of curvature must coincide with the oscu lating circle.
On
the other hand, let us consider two circles of The difference between their radii,
curvature near each other.
which is
equal to the arc of the evolute between the two centers, hence one of greater than the distance between the centers is
;
the two circles must
happen
wholly inside the other, which could not both of them lay wholly on one side of the curve C in
if
lie
the neighborhood of the point of contact. cross the curve C.
It follows
that they
There are, however, on any plane curve, in general, certain points which the osculating circle does not cross the curve this excep tion to the rule is, in fact, typical. Given a curve C which depends upon n + 1 parameters, we may add to the n + 1 equations (30) the
at
;
new equation
provided that we regard t as one of the unknown quantities and determine it at the same time that we determine the parameters I. It follows that there are, in general, on any plane a, b, c, curve C, a certain number of points at which the order of con tact with the osculating curve C is n -f 1. For example, there are usually points at which the tangent has contact of the second order ,
1
;
=
In order to find these are the points of inflection, for which 0. the points at which the osculating circle has contact of the third y"
order, the last of equations (33)
must be differentiated again, which
gives
or finally, eliminating y
b,
(i+yv-*yY"-o.
(34)
The points which satisfy this last condition are those for which dR/dx = 0, i.e. those at which the radius of curvature is a maxi mum or a minimum. On the ellipse, for example, these points are the vertices on the cycloid they are the points at which the tan ;
gent
is
parallel to the base.
PLANE CURVES
450
[X,214
214. Osculating curves as limiting curves. It is evident that an osculating curve may be thought of as the limiting position of a which meets the fixed curve C in n curve 1 points near a fixed
+
C"
A
which is the limiting position of each of the points of intersection. Let us consider for definiteness a family of curves which depends upon three parameters a, b, and c, and let t n + Aj t + hi, and t + 7t 3 be three values of t near t The curve C which meets the curve C in the three corresponding points is
point
of C,
.
,
given by the three equations (35)
F(t
+
AO
=
,
+
F(t
A2 )
=
F(t
,
+
A,)
=
.
Subtracting the first of these equations from each of the others and applying the law of the mean to each of the differences obtained,
we
find the equivalent system
=
(36)
F(^
+
where & t
lies
between h l and
A 1)
0,
F
+
(f
7/,
2
,
*i)
=
F
0,
(*
= 0,
+fc 2 )
and & 2 between A x and h z
Again,
.
subtracting the second of these equations from the third and apply ing the law of the mean, we find a third system equivalent to either of the preceding,
(37)
Fft
where ^
lies
zero, &!,
k2
,
+ AO-O,
F
+
(f
*i)
=
0,
F"ft,
+
/,)
=
0,
between 7^ and 2 As u 2 and h s all approach and ^ also all approach zero, and the preceding equa .
7?
7?
,
tions become, in the limit,
which are the very equations which determine the osculating curve. The same argument applies for any number of parameters whatever. Indeed, we might define the osculating curve to be the limiting position of a curve C which is tangent to C at p points and cuts C at q other points, where 2p + q = n + 1, as all these p + q points approach coincidence. For instance, the osculating
circle is the limiting position of a
which cuts the given curve C
in three neighboring points. It also the limiting position of a circle which is tangent to C and which cuts C at another point whose distance from the point of
circle is
tangency
is infinitesimal.
Let us consider for a moment the latter
property, which is easily verified. Let us take the given point on C as the origin, the tangent at that point as the x axis, and the direction of the normal toward the
EXERCISES
X, Exs.]
451
center of curvature as the positive direction of the y axis. oiigin,
=
y
0.
Hence R
=
and therefore, by Taylor
1/y",
At the s series,
1
where e approaches zero with x. It fol lows that R is the limit of the expres 2 = OP 2 /( 2MP ) as the P oint sion /(2y)
M approaches the origin. hand,
R
let
l
On
the other
be the radius of the circle
is tangent to the x axis at the and which passes through M. origin Then we shall have
Ci which
OP =
Fia
Mm = MP(2R - MP) l
,
or ;2
OP 2MP hence the limit of the radius
M_P. ^
-tti
7
a
2
is really
equal to the radius of
R
curvature
EXERCISES Apply the general formulae
1.
bola 2.
;
to find the evolute of
an
ellipse
;
of an hyper
of a parabola.
Show
that the radius of curvature of a conic
is
proportional to the cube
of the segment of the normal between its points of intersection with the curve and with an axis of symmetry. 3.
Show
that the radius of curvature of the parabola
segment of the normal between the curve and the 4. Let F and F be the normal at that point, and
is
equal to twice the
M
MN
an ellipse, a point on the ellipse, the point of intersection of that normal and the Erect a perpendicular to at jV, meeting foci of
N the
major axis of the ellipse. at K. At erect a perpendicular
K
is
directrix.
NK
KO
to
MF, meeting
the center of curvature of the ellipse at the point
MN MN at
MF
0.
Show
that
M.
For the extremities of the major axis the preceding construction becomes Let .4 CM/ be the major axis and BO B the minor axis of the ellipse. On the segments OA and OB construct the rectangle OA EB. From E let fall a perpendicular on AB, meeting the major and minor axes at C and D, respec Show that C and D are the centers of curvature of the ellipse for the tively. points A and B, respectively. 5.
illusory.
6.
Show
given spiral.
that the evolute of the spiral p
=
ae mu
>
is
a spiral congruent to the
PLANE CURVES
452
[X, Exs.
7. The path of any point on the circumference of a circle which rolls with out slipping along another (fixed) circle is called an epicycloid or an hypocycloid. Show that the evolute of any such curve is another curve of the same kind. 8. Let AB be an arc of a curve upon which there are no singular points and no points of inflection. At each point m of this arc lay off from the point m along the normal at m a given constant length I in each direction. Let wtj and
m
m
As the point describes the arc AB, 2 be the extremities of these segments. the points mi and 2 will describe two corresponding arcs AiBi and t B2 Y Derive the formulae Si = S and 2 are the 18, S 2 = S + W, where S, i is the angle 2 2 respectively, and where lengths of the arcs AB, A\B\, and between the normals at the points and B. It is supposed that the arc -Ai-Bi
m
A
<S
,
A B A
lies
on the same side of
AB as the
.
<S
,
evolute,
and that
it
does not meet the evolute.
[Licence, Paris, July, 1879.]
M
9. Determine a curve such that the radius of curvatures p at any point measured from any fixed point A on the curve and the length of the arc s 2 2 where a is a given constant length. satisfy the equation as = p + a
AM ,
[Licence, Paris, July, 1883.] 10.
A
at 0.
C
be a given curve of the third degree which has a double point revolves about the point O, meeting the curve C in right angle
Let
two variable points
MON
M and N.
Determine the envelope of the straight line MN. = x 3 and z3 + y s = f*xy.
In particular, solve the problem for each of the curves Xy 2
[Licence, Bordeaux, July, 1885.] 11.
Find the points x
at
which the curve represented by the equations
a (nw
sin u),
y
=
a (n
cos w)
has contact of higher order than the second with the osculating
circle.
[Licence, Grenoble, July, 1885.]
m
12. Let m, mi , and Find 2 be three neighboring points on a plane curve. the limit approached by the radius of the circle circumscribed about the triangle formed by the tangents at these three points as the points approach coincidence. 13. If the evolute of a plane curve without points of inflection is a closed curve, the total length of the evolute is equal to twice the difference between the sum of the maximum radii of curvature and the sum of the minimum radii of
curvature of the given curve. 14.
At each point
with the normal.
of a curve lay off a constant segment at a constant angle that the locus of the extremity of this segment is a
Show
curve whose normal passes through the center of curvature of the given curve. 15. Let r be the length of the radius vector from a fixed pole to any point of a plane curve, and p the perpendicular distance from the pole to the tangent. Derive the formula R = rdr/dp, where R is the radius of curvature. 16. Show that the locus of the foci of the parabolas which have contact of the second order with a given curve at a fixed point is a circle.
Find the locus of the centers of the ellipses whose axes have a fixed direc and which have contact of the second order at a fixed point with a given
17. tion,
curve.
CHAPTER XI SKEW CURVES I.
OSCULATING PLANE
M T be the tangent at a point M MT and a point M of limiting position as the point M
Let
215. Definition and equation.
of a given
F near
skew curve
A
F.
M
plane through
in general approaches a If it does, the limiting position of the approaches the point M. plane is called the osculatiny plane to the curve F at the point M.
We
shall proceed to find its equation.
Let
*=A9
(i)
y
=
*-rtO
*(0
be the equations of the curve F in terms of a parameter t, and let t h be the values of t which correspond to the points and and
M
+
Then the equation
&f, respectively.
A(X where the
of the plane
MTM
-*)+ B(Y-y) + C(Z- 3) =
coefficients
^4,
B,
1
is
0,
and C must satisfy the two relations
(2)
(3)
Expanding f(t
+
h),
<j>(t
+
A) and
\j/(t
-f A)
by Taylor
s
series,
the
equation (3) becomes
After multiplying by 7^, let us subtract from this equation the equa and then divide both sides of the resulting equation by
tion (2),
A 2/2.
Doing
so,
we
find a
system equivalent to
<)
where d,
e2
,
and
c3
+
a]
+
(2)
^[^"(0
approach zero with
A.
+
and
cs
]
(3)
=
^/(0 +
**"(0
453
.
In the limit as A
approaches zero the second of these equations becomes (4)
0,
:
+ cy (o = o.
SKEW CURVES
454 Hence the equation
of the osculating plane
[XI,
is
A(X-x) + B(Y-y-) + C(Z-z) =
(5)
where A, B, and C
Q,
satisfy the relations
(Adx + Bdy + C dz =
Co)
215
<
(Ad
2
x
+ Bd
2
y
+
Cd*z
,
= 0.
The coefficients A, B, and C may be eliminated from (5) and (6), and the equation of the osculating plane may be written in the form
X -x Y
Among is
plane
tangent, and
dy
dz
d*
d*z
which pass through the tangent, the osculating which the curve lies nearest near the point of tan-
+
us consider any other plane through the be the function obtained by substituting F(f) + h) for X, F, Z, respectively, in the left-hand h), \li(t this, let
let
f(t h), $(t -f side of the equation (5), which tion of the new tangent plane.
where
77
M
= 0.
dx d?x the planes
the one
To show
gency.
Z -z
y
approaches zero with
we shall now assume Then we shall have
The
h.
M
to be the
equa
distance from any second
of F near to this plane is therefore an infinitesimal of point the second order; and, since F(t) has the same sign for all sufficiently small values of h, it is clear that the given curve lies wholly on one
side of the tangent plane considered, near the point of tangency. These results do not hold for the osculating plane, however. For
+
+
that plane, Af" coordinates of a point of order.
Doing
so,
=
B"