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> x always, the series 2a H will converge or diverge according as we have is ' (0) exists** and therefore 4
(
<.
1
5
l
for all sufficiently large x s
we have command
With Ermakoffs test and Cauchy's integral test, the most important tests for our present series.
over
General remarks on the theory of the convergence and divergence of series of positive terms.
41.
whole of the 19 th century was required to estabconvergence tests set forth in the preceding sections and to elucidate their meaning. It was not till the end of that century, and in particular by Pringsheini's investigations, that the fundamental questions were brought to a satisfactory conclusion. By these researches, which covered an extiemely extensive field, a scries of questions were also solved, which were only timidly approached before his time, although now they appear to us so simple and transparent that it seems almost inconceivable that they should have ever presented any 20 still more so, that they should have been answered m a comdifficulty pletely erroneous manner. How great a distance had to be traversed before this point could be reached is clear if we reflect that Eider Practically the
lish the
,
all about questions of convergence; when a he would attribute to it, without any hesitation, the value of the expression which gave rise to the series 30 Lagrange in 17 70 31 was still of the opinion that a series represents a definite 32 To refute the latter value, provided only that its terms decrease to O
never troubled himself at series occurred,
.
.
= 0, if we interpret log^x to mean e*. This also holds for As a curiosity, we may mention that, as late as 1885 and 1889, memoirs were published with the object of demonstrating- the existence 28
29
vergent series 80
Thus
J
c * for
-"-^" 1
which
cn
in all seriousness
did not tend to a limit! (Cf. 159,
he deduced from
-
-
1
x
=
1 4-
x
-f-
several of con3.)
x2 -f-
,
that
and
l=l o Cf the
first
few paragraphs of
81
V. CEuvres, Vol.
ia
In this,
59.
3, p. 61.
however, some traces of a sense for convergence
may be
seen,
41.
General remarks on series ot positive terms. the
to
assumption expressly by referring
fact
time already
that
(at
299
appears to us at present known) of the divergence of J? superfluous, and many other presumptions and attempts at proof current in previous times are in the same case. Their interest is therefore for the most part historical. A few of the questions raised, however, whether answeied in the affirmative or negative, remain of well
,
A
for us to give a rapid account of them. conproportion of these are indeed of a type to which anyone occupies himself much with series is naturally led.
interest
sufficient
siderable
who
The source
questions which
the
of all
we propose
discuss
to
Those which are necessary
resides in the inadequacy of the ciiteria.
and
sufficient for convergence (the main criterion 81) are of so general a nature, that in particular cases the convergence can only rarely be ascertained by their means. All our remaining tests (comparison tests or trans-
formations of comparison tests) were sufficient criteria only, and they only enabled us to recognise as convergent series which converge at least
The
as rapidly as the comparison series employed. arises
Does a
1.
This question
which converges less rapidly than any other/ already answered, in the negative, by the theorem
series exist is
In fact,
4.
175,
question at once
:
when Zc n converges,
so docs
2c n
'
=
2 -r -* though, V r
2c n
obviously, less rapidly than
The
who
question
is
takes the series
the ratio c n
'' = VV
as c :c n n
,
'
= r_
n-l
l
> 0.
answered almost more simply by
~c n n-1
'
= ^(l/rn _
+ Wn
*
V
1
r n ).
Since
The accented
0.
/.
cn
Hadamard 2
= rn _
1
*',
rw ,
conver-
series
ges less rapidly than the unaccented series. The next question is equally easy to solve: 2. Does a series exist which diverges less rapidly than any other? Here again, the theorem of Abel-Dini 173 shows us that when 2dn diverges, so does
'
In fact as d n
the negative. for
2dn = 2j~>
each given divergent
:
dn
'
series,
an d hence the answer has to be in
= Dn
*
-|-
oo, the theorem
another whose divergence
provides, not so
is
rapid.
These
show
circumstances,
together
with
our
preliminary
with
all series.
remarks,
that 3.
No comparison
test
can be
Closely connected with this, and also answered, by Abel**: 83
Actti ).
f.
effective
we have
the following question, raised*
mathematica, Vol. 18, p. 319. 1894. reine u. angew. Math., Vol. 3, p. 80.
d.
1828
178.
300
Series of positive terms.
Chapter IX. 4.
Can we
a/) P~ r n &n n aw
b)
*
find positive numbers pn , such that, simultaneously, I tj A 3 ., f convergence \ \ are su'ff t cient conditions for \ T } a J divergence J .
.
.
.
'I
^ >
.
Vy;y possible series of positive terms? It again follows from the theorem of Abel-Dini that this
o/
the case.
In fact,
diverges,
ily
if
we
put a n
= --, a > 0, r
and hence so does
But, for the latter,
pn
an
/
an
the series
^0n
is
not
necessar-
/I
'^~
>
where
sn
=a
+ #n
L -}-
.
=~^0.
The object of the comparison tests was, to some extent, the construction of the widest possible conditions sufficient for the determination of the convergence or divergence of a series. Conversely, it might be required to construct the narrowest possible conditions necessary for the convergence or divergence of a series. The only information we is necessary for have so far gathered on this subject is that a n > It will at once occur to us to ask: convergence. tend to zero with 5. Must the terms a n of a convergent series any particular rapidity? It was shown by Pringsheim** that this is not the case. However slowly the numbers p n may tend to -{- oo, we can invariably construct conveigent series 2 c n for which
KPn C n= + 00
'
Indeed every convergent series -Tc /, by a suitable rearrangement, will 36 cn to support this statement produce a series Proof. We assume given the numbers p n increasing to -|- oo, and the convergent scries 2c n'. Let us choose the indices x n 2 ..., odd and such that n 7
2
.
,
,
,
and
.
,
.
.
let '
us write c n
= cj
r -i,
in their original c/, But c n '. arrangement of c4
y
...
filling in
order.
Pn
whenever n becomes equal
to
C
the remaining c 's with the terms n
The
series
2 cn
is
obviously a re-
n>*
one of the indices n 9
.
Accordingly, as
asserted,
The underlying or a sequence of the
fact in this connection is
form (pn
to
c n)
bears
no
simply that the behaviour
essential relation to that of
Math. Annalen, Vol. 35, p 344. 1890 Theorem 82, 3, which takes into account a sort of decrease on the average of the terms a n 86
Cf.
,
General remarks on series of positive terms.
41.
2 cn
301
the sequence of partial sums of this though not the former, may be fundamentally altered by a rearrangement of its terms. is necessary 6. Similarly, no condition of the form lim p n dn
the
series
with
e.
i.
since
series,
the
latter,
>
2 dn
however rapidly the positive numbers p n for divergence of 37 to On the contrary, every divergent series 2d n ', increase -f-oo may its tend terms to becomes, on being suitably rearranged, 0, provided Q. a series dn (still divergent, of course) for which \imp n d n the
,
.
2
The proof is easily deduced on the same lines as the The following question goes somewhat further: Does a
7.
comparison tests exist which is sufficient for Given a number of convergent series
scale of
More
cases?
all
preceding.
precisely:
v
-^ r
^y Crn
(i)
n
>
each of which converges
(2) '
'
>
^y Crn
(*) >
'
'
less rapidly than the preceding, with e. g.
>
()
+ ao,
for fixed k.
(The logarithmic scale affords an example of such sible
7s it posseries.) construct a series converging less rapidly than any of the given The answer is in the affirmative 38 The actual construction
to
series?
.
With a
of such a series is indeed not difficult.
n 19 w 3
indices
is
itself
,
...,
w ,..., the
of the kind required.
large that the series
if
we
denote by
2 c^ \
for every
k
n I> n l
w^n >w fc
r^
We
fc
>
.1
need only choose these indices so
the remainder,
we have
,
^. Wt i* ->
-
suitable choice of the
series
fe
rn
^
after the
n th
(3) <[ -^ with c n 2
^^ <
*
term, of
(l) ^> 2 cn
-^ o ^ -^
M W
M "
y (3 ^ r n
02
M W
r (3) C tt
n
w
C^'<~
n
c^ >2cn w
, (
A|
The it
series c n is certainly convergent, for each successive portion of (fc) is certainly less than the belonging to one of the series -2*cn 87
J. is
Pringsheim, loc.
cit. p.
357
For the logarithmic scale, this was shewn by P. du Bois-Reymond (J. f. reine u. ungew. Math., Vol. 76, p. 88. 1873). The above extended solution due to /. Iladamard (Acta math., Vol. 18, p. 325. 1894). 88
302 remainder of
<
Series of positive terms.
Chapter IX.
(k
^
this
= 2,
3, ...).
with
starting
series,
On
> n q (q > k)
n
that was required. more slowly than all the
all
We
8.
term,
i.
e
we have
+ OOJ obviously
^>
2*
This proves
*.
In particular, there are series series of our logarithmic scale 39 .
converging
show, quite as simply, that, given a number of di(k 1, 2, ..., each diverging less rapidly than \ &
may
vergent series
initial
the other hand, for every fixed k,
2--* in fact for
same
the
=
2dn
^
the preceding, with, specifically, d n diverging divergent series
2
+1)
-f-d n
less
(fc)
say, there are
+0,
rapidly than
every
always one of the
2 d^
series
'.
All the above remarks bring us near to the question whether and what extent the terms of convergent series are fundamentally dishn guishable from those of divergent series. In consequence of 7. and 8., we shall no longer be surprised at the observation of Stieltjesi to
an arbitrary monotone descending seDenoting by (e l , e 3 .) a with limit 0, convergent series 2c n and a divergent series quence 9.
2dn
,
.
.
can always be specif ied, such that
monotonely, p n
= --
>
-f-
cn
= en dn
==
n=l also
divergent
*0
divergent.
By
if
series
ff
whose partial sums are the numbers p n the theorem of Abel-Dini, the series
is
The
oo monotonely.
en
In fact,
.
^
is
,
therefore
v P+I""P + ii i
But the series
2cz==2s d^~y]\\
)
"^""^
**
"Fn
4-
is
1
convergent by 131.
The
following remark
is
only a re-statement in other words
of
the above: 10. However slowly *+oo, there is a convergent series 2c n n and a divergent series 2 dn for which ^n pn cn two the remarks due to Pringsheim, given in 5. In this respect, and 6., may be formulated even more forcibly as follows:
=
89
The
missing- initial
replaced by unity.
Urms
of these series
.
may be assumed
to
be each
General remarks on series of positive terms.
41.
303
However rapidly 2cn may converge, there are always divergent indeed divergent series with monotonely diminishing terms of limit 0, for which 1J.
series,
2 dn
Thus
must have an
However
rapidly
2 dn
may
We have
2 cn
of terms
essentially smaller
Conversely:
.
diverge,
2cn
are always convergent series
of the
number
infinite
than the corresponding terms of
provided only dn
which
for
there
scries
2 dn
= +00.
lim^
Here a
only to prove the former statement.
*0,
form
V
d =-
4-
-4-
-1-
,1
I-
,1
-
-\~
-
-
,1
,1
,
4-
-4-
is of the required kind, if the increasing sequence of indices n ... I9 w 3 be chosen suitably and the successive groups of equal terms contain n1 ), (n 3 ~ w a ), ... terms. In fact, in order that respectively n 19 (n. this scries may diverge, it is sufficient to choose the number of terms in each group so large that their sum 1, and in order that the sebe monotone, it is sufficient to choose quence of terms in the series so that r c nk > n k ^ l 1, 2, ; n large 1) as is always _ (ft ,
>
nfc
possible, since c n it
follows that
*0.
As
= 0,
km n
<
=
.
. .
=
,
n/
j
has the value
the ratio
-
n
for
=n
k
,
as required.
In the preceding remarks we have considered only convergence or divergence per se. It might be hoped that wiih narrower requirements, e. g. that the terms of the series should diminish monotonely,
a correspondingly greater amount of information could be obtained. Thus, as we have seen, for a convergent series 2c n whose terms diminish monotonely, we hiwe nc n *0. Can more than this be asserted?
The answer 12. there
for
is
in the negative (cf. Rem. 5): the positive numbers
However slowly
are
always convergent
series
of
+
oo, p n may increase to monotonely diminishing terms
which
n Pn
cn
not only does not tend to 0, but has
+00
for
40 upper limit
.
40 Pnngsheim, loc. cit. In particular it was much discussed whether for convergent series of positive terms* diminishing- monotonrly, the expression nlogn>cn must -*(); the opinion was held by many, as late as I860, that n log n*c n * was necessary for convergence.
Chapter IX.
304
The
is
proof
again
Series of positive terms.
Choose
quite easy.
n
indices
<
n.2
<
such that
(*=1,2,.
P*,>** and write
=
c
= ...==
ca
=
c ni
1
-^
"'
,
>M'YC i
The groups 2
2*
>*
of terms here indicated
On
was
the other hand, for each
required,
limn-pn .c n These remarks may
13.
possible
2c
the series
of
2"
converge.
so that, as
sum
to the
>
contribute successively
directions.
so that this series
= w,
n
we have
= +00. be multiplied and extended in
easily
They make
,
less than
it
clear
that
it
all
quite useless to
is
of a boundary between was suggested by P. du BoisReymond. The notion involved is of course vague at the outset. But in whatever manner we may choose to render it precise, it will never correspond to the actual circumstances. We may illustrate this on the 41 following lines, which obviously suggest themselves
attempt to introduce convergent and
anything
of
divergent series,
the nature
as
.
a)
to
no
As long
as the terms of the series -S"c n
restriction
of assuming
all
-
lim -^ an
(excepting that of being
> 0),
and
2 dn
the ratio
aie subjected
~
is
capable
possible values, as besides the inevitable relation
=
we may
also have
lim-^ an
= +00.
The
polygonal graphs by which the two sequences (cn) and (dn ) may be represented, in accordance with 7, 6, can therefore intersect at an indefinite number of points (which may grow more and more numerous,
to an arbitrary extent). 41
A
detailed
and careful discussion of all the questions belonging to the submentioned on p. 2, and also in his writings Munch. Ber. Vol. 26 (1890) and 27 (181)7),
ject will be found in Pringsheim*s work in the Math. Ann. Vol. 35 and in the
to
which we have repeatedly
referred.
42.
Systematization of the general theory of convergence.
305
b) By our remark 11, this remains true when the two sequences and (cn ) (dn ) are both monotone, in which case the graphs above referred It is therefore certainly to are both monotone descending polygonal lines. not possible to draw a line stretching to the right, with the property that
every sequence of type (c n ) has a graph, no part of which lies above the line in question, and every sequence of type (dn ) a graph, no part of which lies even if the two graphs are monotone and are considered below this line,
only from some point situated at a sufficiently great distance to the right.
Notes 11 and 12 suggest the question whether the statements
14.
made remain unaltered if the terms of the constructed scries 2 cn and 27 dn are not merely simply monotone as above, but fully monotone
there
in the sense of p. 263.
This question has been answered in the affirmative
by H. Hahn^.
42.
Systematization of the general theory of convergence.
The element
of chance inherent in the theory of convergence as rise to various attempts to systematize the criteria
developed so far gave
from more general points of view. The first extensive attempts of this made by P. du Bois-Reymond 43 but were by no means brought to a conclusion by him. A. Pringsheim** has been the first to accomplish this, in a manner satisfactory both from a theoretical and a practical stand-
kind were
,
We
point.
propose to give a short account of the leading features of the
developments due to him
and the
.
All the criteria set forth in these chapters have been comparison tests, their common source is to be found in the two comparison tests of first
and second kinds, 157 and 158.
=
(I)
is
45
s
e>
The
an
former, namely
^dn
:
V,
undoubtedly the simplest and most natural test imaginable; form
not so
that of the second kind, given originally in the
42 f.
//.
Math.
Hahn, Dber Reihcn mit monoton abnehmenden Ghedern, Monatsheft
u. Physik, Vol. 33, pp.
43
J. f. d.
44
Math. We have
45
121134,
1923.
reine u. angew. Math. Vol. 76, p. 61. Ann. Vol. 35, pp. 297394. 1890.
1873.
all the more reason for dispensing with details in this connexion, seeing Pringsheim's researches have been developed by the author himself in a very complete, detailed, and readily accessible form.
306
Chapter IX.
Series of positive terms.
In considering the ratio of two successive terms of a series we are already going beyond what is directly provided by the series itself. might therefore in the first instance endeavour to construct further
We
types of tests by means of other combinations of two or more terms This procedure has, however, not yielded any criterion of the series. of interest in the study of general types of series. If we restrict our consideration to the ratio of two terms, it is possible to assign a number of other forms to the criterion of the second kind; e. g. the inequalities may be multiplied by the positive still
We
shall return to without altering their significance. these for relatively unimportant transformations, Except must regard (I) and (II) as the fundamental forms of all
factors a n or cn this point later.
however,
we
46 All conceivable special comconvergence and divergence will tests be obtained by introducing in (1) and (II) all conceivparison able convergent and divergent series, and, if necessary, carrying ovit transformations of the kind just indicated.
criteria of
.
The task of systematizing the general theory of convergence will accordingly involve above all that of providing a general survey of all conceivable convergent and divergent series. This problem of course cannot be solved in a literal sense, since the behaviour of every series would be determined thereby. can reduce it endeavour to to factors in themselves easier to only survey and therefore not appearing so urgently to require further treatment.
We
and this is essentially the starting point of his that a systematization of the general theory of convergence carried out when we assume as given the totality of all
Pringsheim shows investigations
can be fully monotone sequences of (positive] numbers increasing
Such a sequence
will
principle,
the
problem
+00.
be denoted by (pj; thus
< Po ^ Pi ^ P2 ^ In
to
solved
is
Pn -*
and
by the
two
+ following
simple
remarks: a)
series
Every divergent
2dn *zpo + (Pi
n=0 (each
in one
type (p n ).
is
JP
)
2 dn H
is expressible in
h (Pn
and only one way) in terms
of
the
form
Pn-l) H a suitable sequence of
Also, every series of this form is divergent.
46 Thus since (as seen in 16O, 1,2) (II) ultimately from (1) that all the rest follows.
is
a consequence of
(I)
-
it
V
^
/ C c= /
I
(each in
type
is
*
1
1
!
1
I
Q
{
-
expressible in the
-
+./ ^\P n I
*
form \ I
+ ,
*
Pn+l)^
one and only one way) in terms of a suitable sequence of Also, every series of this form is convergent.
w ).
(
2c n
47
Mi/ - + ^ - - \i+ P,)^ Pj \P
~\P
H
series
Every convergent
b)
307
Systematization of the general tneory of convergence.
42
when
In fact,
these statements have been established,
only to substitute, in the
two comparison
tests (I)
and
we have
(II),
respectively for c n and d n > to obtain in principle all conceivable tests of the first and second kinds: All particular criteria must necessarily follow by more or less obvious transformation from the tests so obtained; for this very reason, the former can never present anything fundamentally new. They become of considerable importance, howthat they give deeper insight into the connexion between the ever, and state the latter in a coherent form, and also apply criteria various
m
them in practice. Herein lies the chief value of the whole method. It would accordingly be well worth our while to describe the details of the construction of special criteria exactly; but for the reasons given, shall abide by our plan of giving only a brief account.
we
The
and b) must be regarded as undoubtedly 180. for convergent and divergent series. But we can obviously replace them by many other forms, thereby altering the outward form of the criteria in various ways. For instance, by the theorem of Abel-Dini 173, 1.
the
typical
forms
a)
imaginable forms
simplest
diverge with
2(p n
/> n -i)>
while at the
same
time,
by Pringsheim's
theorem 174,
2~n-~^-
and
2 -",,
With a few restrictions of little importance, all for Q > 0. series are also expressible in one of these and convergent divergent
converge
new
forms. 2.
Since the only condition to be satisfied by the numbers p 9 forms of divergent and convergent series which we are
in the typical 7
48
them
all from some stage on. oofs of these two statements are so easy that
Unless the terms are
The
pi
further.
we need
not
go
into
308
Chapter IX. Series of positive terms.
considering, is that they are to increase monotonely to f- oo, we may of course write \ogp n iog 2 /> n ... or generally F (p n ) instead of /> n where and increasing monotonely (x) denotes any function defined for x ,
,
,
F
>
+0
(in the strict sense) to
with
This again leads to
x.
criteria
which,
though not essentially new, are formally so when the /> w 's arc specially chosen. It is easy to verify that the first named types of series diverge or converge more and more slowly, as />-> oo more and more slowly;
+
we therefore by replacing p n successively e. g. by logp n Iog 2 /> n 4a criteria The n scales case p n obtain a means of constructing of on of its account the peculiar simplicity; naturally calls for consideration ,
,
.
.
.
,
=
.
development of the ideas indicated above for 37 and 38. the main contents of
particular case forms
this
3. A further advantage of this method is due to the fact that one and same sequence (p n ) will serve to represent both a divergent and a convergent series. The criteria therefore naturally occur in pairs. E. g. every comparison test of the first kind may be deduced from the pair of tests:
the
<; 'Pn
~
PnPn-l n
= .
and similarly
The
4.
for other typical
right
hand
* Pn-l A.-I
forms of
sides can
series.
be combined to form a single
disjunctive
we
introduce a modification, arbitrary in character in so far as it is not necessarily suggested by the general trend of ideas, but otherwise of a simple nature. We see at once, for instance, that the series criterion, if
_
Pn
n
>
^
] 1 and diverge when a For the first of these scries when a the proof has just been given; and the second has all its terms less than the first if a 1, while if a 1, and hence for all a 2> 1, it is immediately
converge
.
=
>
seen to be divergent. The pair of criteria set up in be replaced by the following disjunctive criterion:
49
The
usual passage from
arbitrary step, of course.
Between instance
e.
g.
p n and
e**^ which
3.
may
accordingly
is again quite an . . , step natural, however. could easily introduce intermediary stages, for
pn direct to log pn Iog 2 pn Theorems 77 and 175, 2 render the
log/> n ,
,
we
,
.
in fact less rapidly than /> n> yet more rapidly any fixed positive power of p n however small its exponent, than every fixed positive power of log p n however large its exponent.
increases less rapidly than ,
,
Systcmatization of the general theory of convergence.
42.
and, in
essentials 50 , also by:
all
-i
.,
with
(>1 {>! ^
remarkable that in the
is
It is
all.
-**-*and
(cc~
p )
p n -j), and hence
2(p n
for arbitrary
H
GC
+
should be monotone. In
sufficient that (p n )
ded, the convergence of
:
criteria of convergence arising through + oo is no longer necessary
these transformations, the assumption p n at
a
e
:
1
I
It
309
> 0,
fact,
it
*
that of J
(p n )
-
follows from that of (p n ),
are also bounded sequences.
These convergence
is
-
boun-
and
as (p~
a )
tests 51 thus
possess a special degree of generality, similar to that of Kummers^ criterion of the second kind, mentioned below in 7. as indeed in general from any 5. From this disjunctive criterion others may again be deduced by various transformations, criteria so obtained can be new only in form. For these the though transformations we can of course lay down no general rule; new ways criterion
may always be found by skill and intuition. This the great number of criteria which ultimately remain
is
the reason for
outside the scope
of
any given systematization. It is obvious that every inequality may be multiplied by arbitrary positive factors without altering its meaning; similarly we may form the same function F(x) of either member, provided F(x) be monotone
in particular we may take logincreasing (in the stricter sense), E. of cither side. etc. roots, arithms, g. the last disjunctive criterion the form be into therefore may put
or
^# !*/__ P f
V
We
see at a glance that
work
===
n
:
&
-Pn-l\ by
this
means we obtain a general framewere set up by
of the preceding sections which or == log n.
for the criteria
assuming p n the
n
L
50 The equivalence is not complete, i. e. with the same sequence (pn ) as basis, new criterion is not so effective as the old one; in fact, the divergence of
-
5?
"" ,
for instance,
may
be inferred from the old criterion, but not
Pn from the new one 61
Pnngsheim: Math. Ann Vol. 35, p. 342. 1890 f. d. reine u. an^ew. Math., Vol. 13, p. 78. 1835 ,
62
11
Journ.
(o5l)
310
Series of positive terms.
Chapter IX.
same remarks remain
Substantially the
6.
"" *
stitute
for cn
Pn'Pni of the
terion
and p n
second kind
p n -i
dn
f r
valid,,
m me
when we
sub-
fundamental
cri-
or perform any of the other typical way we obtain the most general
(II),
substitutions for c n and dn there. In this form of the criteria of the second kind.
We
1. may observe (cf. Rem. 4.) that here again, after carrying out a simple transformation, we may so frame the convergence test that it combines with the divergence test to form a single disjunctive criterion. The convergence test requires in the first instance that, for
every sufficiently large n, or
If
here
we
~
replace cn by
j>~^"
^ie
f
rmer
an
pn-1
inequality reduces to
-Q
.
'
p n cancels out, the typical terms of a divergent series automatically appear, so that the convergence test reduces to
as
or /o
.
if
Finally,
we
take into account the fact that
with
2dn
Now
the original criterion
,
2 Q dn
(Q
>
0) diverges
the criterion takes the form:
_i n
is
certainly satisfied a n+\
an
by the assumption
n -^ *t> v O* -^ tf **>
cn + l
thus appears that in this form slightly less general than the the form of convergence test, it is absolutely indifferent original whether a convergent series or a divergent series is introduced as comparison
It
series.
Hence,
still
of the criterion
we may
write:
more
may be
generally, the cn 's
and dn 's
in the
above forms
replaced by any (positive) numbers bn ; thus
Exercises on Chapter IX.
This extremely general criterion On the other hand,
is
due
^
to E.
311
Kummer. 181
( 3)
represents a disjunctive criterion of the second kind which immediately follows, as the part relative to divergence is merely a slight transformation of (II) All further details will be found in the papers and treatise by A. Pringsheim. The sequences of ideas sketched above can of course lead only to criteria having the nature of comparison tests of the first or second kinds, though all criteria of this character may be developed thereby. The integral test 176 and Ermakoff's test 177 of course
could not occur in the considerations of
this
section, as they
do not
possess the character in question.
Exercises on Chapter IX. 133. Prove
in the case of
each of the following series that the given
indications of convergence or divergence are correct:
'
2-4... (2n)
>2 <2
d)
e/
v(__i___lo g -!^^l>) ^-j
7*i _L_ i \
/o
-
_i_ 1 \
:
:
C,
:
S>;
S;
/^nrzriT
was given by Kummer as early as 1835 (Journ. f. d. reinc u. angew. Vol. 13, p. 172) though with a restrictive condition which was first recognized as superfluous by U.Dmi in 1867. Later it was rediscovered several 68 It
Math
,
1888, to v.olent contentions on questions of O. Stoh (Vorlesungen liber allgem. Arithmetik, Vol. 1, p. 259) was the first to give the following extremely simple proof, by means of which the criterion was first rendered fully intelligible:
times and gave rise, as late as priority.
Direct proof: The
It
criterion
is
that from
some stage on
follows in particular that the products an bn diminish monotonely and
therefore tend to a definite limit
y>0. By
thus a convergent series of positive terms the corresponding terms of ~a n this series ,
131,
And is
as
(<*& its
an + ibn + i)
is
terms are not less than
also convergent.
Chapter X. Series of arbitrary terms.
312
134* For every
fixed p, the expression
n \
Cp when n > -f- oo integer for which log p n^>l.
has a definite limit the
first
135* For every
fixed Q in
when w
has a definite limit y
136.
If
<< g
#->?,
it
-f
,
<
summation commences with
if
the
1
the expression
,
OO.
follows that
where p, />', and q denote given natural numbers. and if the 137. If 2dH is divergent, with dn -> we have ,
Dn 's
are
its
partial suras
r=l
138.
when
If
p-a>pn
139.
If
2a n
has monotonely diminishing terms, it is certainly divergent for a fixed p and every sufficiently large n.
^ < dn
two series
for every n, the
are convergent, for every Q ^>
.
14O. Give a direct proof, without the use of Ermako/f's
test
and without
the help of the integral calculus, of the criterion
__ 2a.2W
<1 l>2
for series of
141.
If
:
J
~^
:
monotonely diminishing terms the convergence of a series 2an follows from one of the criteria 164, II, then, as n >-}-oo,
of the logarithmic scale
[n log n loga n
.
.
.
log k n\-a n ->
and diminishes monotonely from a certain stage on, whatever the value of the positive integer h
may
be.
Chapter X. Series of arbitrary terms. Tests of convergence for series of arbitrary terms. With series of positive terms, the study of convergence and divergence was capable of systematization to some extent; in the 43.
case to
of
series
of
be abandoned.
arbitrary
terms,
The reason
lies
all
not
attempts of so much in
this
kind
have
insufficient de-
43.
Tests of convergence for series of arbitrary terms.
velopment of the
as
theory,
essence
the
in
of
the
313
matter
itself.
A series of arbitrary terms may- converge, without converging abso1 Indeed this is practically the only case which will interest us lutely .
here, as the question of absolute convergence reduces, by 85, to the therefore need only consider study of a series of positive terms.
We
the case in which either the series
gent or its absolute convergence the previously acquired means.
actually not
is
absolutely conver-
cannot be demonstrated by any of If a series is conditionally conver-
gent, however, this convergence is dependent on the mode of succession of the terms as well as on their individual values; any comparison test which we might set up would therefoie have to concern the series
and not merely its terms individually, as before. This means that each series has to be examined by itself and we cannot obtain a general method of approach valid for them all. as a whole,
ultimately
Accordingly
more is
we have
be content
to
The
restricted field of validity.
the formula
known
+
<*0
^
criteria
with a
<*1
.
// a Q ,a 19 ...
and
= An
H-----h
b Q , b^, ... denote
("
Proof.
k 0)
and every k^>l,
We
have
by summation from v follows
=n
-\-
1
to v
n
-f-
k,
the statement at once
3 .
The formula continues
1.
Supplements.
we put
A_i = 0.
to
hold
when n=^
1,183,
1 The case in which the series may be transformed into one with positerms only, by means of a "finite number of alterations" (v. 82,4) or by a change of sign of all its terms, of course requires no special treatment.
tive
2
182.
n+fc
n+fc
if
establish
as
Abel's partial summation 9 arbitrary numbers, and we write
then for every n
to
chief instrument for the purpose
Journ
1 It
is
f. d. reine u. angew. Math Vol. 1, p. 314. 1826. sometimes more convenient to write the formula in the form
n + fr-1
n+k
^ v=n M
a v bv
^-
^ n
f
^v(V-^ + i)~^A* 1
Chapter X. Series of arbitrary terms.
314
If c denotes an arbitrary constant, and
2.
n+k
E
'
we have
v
v
= E AJ (b - b - AJ b n+1 + A'n+k n+1 = A A _i = A A _^ a, 6,
l>+l )
v
v-w + l
also:
.
b n+1e+l
v
f
'
for a v
A = A + c,
n+k
v
v
v
v
Accordingly, in Abel's partial summation we "may" increase or diminish all the A^s by any constant amount. This is equivalent to altering a Abel's partial summation enables us to deduce a number of tests of a v b v almost immediately 4 . In the convergence for series of the form .
2
first
provides the following general
it
place,
Theorem.
184.
1) the series
2)
lim Aj>
p >+x
Proof.
-
The
2A b p+1
series
2 ab
exists.
summation
Abel's partial
k
k
Za
certainly converges, if
v
b v+l ) converges, and
(b v
v
bv
v
= 2A
v
(b v
'=0
v-^O
gives for n
=
1
- b, +1 + A k b M )
:
,
+
for every fcJjgO; making &-> oo, the statement follows, in view of The relation just written down shows further that the two hypotheses.
= +I b v+1 = (b = 0. only s
Sa
where
v
In particular,
bv 5
2A
s,
=
5'
and
if,
s'
v
v
)
if,
s',
lim
A9 b
M=
/.
/
The theorem
Ea
series
v
bv
,
does not solve the question as to the convergence of the since it merely reduces it to two new questions; but these
The result is in any case a far-reaching us enables it and one, immediately to deduce the following more special which are criteria, comparatively easy to apply. are in
many
1. is
Abel's test
monotone 4
cases simpler to treat.
6
6 .
and bounded 7
Za
v
bv
is
convergent if
2a
v
converges
and
(b n )
.
We
can of course reduce any series to this form, as any number can be Success in applying the above expressed as the product of two other numbers. theorem will depend on the skill with which the terms are so split up. 6 Abel's test provides a sufficient condition to be satisfied by (6n), loc. cit. in order that the convergence of 2 a n may involve that of Z an b n J. Hadamard .
1903) gives necessary and sufficient conditions; cf. (Acta math., Vol. 27, p. 177. E. B. Elliot (Quarterly Journ., Vol. 37, p. 222. 190(5), who gives various refinements. 6 In anticipation of the extension to complex numbers (v. p. 397) it may be emphasized already that a sequence of numbers assumed to be monotone is necessarily real.
A
7
In other words: convergent series "may" be multiplied, term by term, by Theorem 184 and the criteria factors forming a bounded and monotone sequence. deduced from it all deal with the question: By what factors may the terms of a convergent series be multplied so that a convergent series results? And by what factors must the terms of a divergent series be multiplied, so that the resulting series
may be
convergent?
Tests of convergence for series of arbitrary terms.
43.
Proof. By hypothesis (A n ) and
On
(b n ), (v. 46),
and hence
also
315
(A n
n+1 ),
the other hand,
b v+1 ) is by 131, the series (b v and indeed as its all terms have the absolutely convergent, convergent, same sign, in consequence of the monotony of (b n ). It follows, by 87, are convergent.
2,
EA
that the series
v
(b
b^ +l ) is also convergent, since a
lt
The two
certainly bounded. sequence av bv and fulfilled accordingly is
S
is
convergent.
Dirichlet's test 8 Za v b v sums and (b n ) is a monotone null
2. partial
convergent
conditions of theorem 184 are
is
.
2a
convergent if
v
has bounded
sequence.
same reasoning as above, 2 A v (& b v+l ) is conb as is is a null bounded, (A n n+l ) (A n ) sequence if (b n ) vergent. Further, The two conditions of 184 are again is is, i. e. it certainly convergent.
Proof. By
the
fulfilled.
3.
Tests of
2a
a)
v
bv
is
du Bois-Reymond* and Dedekind
2 (b
convergent if
^.+i) converges absolutely
v
and
av
converges, at least conditionally.
Proof. By bounded.
tainly
(*o
ZA 1
87,
2,
(b v
- *i) + (*i ~ *2) +
tends to a limit
when n ->
+ GO,
and the existence of lim
thesis,
b v+l ) also converges, as (A n )
-
v
is
-
-
+ (ft-i - 6) = *o - bn
so does b n
A n b n+1
itself;
lim
An
exists
by hypo-
follows.
Z
2
a v b v is convergent if 6, +1 ) converges absolutely and (b v b) has bounded partial sums, provided b n -> 0.
Proof.
cer-
Since further
2A
v
b y+1 )
(b v
is
again convergent and
E av
A n bn+l -> 0. 185
Examples and Applications. 1.
The convergence
2.
J?(
1)"
of
2 an
involves,
by Abel's
has bounded partial sums.
test, that
Hence
if
of
E
(6 n ) is a
n ,
monotone
null
sequence,
8 9
101. Vorlesungen uber Zahlentheorie, l sfc edition, Brunswick 1863, The designation Antnttsprogramm d. Univ. Freiburg, 1871.
above
adopted for the three tests is rather a conventional one, as all three are substantially due to Abel. For the history of these criteria cf. A. Pringsheim, Math. Ann., Vol. 25, p. 423. 10
1S85.
143 of the work referred to in footnote 8.
Chapter X. Series ot aroitrary terms.
316
converges by Dmchlefs
This
test.
series with alternately positive
kn has bounded k 2 , ... such that 2"( \) number of even integers over that exponents fc lf & 2 , . .
Given positive integers &
3.
sums
,
fc
1}
for this the excess of the
partial of odd integers
as n -> -f
a fresh proof of Leibniz's criterion for
is
and negative terms (82,5).
the n the series
among
oo
first
.
denotes any null sequence. an x n is convergent for convergent, the power series n 0<jo;<-f-l, since the factors x form a monotone and bounded sequence. If E a n merely has bounded partial sums, the power series at any rate conn monotoncly. verges for every x such that 0
converges, 4.
If
(&n )
2 an
2
is
--
sin sin (a
+ x) -f sin (a -f 2 x)
-f-
---H sin (a -f
n
x)
=
n
x
/
sin
f a.
sin
The proof
of the formula is given in
x -f
sin
2
a; -f-
-f-
sin
=
nx
we
0,
sin (n
-
sin
,
.
and for a
JT
=
,
a;
sinn cos
a;
+ cos 2 + cc
-f
cos w
a;
=
this the
Thus
if
~J
.
-f-
1)
(x
,
=j=
2
fc
rc)
-
-cos (n sin
From
1)
get
--n
x\
-f-
|
-
201. For a sin
sin
=
+ (n
a;
+l)-(x
,
-]-
2 A n)
.
|
boundedness of the partial sums can be inferred at once. 2(b n & n +i) converges absolutely and & ->0, we conclude from /t
the criterion 3b that ^T bn sin
nx
2bn cos n x In particular 6. If
where
3
fact, if
a
>
>
12 ,
this is the case
converges for every x converges for every x
when
the bn 's are positive, and
o and 0, it
n
(/? )
is
,
=f-
2&
^r
.
b n diminishes monotonely to 0. if we may write
bounded, then 2"(--
l)
n b n converges
follows from these hypotheses that
if,
--!<;
and only 1
t/ f
a
> 0.
from some stage on,
decreases monotonely, and the convergence of the series in question therefore secured by 2., if we can show that 6n ->0. The proof of this i.
e. (6n)
similar to that of the parallel fact in 17O, sufficiently large v say
1
:
Jf
< a' < a,
we have
11
For x=2k7t, the sum has obviously the value n
18
Malmsten, C.
/.:
Nova acta
Upsaliensis
(2),
sin a, for all n's.
Vol. 12, p. 255.
is
is
for every
v>w,
t
In
1844.
Tests of convergence for series of arbitrary terms.
43.
Writing
down
together,
we
inequality for v
this
= m,
m-f-1, ..., n
1
317
and multiplying
obtain
the divergence of the harmonic series, it follows as in 170, 1 that 6 n ->0. In the case a < 0, b n must for similar reasons increase monotonely from n b some stage on, so that n certainly cannot converge. Finally, when ( l) a = 0, we deduce in precisely the same way as on p. 289, that bn cannot tend to
From
2
and the series therefore cannot converge. If
7.
series;
we
such series are known as Dirichlet
a series of the form ** ,57^ X YI shall investigate
them
in
more
x>x09
it
,
also
is con58, A) ( converges for every
This simple application of is a monotone null sequence. x~x \n J by reasoning quite similar to that employed for power series (93),
for
Abel's test,
=x
on
later
detail
vergent for a particular value of x, say x f
)
theorem: Every
leads to the
of convergence
with the
JL
diverges whenever
series of the
that
property
the
form
possesses a definite abscissa
converges whenever
series
(For further details,
x<Ji.
^~
v.
x
>
Jl
and
58, A.)
186.
General Remarks.
We
have already mentioned the
fact that the magnitude of the indivarbitrary scries is not conclusive with regard to convergence. a n and Zb n whose terms are asymptotically equal, In particular, two series 1.
idual term in an
2
i.
e.
*
such that
vergence
7O,
(cf.
Thus
-
1
,
need not exhibit the same behaviour as regards con-
,
4).
for
e. g.
we have ~~
But
Sb n
positive
convergent and
is
// the series
2.
and
precisely,
let
<
when a n
it*
~
an
~*
bn
log n
S an
divergent, since
2(a n
bn ) diverges
is non-absolutely convergent^ (cf. p. 136,
negative terms,
an when a n pn and =0 when
0,
by 79,2.
footnote
9), its
taken separately, form two divergent series. More when a n an 0, and similarly let qn = 0, and = aw ^>0. 18 The two series n and qn are scries
<
>
2p
2
2
an and the containing only the positive terms of second only the absolute values of the negative terms of 2a nj in either case with the places unchanged, while their other terms are all 0. Both these series are divergent. In fact, as every partial sum of 2 an is the difference of two suitable partial sums of H and qn , it follows at once that if pn and qn were both convergent, so would 2"|a w be (by 70), contrary to hypothesis; and if the one were convergent, the other divergent, the partial sums of an of positive terms,
the
first
2p
2
2
2
|
2
1 11US p Thus i> n
it*
-
'
*"
'
(051)
Chapter X. Series of arbitrary terms.
318
2 pn
would tend to oo or -f oo (according as which is again contrary to hypothesis. -
3.
2 4n
01
is
assumed convergent),
the preceding remark, a conditionally convergent series, or rather by its partial sums, is exhibited as the difference of two
By
the sequence formed
monotone increasing sequences of numbers tending to the rapidity with which these increase, we may easily The partial sums of
Theorem. In fact,
2 pn
and
2 qn
infinity
14 .
establish
As regards the following
are asymptotically equal.
we have
since the numerator in the latter ratio remains bounded, while the denominator increases to -J-CX) with n t this ratio tends to 0, which proves the result. 4.
The
relative frequency of positive and negative terms in a conditionally diminishes monotonely is subject to the a n for which \an
convergent series
2
\
following elegant theorem, due to E. Cesaro: The limit, ratio
pn -
of
Qn for
v<w,
44.
PM is
the number o/ positive terms to }
necessarily
it
of the
exists,
the number of negative terms a vt
1
of conditionally convergent series.
Rearrangement
The fundamental
QM
if
distinction
between absolutely and non-absolutely
convergent series has already been made clear in 89, 2. This is, that the behaviour of non-absolutely convergent series depends essentially on the order of the terms in the series, so that for these series the
commutative law of addition no longer holds. The proof consisted in showing that a non-absolutely convergent series could, by a mere rearrangement in the order of its terms, be transformed into a divergent series. This result may now be considerably elaborated. In fact it
may be shewn iour,
theorem which
187.
that
as regards
by a
suitable rearrangement
convergence
we
obtain
or divergence,
any prescribed behav-
may be
induced.
The
is
Riemann's rearrangement
theorem.
//
2a
n
is
a conditionally
convergent series, we may, by a suitable rearrangement (v. 27, duce a series 2a^ with any one of the following properties:
3),
de-
14 It is best to avoid, as being far too superficial in character, the mode of expression which may be found in some writings: "the sum of a conditionCO." ally convergent series is given in the form CO
Cf. a Note by Rom. Ace. Lincei Rend. (4), Vol. 4, p. 133 1888. G. H. Hardy, Messenger of Math. (2), Vol. 41, p. 17. 1911, and one by H.Radt macher. Math. Zeitschr., Vol 11, pp. 276288. 1921.
44.
a)
to
Rearrangement
converge
an
to
b) to diverge to c)
-f-
arbitrary numbers
p
16
arbitrary
oo or
and
x, with
sum
prescribed
oo
to
convergent s'
319
series.
;
;
upper and lower limits
exhibit as
to
of conditionally
its
of
sums two
partial
ju^>x.
Proof. It suffices to prove c), since a) and b) are particular cases s' and the latter for x the former for -f- oo or p, p, c), oo. To prove c), let (xn) be any sequence tending to x and (/*n ) any xn and 17 0. sequence tending to p, y with jjin
= =
= =
of
=
^>
>
2
the terms in an E= a 1 ^- # 2 H---Let us denote by p^ 3 , in in the which order which are 0, they occur, and by q^ 9 q^, the absolute values of those which are 0, again in their proper .
.
.
^
.
.
.
<
The series thus slightly modifying the definition in 186, 2. absence of a the differ from those in 2 186, by q n only number of zero terms, and are accordingly both divergent, with posiorder,
2p
tive
and
2
We
terms which tend to 0.
show
to
proceed
that
a series of
the type
Pl
+ P* ----h Pm
fe+i
-----
ft
?i
l
-----
+ Pm^l H-----h
?*,
+ #+i H----
fc
Such a
series is clearly a reindeed one which leaves unterms relatively to one another and that of the negative terms relatively to one another. Let us choose the indices m^ m 3 -. ..., k in the k 2
all
satisfy
the
requirements.
arrangement of the given series, altered the order of the positive
and
is
<
<
1)
the
partial
sum whose
2) the partial
sum whose
partial
sum whose
16
term
.
p mi has a value
>
// x
,
;
q^ has a value
is
< ^,
^> x x ;
is
last
while that ending one term earlier
is
^ /^
is
last
while that ending one term earlier 3) the
term
last
while that ending one term earlier
.
term
is
<^
;
is
/v 2
p m^ has a
value
>
yu
,
Riemann.B.: Abb. d. Ges. d. Wiss. z. Gottingcn, Vol. 13, p. 97. 186668. b) and c) are obvious supplementary propositions. This is clearly possible in any number of ways. In fact, if * = /* with
The statements 17
a
finite
value
s',
say, take *
= sf
n
and
ftn
=
s'H
x = ^ = -f OO ( oo), take * w = n ( necessary. finally, *
If
n
,
taking ^. even larger,
and // n = x w + 2. If, and p\ from some stage we can arrange that this n)
^^O.
Chapter X. Series of arbitrary terms.
320
sum whose
4) the partial
term
that ending one
earlier is
term
last
^x
2
q kl has a value
is
<x
2,
while
;
and so on. for by taking a sufficient number of sum the may be made as large as we please, and partial positive terms, of negative ones to follow, the partial number a sufficient by allowing sum may again be depressed below any assigned value. On the other
This can always be arranged;
<
so hand, at least one term must be taken at each stage, since x n /z n in new occur the series. series does of the term really original every a n so obtained; a n denote the definite rearrangement of Let
H
the partial
E
'
sums of
for brevity fact, are terms /> Wl , /> m2
fc,
Since
,
,
.
.
'
have the prescribed upper and lower limits. the partial sums whose denote by r lf r 2
and by
cr
l9
o-
2,
.
.
,
.
.
.
.
.
In last
,
those whose last terms arc
,
we have and q n ->
p n ->
x and /* sums of 27 an
S an we
if
fe
;
it
0,
that a, ->
follows
x and r v ->
/z,
so
certainly represent values of accumulation of the partial Now a partial sum s n f of an \ which is neither a a v nor
that
2
.
T,,, has necessarily a value between those of two successive partial sums ' of this special type; hence s n can have no value of accumulation outside
a
the interval
x
.
.
.
/i,
(or different
common
from the
value of x and
IJL
if
In other words, /x and x are themselves the upper and these coincide). the lower limit of the partial sums, q. e. d. as a
Various researches of an analogous nature were started in different directions 18 and O. Schlomilch 19 investigated consequence of this theorem. M. Ohm
the effect of rearrangement on the special series
1
^
+
-
-+
..., in par-
which p positive terms are followed by q negative terms throughout Exercise 148). A. Pringsheim 20 was the first, however, to aim at general results for the case in which the relative frequency of the positive and negative terms in a conditionally convergent series is modified according to definite prescribed rules. E. Borel 21 investigated the opposite problem, as to what rearrangements in a con22 showed ditionally convergent series do not alter its sum. Later, W. Sierpinski s converges conditionally and s' s the series can be made to have that if 27 a n the sum s' by rearranging only the positive terms in the series, leaving all the negative terms with unaltered place and order while similarly it can be made to have any sum s" > s by rearranging only the negative terms. (The proof is not so simple.) ticular the case in (cf.
=
<
t
>
45.
Multiplication of conditionally convergent series.
We
showed in the preceding section, thus completing the considerations of 89, 2, that the commutative law of addition no longer holds for series which converge only conditionally. have also seen
We
18
19 Zeitschr. f. Math. u. Phys., Vol. 18, p. Antrittsprogramm, Berlin, 1839. 20 21 Math. Ann., Vol. 22, p. 455. 1883. 1873. Bulletin des sciences mathcm. 22 1890. Bull, internat. A'c. Sciences Cracovie, p. 149. 1911. (2), Vol. 14, p. 97.
520.
45.
321
Multiplication of conditionally convergent series.
in an example due to Cauchy, that the dis17), law does not in general subsist, so that the product of two a n and 2b n may no longer be formed according to the such series The question remained unsolved, however, whether rules. elementary
already (end of tributive
2
=
+
a bn the product series Sc n (with cn (- a n 6 ) might a^ 6 W _ 1 -j an A not continue to converge under less stringent conditions for and 2bn 17, it was required B, and to have the sum A-B. In that both Za n and 2'6 M should converge absolutely.
=
2
=
we have
In this connection,
Theorem oiMertens
Z
=A
an
and
2 bn = B
Proof. We
23
the
first
If at least
.
one of the
E cn
converges absolutely,
have only to show
that,
two convergent
series
and = A
converges
with increasing
188.
B.
n, the partial
sums
Cn
= <0 + c -!-...+* = *o h + K *i + b + l
o
tend to
A B
We may
as limit.
Bn
by
,
Cn or,
we
if
Since
^4
put
n
-B
Hb
those of
>-yl
-5,
nj
<*
1
<*i
b n -i
+ an b
+
)
Z an
is, of the two series, the denote by A n the partial sums of
+
Bn_ 1
/?+
only /?n
+
B
,
1
/.-!+-+
remains
to
show
A)'
when
lhat
2an
is
*(), the expressions
.^o +
form a null sequence. But we have only to put xn
.-iA+- +
o/
an immediate consequence of 44, 9 b; and yn a n there. Thus the theorem
this is
=
is
we
If
+
,
(
and
=
w
fln
it
absolutely convergent
ao b n
we have
= <*o-B n +
Bn = B -\= *n B +
(
assume that
one that converges absolutely. 27 # n
+
-
o)
<*i
f)n
=
proved. Finally,
2cn
we
shall
answer the question whether
if convergent, necessarily has the
,
The answer Theorem
2c n = 2(a
is
in the affirmative, as the following
Abel**.
of
the
+
b n -}--
n&
If )
are
the
product series
sum A-B.
three
convergent,
series
theorem shows:
2a n
,
2bn
and A, B, and
C
and 189. are
C. sums, we have A-B The theorem follows immediately from Abel's limit 1. Proof. theorem (10O) and was first proved by Abel in this way. If we their
write
S* i
J.
given by 94
J.
f.
d.
*n
=
reine
/"x
u.
d.
2bn x
=
angew. Math., Vol. (Nouv. Annales
/; (x), 79, p 182.
2cn x n = fs (*), 1875.
An
Vol.6, p. 210. 1887). reine u. angew. Math., Vol. 1, p. 318. 1826.
T. /. Stieltjes f.
(*)>
(3),
extension
was
Chapter X. Series of arbitrary terms.
322
these three power series (cf. 185, 4) certainly converge absolutely for 1, and for these values of x, the relation <^ a;
<
A (*)/;(*)
(a)
The assumed convergence
holds.
= /;(*)
of
2a n 2bn ,
and
2cn
implies,
each of the three functions tends theorem 100, when a? * 1 from the left; and that
Abel's limit
by
to a
+
limit
fi(x)-+A=2an Since the relation (by
(a)
Theorem
19,
We may
f,(x)-+B
,
holds for
that l)
it
= Sbn
the values of
all
must hold
= 2c n
f^(x)-+C
,
x concerned,
it
.
follows
in the limit:
and adopt
also dispense with the use of functions
the
following
Proof due
2.
From
this
it
to Cesdro**.
It
was shown above
that
follows that
Dividing both sides of this equality by n -f- 1 and letting n we obtain C as limit on the left hand side (by 43,2) and
=
+
-f-
A-B
oo, as
on the right (by 44, 9 a). Hence A-B C, q. e. d. In consequence of this interesting theorem, with which we shall again be concerned later on, any further elaboration of the question of multiplication of series has only to deal with the problem whether limit
the series
propose
2 c n converges.
Into these investigations
we do
not,
however,
to enter 36 .
Examples and Applications. 1. It
follows from !L
*
= n-rO
^^ = ^
n
~>~
L
1
J. *
+ -L _ I + f)
.
.
.,
by the
pre-
'
cedingf theorem, that
provided the series thus obtained converges. 26
26
Bull, des sciences math. (2), Vol. 14, p. 114. 1890. Theorems of the kind in question have been proved
by A. Pringsheim (Math. Ann., Vol.21, p. 340. 1883), and in connection with the latter's work, by A. Voss (ibid. Vol. 24, p. 42. 1884) and F. Cajori (Bull, of the Americ. Math. Soc., Vol. 8, p. 231. 1901-2 and Vol. 9, p. 188. 1902-3). Cf. also 66 of A. Prings heim's treatise, Vorlesungen Uber Zahlen- und Funktionenlehre (Leipzig- 1916), to which we have already referred more than once. G. H. Hardy (Proc. London Math. Soc. (2), vol. 6, p. 410, 1908) has proved a particularly elegant example of a related group of much more fundamental theorems.
323
Multiplication of conditionally convergent series.
45.
Now
+ 1)
p
(2
(2
n
+1-2
+ 12n
2ln+l)V2
)
new
so that the generic terra of the
series has the value
_ _1_\
,
Since ^ 4 W
t 1
3
V
tends monotoncly
--
-f"
and the new series therefore have
to zero,
does
=
The
3.
+ ---h
-
1
result obtained
00
1
=
7T
'
w-M,/ its
mean
arithmetic
we deduce
(v.
We
82, 5
Leibniz's test
converge by
as series log 2
2
so docs
a precisely similar manner,
2. In
r
,
,
n+1
ISO), by squaring- the
,
in
1.
mode
provides a fresh
of
approach to the
3
--, which has occupied us repeatedly before now J>!\ b *=!*" To see this, we first prove the followingand 15ft) equation
f>
Theorem. for which
2 an 2
Let (aot a lt 2 , . . .) fo a monotone Then the series ts convergent.
J}(
1-
thus
=
1)"
2
*;
n~0
^
136
sequence of positive numbers.
a n a n+f
n=0
(v.
= S J/f
p=1.2,...,
anc2 3.
(-1)
P
=J,
converge, with
J>
(c)
M
*
= s*-2J.
n=o
^
Proof. Since -S 9 converges, a->0; accordingly the series 1 cona aw s for every ^ 1, and ^"an verges by Leibniz's test. As a n a n +p converges, for >1. Further, as a n + p + l -< an + p9 we the series 2 are also convergent The series 3 will accordingly converge if cJ^-^O. Now have d p + i<_dp 1
<
^
.
given ra
e
^> 0,'
we can choose
ciently large p, p
Hence
O
op
a/>
4-
^*i
we a + /;
m
shall then i
+
'
-f
so that aji 77
.
T"
1 -j- aj* A l
a
a m ap + m
2
-f-
,
^-pr-: for every suffi-
2
have 8
+ "2" < ap
and the series 3 also converges.
a,
- -
T aa -J-
a/
a, a, -f
^ a,
(
aO
"I"
ai H
Let us
H
.
h
O
now form
S -f -g-
<
-
the array
324
Series of arbitrary terms.
Chapter X.
^
al aft for which A and fi are <j n. and let Sn denote the sum of the products These obviously fill up a square in the upper left hand corner of the array, and
Sn =
-a1 + ---- + (_!)
(a
)_,!.
other hand, the sum of all the (primary) diagonals which contain at least one product a^ a^ belonging: to that square, is clearly
On
the
r-O Hence, to obtain (cj, it now suffices to prove that out the above array in a more detailed fashion, we
(-
n
l)
- Sn
(Tn
2
==
)
+
1
-1
This
we
^
a,
r
-
a,
>
a
4.
+ 2 -f ---- ] f-2
H---- ]
,fl n
+1
+ a^
-2[a
If,
+
>a \
was
thus, as
an
an
j
hl
-f-
3
+ 3 -\---- ]
---I
t-i
+ ... + (_ we have
+ (- 1)"
l)--i
(of.
81
c,
/?n
,
1^)
Tn~ ^[^^-r-^^^-f/?!,;
we now
in 3.,
0,
n
Tn
asserted,
,-
^
5 ->0 and 7l
=
take an
therefore
--
-
2i
11 -f-
^
M
s3
2
2
/f
.
the hypotheses are obviously
-,
1
all
and we have
fulfilled,
But
- 2 [a a an
-
Sn *0. By writing moreover, that
write for brevity
and as
3 4
+ +
a n +1
[a a
Tn see,
we
in this case,
have, by 133,1,
00
*
1
y^
# f for
every p
>
1
__ -_, 16
_
n=0^2w
By used
the
yi
and hence
,
+
1)
a
equality (a) proved in
1.,
V
,
3^.....+ 2n
hand
the right
137 from 136,
deduce
to
"+ 1
n-0
1 - -
equality ^=-l
The
fresh
By
-=-.
o
op
the
=
side
=
^
^2 b
the method
follows at once
proof thus obtained for this relation
most elementary of
all
known
proofs, since
of functions except the Leibniz series 122. back to Nicolaus Bernoulli 21
may be regarded as the borrows nothing from the theory The main idea of the proof goes
it
.
Exercises on Chapter X. 142. Determine the behaviour
of the following- series:
n=l '
87
sin
\
Comment. Ac. Imp
scient. Petropolitanae, Vol.
X,
p. 19.
1738.
Exercises on Chapter X.
27(-l)"
e)
sin-?-,
g) 2&\n(n*x),
325
f)
27sini
h)
2 sin (w! nx),
4-
+ '" +
n
It
In the last series,
"*
s *" 2
(~ 1)n
-,
2
m)
n
J
f
n
a monotone null sequence. The series g) does not converge
(ct n ) is
x-kn\
unless
v = e, =
the series h) converges for all rational values of x, also 2 k = sin 1, = cos 1, and for (2fc-f-l)0, = ,
1_ 5! +
1
X
cli
1
2__
!_
"24!
and many other special values of tainly
g. for
e.
2~6!
1_
7
"*"
!
1
"*""" !
values of x for which
Indicate
a?.
1
2 8
cer-
it
\cTges.
*.
+
J7 L-qraV-i
x ;> 144. If (wa w ) and
for every
F+ aU
~
^] "
log 2
.
2"
w
an + 1)
(a n
converge, the series
Z an
also
con-
verges.
145. p 2>
a) If
sums,
partial
2an
2
/> 1
\
1
2a H b H *
the series
1
and 2\bn b n + l both converge, or b), if 2a n has bounded bn + 1 converges and 6n -0, then for every integer is
146. The conditions
convergent.
184, 3 are in a certain sense necessary, as well as sufficient, for the convergence of 2an b n If it be required that for a given (&), 2a n b n always converges with 2a nt the necessary and sufficient bn + l condition is that 2\bn should converge. Show also that it makes of the test
:
\
little
difference in this connection whether
verges or merely that (&)
147. a
way
2an
If
2pn
that
is
we
require that
^\bn
bn
+l
con\
monotone.
converges, and if pn increases monotonely to -f-oo in such is divergent, we have
~l
n
148. Let an tend 00
we
write
J?
W (
1)
M
=
monotonely, and assume that
to 5>
an ^
now rearrange
hm n an
this series (cf.
n=o
exists.
If
Ex. 51) so as to
have alternately p positive and q negative terms: the
sum
s'
a o-i-^
+ -"+ a 2i-a--0i-*3 ----- a a ?- !+**+.
of the
new
series satisfies the relation s'
140.
A
product series
of
= s -f
*
hm (n an )-log
?-. 9
necessary and sufficient condition
v
CB
= l'(a
two convergent series -Tan ,
6B
2bnt
fib=-
should form a null sequence.
+a
l
is
6,1
for
_ 1 +...
that the
the convergence of the
+ aB 6
)
numbers
+ ^-l + --- + ^-r4.|)
326
Chapter
XL
Series of variable terms.
150. If (an ) and (bn) are monotone sequences with limit n n product series of 2( l) a n and 2( l) bn is convergent if, = b and r b numbers on an (bQ -f t -f+ n) n = b n (a -f- a t ^-----j-
0,
the Cauchy's if, the also form a
and only an )
null sequence.
151. The two series
multiplied together
and
^~^"
by Cauchy's rule
-^7
and only
if,
>>
'
'
"/si
>0> may
be
a-J-^>l.
if,
152. If (#) and (& n ) are monotone null sequences, Cawc/ty's product of and J( l)"&n certainly converges if 2a H bn converges. the series 2( !)" A necessary and sutficient condition for the convergence of the product series that
is
2(an
153.
bn )
If,
l
'*~
e
should converge for every
for every sufficiently large
an 6n
and
2b n
if
= n '.(logn)' = w^.(logw/'
converges,
tt
t
we can
write
(log, n)"
(log r
n)' ,
nA
(log,
nf*
(Iog3
we have, provided
-f a,
n
an
is
,
not equal to bn for every n
&_,
t
r-0
Chapter
XI.
Series of variable terms (Sequences of functions). 46.
Uniform convergence.
Thus far, we have almost whose terms were given
exclusively
taken
into
consideration
It was only in (constant) numbers. the value cases that of the terms depended on the particularly simple choice of a definite quantity, or variable. Such was the case e. g. when
series
we were
y
1
na
;
considering the geometric series
their
2an
or the harmonic series
behaviour was dependent on the choice of a or of a.
2
A more
n
is that of the power series where the number an x be given, before we could attack the problem of its convergence or divergence. This type of case will now be generalized in the following obvious way: we shall consider series whose terms depend
general example
x had
in
any manner on a
We the
,
to
variable
x,
i.
e.
are functions of this
variable.
and consider
series of
accordingly denote these terms by fn
form
(x)
2fn (x).
A
function of x, in the general case, is defined only for certain values of x (v. 19, Def. 1); for our purposes, it will be sufficient to assume that the functions fn (x) are defined in one or more (open or closed) intervals
For the given series to have a meaning
for
any value
327
Uniform convergence.
46.
of x at all, we have to require that at least one point x belongs to the intervals of definition of all the functions fn (x). shall, however,
We
once lay down the condition that there exists at least one interval, in which all the functions fn (x) are simultaneously defined. For every particular x in this interval, the terms of the series 2fn (x) arc in any case all determinate numbers, and the question of its convergence can be raised. We shall now assume further that an interval / (possibly smaller than the former) exists, for every point of which the at
series
found
fn (x) is
Definition
.
of the series
neither
or
both,
and
An
1
to converge.
interval
2fn (x)
of
its
will be called
J
an interval of conver- 190.
at every one of its points (including one,
if,
endpoints), all the functions
fn (x) are defined
the series converges.
Examples and
2
Illustrations.
n
<
+
1. For the geometric series x the interval 1 a;
A power series ~ an (x x ) , provided it converges at one point at other than x09 always possesses an interval of convergence of the form When r is r) (x (Xfi-i-r), inclusive or exclusive of one or both endpoints. properly chosen, no further interval of convergence exists outside that one. n
2.
least,
.
8.
axis
x
.
.
The harmonic
>
1
,
1
series
has as interval of convergence the semi-
2j
n*
with no further interval of convergence outside
it.
4. As a series is no more than a symbolic expression for a certain sequence of numbers, so the series 2fn (x) represents no more than a different symbolic form for a sequence of functions) namely that of its partial sums
In principle,
it
is therefore
immaterial whether the terms of the series or
its
partial
sums are assigned, as each set determines tht other uniquely. Thus, in principle, it also does not matter whether we speak of infinite series of variable terms or of sequences of functions. We shall accordingly state our definitions and theorems only for the case of series and leave it to the student to formulate them for the case of sequences of functions*. 5.
For the
series
r+
j
For the case of complex numbers and functions, we have here to substitute throughout the word region for the word interval and boundary points of the region for endpoints of the interval. With this modification, the sign o has the same sig1
nificance in this chapter as previously. 2 Occasionally, however, the definitions
to sequences of functions.
and theorems
will also
be applied
328
Chapter XI. Series of variable terms.
we have
The
series converges for every real x. a)
sH
b)
sn (x)
c)
s*(x)-+l,
6.
On
defines
lim sn
(x)
a
x
if
,
we have
|*,<1,
if
(x)-+0, -> 1
Clearly, indeed,
|
|
if
|0|
>
1
=
1.
and
,
the other hand,
with an infinity
series
obviously exists
if,
of
separate
and only
if,
intervals
of
<
75- ,
-
'
or
^
convergence; i.e.
foi
if
if a; lies in an interval deduced from these by a displacement through an throughout the interior integral multiple of 2vt. The sum of the seties = 1 at the included endpoint. of the interval and
or
=
7.
,
Ihe
sin 3 sin 2 x -- ---h --
x
~\
-^
1
X
=f=
converges, by 185,
O
fj
real
o *-
a:
.
series sin
cos -3 -- ----O
cos 2 x\ the series cosxH--- ^ 6
a;
a:
,
,
1
1
r
r
5. for
.
converges
every .
for
every
real
2kyt.
2
If a given series of the form fn (x) is convergent in a determinate interval /, there corresponds to every point of / a perfectly definite value of the sum of the series. This sum accordingly ( 19, Def. l) is itself a function of x, which is defined or represented by
the series. is
we
also
When
said
to
the latter function
be expanded in
is
the chief centre of interest, it In this sense,
the series in question.
write
n=0
power series and of the functions they represent and VI), these ideas are already familiar to us. (v. Chapters The most important question to be solved, when a series of variable terms is given, will usually be whether, and to what extent, properties In
the
case
of
V
belonging to
all
the functions fn (%),
series, are transferred to its
i.
e.
to the
terms
of the given
sum.
Even the simple examples given above show that this need not be the case for any of the properties which are of particular interest in the case of functions. The geometric series shows that all the functions fn (x) may be bounded, without F (x) being so; the power series for sin a;, x 0, shows that every fn (x) may be monotone, without F(x) being so; example 5 shows that every fn (x) may be continuous,
>
Uniform convergence.
46.
and the same example
without F(x) being so, for
fact
ponding showing
{
Then
for
1
illustrates the corres-
easy to construct
that the property of integrability
For instance,
=
differentiability.
is
It
329
may
an example
also disappear.
let
every rational x expressible as a fraction with denominator
<
n, (positive and) for every other x
=
.
for each n, and consequently fn (x), for each n, is intcover any bounded interval, as it has only a finite number of discontinuities in such an interval (cf. Also lim sn (x) = F (x) exists 19, theorem 13) s n (x),
grable
for every x.
In fact,
if
x
is
rational, say
=
(?>0, p
and q prime
sn (x) = l and hence F(x) = l. another), we have, for every the other hand, x is irrational, s n = for every n and so F(o;) = 0. (a;) ~fn ( x) = lim sn (x) defines the function
n>q,
= = This function
is
Even by
not integrable,
to
one
on Thus
If,
for a rational x, an irrational x.
1
for
for
it
is
discontinuous
we
these few
3
for every x.
to see that a quite examples, of problems arises with the consideration of series of variable terms. have to investigate under what supplementary conditions this or the other property of the terms f n (x) is transferred to the It is clear from the F(x). examples cited that the mere fact of
are led
new category
We
sum
the cause must reside in the convergence does not secure this, mode of convergence. A concept of the greatest importance in this respect is that known as uniform convergence of a series fn (x) in one of its intervals of convergence or in part of such an interval. This idea is easy to explain, but its underlying nature is not so
2
intuitively,
We
shall therefore first illustrate the matter before proceeding to the abstract formulation:
readily grasped.
somewhat
CO
2 fn n=0
Let
(x)
converge, and have for
sum
F(x), in an interval/, a
<x
we shall speak of the graph of the function y = s n (x) = f (x) -\ h fn (x) as being the nU* curve of approximation and of the graph of the function y = F(x) 8 We may modify this definition a little by xs whose denominators are factors of n^ and
taking sn (x) = 1 for all rational elsewhere; the rational a?'s in question comprise, for each n, a definite number of other values besides the integers used above. then obtain as limsn (#) the same function F(x) as above. In this case, however, both s n (x) and F(x) may be represented in terms of a closed expression, by the usual means; in fact, we have 2 k $n (x) = lim (cos nl nx) t and therefore
<w
=0
We
F(x)
=
lim
[
lim (cos 2 n! ytx) k}.
This curious example of a function, discontinuous everywhere, yet obtainable by a repeated passage to the limit from continuous functions, is due to Dirichlet.
Chapter XI. Series of variable terms.
330
00
as
The
the limiting curve.
fact of the
convergence of
n=o
fn
to
(x)
F (x)
in
J
then appears to imply that for increasing- n, the curves of approximation lie closer and closer to the limiting curve. This, however, is only a very imperfect description of what actually occurs. In fact, the convergence in / implies only, in the first instance, that at each individual point there is convergence; all we can say, to begin with, is therefore that when any definite abscissa x is singled out (and kept fixed) the corresponding ordinates of the curves oi approximation approach, as n increases, the ordinate of the limiting curve for as a whole, the same abscissa. There is no reason why the curve y = sn (x) should lie closer and closer to the limiting curve. This statement sounds rather paradoxical, but an example will immediately make it clear. ,
The
whose
series
n=
for
in the interval 1
certainly converges
The limiting curve
sums
partial
have the values
1, 2, ...
<x<2
In fact, in that interval,
.
< <
2 on the axis of x. The n** therefore the stretch 1 a; lies above this stretch and, by the above inequality, at a
is
curve of approximation distance
of less than
interval 1
<x<2
from the limiting curve, throughout
For large
.
n's,
the whole of the
the distance all along the curve
is
therefore
very small. In this case, therefore, matters are much as we should expect; the position is 1 x entirely altered if we consider the same series in the interval at every point of this interval 4 still have lim sn (x) = so that the limiting curve is the corresponding poition of the a;- axis. But in this case the
< <
.
We
,
n**1
approximation curve no longer
out the interval, sn
(a:)
=
,
lies
any n (however
for
close to
the
limiting curve
For #==
large).
,
through-
we have always
so that, for every n, the approximation curve in the interval from
t
to 1 has a
hump
of height
~2
!
!
The graph
of the curve
y
=
S 4 (x)
has the
following appearance:
Fig. 4. 4
In
every n
>
for
fact,
X
;
for
x
>
x
we have ,
sn (x)
=
<
sn (x)
< nx
as before,
even permanently.
i.
e.
<
e
for
46.
The curve y
),
Uniform convergence.
331
however, corresponds more nearly to the following graph:
Fig. 5.
For larger
the
n's,
hump
without diminishing in height beThe approximation
in question
comes compressed nearer and nearer
to the ordinate-axis. 5
curve springs more and more steeply upwards
from the origin
to the height
,
Ci
which
for x
attains
it
n
,
within a very small distance of the
The beginner,
whom
down again almost
only to drop a;
as rapidly to
-axis.
phenomenon will appear very odd, should take care to get it quite clear in his mind that the ordmates of the approximation curves do nevertheless, for every fixed x, ultimately shrink up to the point on the tf-axis, so that we do have, for every fixed x, lim sn (x) = 0* If x is given a fixed value (however small), the disturbing hump of the curve y=sn (x) to
this
will ultimately, i. e. for sufficiently large w's, be situated entirely to the left of the ordinate through x (though still to the right of the y-axis) and on this ordinate the curve will again have already dropped very close to the a;- axis*.
Therefore the convergence of our series will be called uniform 2 but not in the interval 1 x x
< <
interval 1
We now possesses an
< <
,
to
proceed
the abstract formulation:
interval of
convergence /; it XQ idual point of /, for instance at X
=
F(x) = sw (#) + rn
number nQ
x and assume > ( ] such that, for every n > n
Suppose
2fn (x)
convergent for every indiv-
is \
in the
.
this
means
arbitrarily
that
given,
if
we
there
write is
a
,
Of course the number nQ as was already emphasized (v. 10, rem. 3), depends on the choice of e. But n Q now depends on the choice of XQ also. In fact for some points of / the series will in general converge more ,
6
8
At the If
we
point of the to a height
origin, its slope take,
hump
<
say, is
.
x
=
is s n '(Q)
y^r~
ffinnftflft'
and
=
n.
and n at
=
1000000, the abscissa of the highest
our point * the curve has already dropped
Chapter XI. Series of variable terms.
332
By analogy with 10, 3, we shall therefore and more simply, dispensing with the index with the special emphasis on the dependence on e, we shall say: Given e > and given x in the interval /, a number n (x) can always be assigned, such that for every n > n (x) 7 rapidly than for others write w w (e,a: ); or
.
=
,
we now assume n(x) still for the definite given e chosen, say as an integer, as small as possible, its value is then uniquely In a defined by the value of x; as such it represents a function of #. certain sense, its value may be considered as a measure of the rapidIf
convergence of the series at the point
ity of
We now
x.
define as
follows:
191.
Definition of uniform convergence (1 st form). The series 2fn (%) convergent in the interval f, is said to be uniformly convergent in the sub-interval ]' of /, if the function n(x) defined above is bounded in in / each value 8 of e. Supposing we then have n (x) this
N
will
of course
n(x) themselves
depend on
we may
the
choice of
like
c,
A
assigned independently
numbers
2f n (x),
con-
convergent
in a
series
vergent in the interval J, is said to be uniformly sub-interval /' of /, if, given e, a single number of x,
the
also say:
2 nd (principal) form of the definition.
N = N(e)
can be
such that
n>N,
but also for every x in /'. in /'. also say that the remainders rn (x) tend uniformly to
not only (as formerly) for every
We
/',
for
Illustrations and Examples. 1. Uniformity of convergence invariably concerns a whole 9 an isolated point
interval,
nevei
.
2. A series 2 fn (x) convergent in an interval / does not necessarily converge uniformly in any sub-interval of /. X ) n has the positive radius r and if 3. If the power series 2 an (x
the series
is
uniformly convergent in the closed sub-interval /' of
The student should compare, for instance, the rapidity of convergence n geometric series 2x (i. e. the rapidity with which the remainder diminishes 1 99 as n increases) for the values 3 = -^ and 05 = ^^^. IUU 1UO 8 the above-mentioned measure of the rapidity of conIf, that is to say, In parvergence evinces no unduly great irregularities in the interval /'. ticular cases /' may of course consist of the complete interval /. 9 More generally, it may have reference to sets of points more than finite in number. 7
of the
Uniform convergence.
46.
<
333
<+
interval of convergence, defined by x x Q. In fact as the point tne interior of the interval of convergence of the power series, li es #Q-}n But if the latter is absolutely convergent at that point. n Q converges choose so that for e we ;> can, given 0, every absolutely, N=N(e)
its
x
m
*=
,
2a
n> N
n+ * -f +i \-e
I
Also, since
Thus /'
xQ
|a5
\
*+2
I
for every
The
-e
|
rn (x)
-
M -h
<.
we have
in /',
a;
n ^> N, we certainly have
for
may be
1
n
\
O>
whatever the position
x
of
in
.
we have
result
obtained
is
as follows
2an (x~x
o Theorem. A power series formly in every sub-interval of
the
form
\
n )
x
of positive radius r converges unir of its interval of conQ
X
\
< <
vergence. 4. The above example enables us to make ourselves understood, if we formulate the definition of uniform convergence a little more loosely, as follows:
2 fn (x)
is said to be uniformly convergent in J', if it is possible to make a statement about the value of the remainder, in the form "\ rn (x) *", valid for all positions of x simultaneously.
<
\
The
5.
series
-
2
for,
whatever the position
of
uniformly convergent for every value of x\
is
5 *
n=l
x may
be,
~
whence the
rest
be inferred by
may
(>. The geometric series 1 < x of convergence -|-
<
however
<
x
large 1, for
<
N
may be
which
for instance,
If,
is
we
not uniformly convergent
in the
whole
interval
For
1.
chosen,
e. g. rn (x)
4.
>
we can
always find an rn
(x)
with n
">
N
and
1.
choose any fixed n
>
N
t
then as x ->
we have
1
x n+i
Hence r n
(x)
>
1 for all
x in a
definite interval of the
<
form x
x
<
1.
7. The above clears up the meaning of the statement Sfn (x) is not uniformly convergent in a portion J' of its interval of convergence. A special value of e, say the value e<j > 0, exists, such that an index n greater than any assigned may be found, so that the inequality rn (x) < e is not satisfied for some suitably chosen x in J' :
N
|
|
.
=
With
sn (x) t our definition reference to the curves of approximation y clearly implies that, with increasing n t the curve should lie arbitrarily close to the 0, limiting curve throughout the portion which lies above J'. If, for any given e sn (x) will we draw the two curves y e, the approximation curves y (x) ultimately, for every sufficiently large n, come to he entirely within the strip bounded 8.
=F
by the two curves.
=
>
334
Chapter XI. Series of variable terms.
9. The distinction between uniform and non-uniform convergence, and the great significance of the former in the theory of infinite series, were first recognized (almost simultaneously) by Ph. L. v. Seidel (Abh. d. Mimch. Akad. r
1848) and by G. G. Stokes (Transactions of the Vol. 8, p. 533. 1848). It appears, however, from a paper published till 1894 (Werke, Vol. 1, p. 67), that the latter distinction as early as 1841. The concept of uniform
Cambridge Phil. Soc., by K. Weierstrass, unmust have drawn the convergence did not
p. 383,
become common property
much
till
through the lectures of
chiefly
later,
Weierstrass.
Other forms of the definition of uniform convergence.
3 rd form. in whatever
2fn (x)
said
is
way we may
to
uniformly convergent in
be
choose the sequence
10
the
in
(x n )
J
interval
f
^f
)
J\
the corresponding remainders 11 invariably form a null sequence . We can verify as follows that this definition
is
equivalent to the
preceding: nd Suppose that the conditions of the 2 form of the definition are so that r n (x) 6 fulfilled. Then, given e, we can always determine for every n > and every x in /'; in particular
a)
N
|
N
I
hence
rn( x n)
<e
I
*or
\
<
n> N;
ever y
rn (xn )-+0. rd
b) Suppose, conversely, that the conditions of the 3 Thus for every (o;n ) belonging to /' , rn (xn ) *
filled.
of the 2 nd
form must then be
N=
if a number the case, did not exist for every e
>
satisfied also.
In fact,
form are
The
.
this
if
ful-
conditions
were not
-^(e) with the properties formulated there
this would imply that for some had these properties; above any no number number N, however large, there would be at least one other index n such e xn in /', rn (xn ) Let n be an that, for some suitable point x s Above n l there would be another index index such that r n (x n ) e for a suitable corresponding point x n r (x n 2 such that ^ and n2 ^ so on. We can choose (x n ) mj' so that the points xn x n ^ belong to (x n ), in which case
special e,
say e
=
,
N
,
=
|
|
,
^
\
|
|
\
^
li
^
,
10
.
.
The sequence need not
converge, but rn (x)
11
Should each of the functions choose x n in particular so that rn (x n ) |
|
\
.
.
.
may occupy any position in./'. attain a maximum in J' we may t
\
= Max
|
rn (x)
\
;
our definition thus takes
the special form: Zfn (x) is said to be uniformly convergent in /' if the maxima Max rn (x) in J' form a null sequence. does not attain a maximum in _/', it has, however, a rn (x) If the function We may also formulate the definition in the general form: definite upper bound /iw Form 3 a. 2f (x) is said to be uniformly convergent in J' if pn -> 0. (Proof?) |
\
\
|
.
n
46.
Uniform convergence.
335
will certainly not form a null sequence, contrary to hypothesis. Our assumption that the conditions of the 2 nd form could not be fulfilled is inadmissible; nd
the 3 rd form of the definition
is completely equivalent to the 2 In the previous forms of the definition, it was always the remainder of the series which we estimated, the series being already assumed to converge. By using portions of the series instead of infinite remainders (v. 81) .
the definition of uniform convergence may be stated so as to include that obtain the following definition
We
of convergence.
:
4 th form. A e,
given s
if,
for every n
> N,
every
and
|
all
we
find that
then
x
rn (x)
|
n
all
> N and
In the inequality, we may make k tend e for each x in J'. Conversely, if
^
\
x in /', then for
all
+fn+k (x) -
+
|
This shows, however, that the 4 th form,
depending
(e)
all
these n,
k ^>
all
1,
we have
in /',
l/n+i (x)
may
N = TV
every x in /'. For if the conditions of it follows firstly (by 81) that fn (x)
k^.1 and
converges for each fixed x iny'. to QO, and rn (x)
said to be uniformly convergent in the
> 0,
this definition are satisfied,
|
is
we can assign a number and independent of x, such that
interval /'
only on
Efn (x)
series
.
I
Hfn (x)
the series
if
- rn+k (x) ^ 2
rn (x)
|
the conditions of
satisfies
We
also satisfies those of the 2 nd form, -and conversely.
it
finally express this definition in the following
5 th form. A series interval J' if, when positive
Sfn (x) integers
is
form
(cf.
8 la):
said to be uniformly convergent in the
k ly k 2 & 3 ,
,
.
.
.
and points x^ x 2J # a
,
.
.
.
of J' are chosen arbitrarily, the quantities [/nfl (*n) invariably
form a
+/n +2
null sequence
12
(*n)
+
+/nf/r n (*n)
]
.
Further Examples and Illustrations. 1.
The
student should examine afresh the behaviour of the series
a) in the interval 1
b) in the interval 2.
For the
<x< ^ x 5^
series 1
+
(x
-
1)
+
(x*
Zfn (x), with 192*
2, 1 (cf.
the considerations on pp. 330
- x) +
-
+
(x
n
-
~ xn l
)
1).
+.
+A
for the above, write C/^+i (*) + w+* n (*n)] any integers tending to + <. Exactly as in 81, we may speak of a sequence of portions, except that here we may substitute a different value of x in each portion. The statement we then obtain is: A series 2fn (x) is said to be uniformly convergent in J' if every sequence of portions of the series forms a null 12
By
where the
51,
we might even
.
.
.
vn 's are
sequence. Similarly:
A
sequence offunctions
gent inj' if every difference-sequence
is
a
sn (x) is
null sequence.
said to be uniformly conver-
336
Chapter XI. Series of variable terms.
we have val /:
obviously
< < x
1
-f-
s n (x) 1 , in
= xn The
scries accordingly converges in the inter Here x 1 particular in the sub-interval /': .
= =
F(x)
The convergence in this interval x < 1 for here rH (x) = F(x)
<
;
in /') the
(hence
< <
for
have
=
1
I
.
.
s n (x)
.
sequence of points
=1-
I
rn (xn )
I
\
1
1
V
n
I
(n=l \
n
1
29
) )
1
*
)
e
,
so that the series cannot converge uni-
This may be made clear geometrically by examining the position successive curves of approximation, as illustrated by the accompanying 13 .
formly of
x
not uniform. It is not so even in /": = xn We have only to choose in /"
is
xn to
<x<
for 1
.
figure
:
For large values of n the curve y ~ s n (x) remains, almost throughout the whole interval quite close to the #-axis, which represents the limiting curve. Just before the ordmate as = +l, until it reaches its terminal it rises abruptly point (1, 1). However large a value may be assumed for n, the curve y = sn (x) will never remain close to the limiting curve throughout the entire l * inter val (or./'). In the preceding example, we could 3. almost expect a priori that the convergence would not be uniform, as F(x) itself has a "jump" of height 1 at the endpoint of the interval. Fig. 6. The case was different with the example treated on p. 330. An example similar to the latter, but even more striking, is the following: Consider the series for which ,
y
nx
(11=1,2,...). x* the number e~~~* for every n\ for x =)= is , 0, we have sn (0) , positive and less than 1, so that (by 38,1) sn (x) *0. Our series is therefore 0, i. e. the limiting curve coinconvergent for every x and its sum is F(a;) cides with the a; -axis. The convergence is not in the least uniform, however,
=
For a! =
=
if
we
consider an interval containing the origin.
Thus, for xn
=
rr=-
'
V"
which certainly does not ->
For xn =
f
1 g-J
,
.
The approximation curves have a
we even have
rn (xn )
-+
1
similar
.
< <
14
In spite of this, it is easy to see that for every fixed x (in x 1) sn (x) diminish to as n increases, so that the abrupt rise to the 1, providheight 1 occurs to the right of x, however near x may be taken to ed only that n is chosen sufficiently large. the values
+
Uniform convergence.
46.
m
Figs. 4 and 5, with this modification, that the height of increases indefinitely with n ; this is because 15
appearance to those
hump now
the
337
~+ -r-OO.
We
4.
i
mst emphasize particularly that uniform convergence does not fn (x) to be individually bounded. The series
require each of the functions L_
x
1 4-
a;
-f
'
the
sum
.,
x2
+
j^
for instance,
,
since the remainders
.,
< x
uniformly convergent in
is
,
with
have the value
xn == t2n-l
\-x
The first term of this series (as also the limiting function) the interval in question. (Cf., however, theorem 4. below.)
not
is
bounded
in
With a view to calculation with uniformly convergent series, it convenient to formulate the following theorems specially, although the proofs are so simple that we may leave them to the reader: is
Theorem
// the
1.
2fn
series
p
^
(x),
2 fn2 (x),
.
.
.,
2fnp (x)
same interval J, simultaneously, uniformly convergent the series which whole fn (x) for definite number), in
the
2
are,
(p is a
denote ., c uniformly convergent in that interval, if c19 c a , e.: series be Uniformly convergent may any (I. multiplied by constant factors and then added term by term?)
is
also
.
.
constants.
// 2fn (x) is uniformly convergent in J, so is the where g(x) denotes any function defined and bounded in the interval /. (I. e.: A uniformly convergent series may be multiplied term by term by a bounded function.)
Theorem
series
2.
2 g(x}fn (x), Theorem
// not
3.
merely
2fn (x), but H\fn (x}\ is uniformly 2gn (x)fn (x), provided that when m
convergent in /, then so is the series is
the functions g m ^\(x],
suitably chosen,
gm +*(x), ..., are uniformly
and a J, provided we can find an integer m > number G > such that gn (x) < G for every x in J and every n > m. A series which still converges uniformly when its terms are taken (7. e.: bounded in
i.
e.
\
\
in absolute value all
a
but
finite
may
be
number
multiplied term by term by any functions which, at most, are uniformly bounded
of
in J.) 15
y
The point
s n (x),
as
for
may be
which x
~ v*
inferred from
is
s'
n
actually the (n
n9 x 2)
maximum e~~ *
nx
=
point of the curve .
Chapter XI. Series of variable terms.
338
Theorem
m
Sfn (x)
converges uniformly in /, then for a suitable fm+l (x), fm+2 (x), . . . are uniformly bounded in and con4.
If
J
the functions
verge uniformly to 0.
Theorem 5. If the functions gn (x) converge uniformly to inj, so do the functions yn (x) gn (x\ where the functions y n (x) are any functions and with the possible exception of a finite number of them defined in
J
uniformly bounded in J. We may give as a model the proofs of
Theorems 3 and
3. By hypothesis, given e > 0, so that for every n > and every x in
Proof of Theorem
> m
termine nQ
For the same |ft+i/.+i This proves
we can
de-
J
and
n's
+ all
4:
I
5S
that
#'s
I
we
g.+i
y
then have |/.+i
I
< G (!/
+
I
,-i
I
+
.
.
< e.
.)
was required.
Proof of Theorem 4. By hypothesis, there exists an m such that, ^ m and every x in y, rn (x) < \. Hence for n > m and
for every n every x in /,
\
|
I/. (*)
= I
I
r.-i (*)
- rn (x) ^ |
|
r.-!
|
+
|
rn
< 1,
|
>m
so If we now choose nQ part of the theorem. in J, rn (x) i e, (e being previously assigned) the second part follows in quite a similar way.
which proves the that for every n
first
^ nQ
and every x
|
|
<
-**
v
47.
Passage to the
limit
term by term. ^
*
Whereas we saw on pp. 328 9 that the fundamental properties of the functions fn (x) do not in general hold for the function F (x) represented by 2fn (x), we shall now show that, roughly speaking, this mil be the case when the
We
first
series is
uniformly convergent
16
give the following simple theorem, which
.
becomes particularly
important in applications:
193.
Theorem interval
and
function
F (x)
16
We
1.
// the
if its terms
fn
series
2fn
(x) is
(x) are continuous at
represented by the series
may, however, mention
at
is
uniformly convergent in an
a point xn of
this interval^ the
also continuous at this point
17 .
once that uniform convergence still only and is not in general
represents a sufficient condition in the following theorems necessary.
17 If x is an endpoint of the interval y, only one-sided continuity can of course be asserted at XQ for F (x), but of course only the corresponding one-sided continuity need be assumed at x for /n (x).
Passage to the limit term by term.
47.
>
Proof. to
show
that
|
Given s 0, we have a number d d (e)
F (x ) < e
F (x)
Now we may
-
F(x) the
assumed
large that,
)=
F(*
fact of
for every
x
19, Def.
accordance with
(in
>
6b)
such that
exists
x with
for every
|
in the interval.
By
=
339
|
z
XQ
\
<8
write
- sM + rn (x) - rn (xQ).
s n (x)
we can choose n
uniform convergence, in the interval,
|
rm (x)
<
\
.
=m
so
Then
The integer m being thus determined, s m (x) is the sum of a fixed number of functions continuous ata? and is therefore (by 19, Theorem 3) ,
for
We
can accordingly choose 6 so small that for which )# XQ interval in the x (5, we have every
itself
continuous at XQ
.
<
\
For the same
x's
we
then have
which establishes the continuity of F(x) Corollary. //
and
if
2fn (x]
= F(s)
the functions fn (x\ are
all
at
a;
.
uniformly convergent in an interval, continuous throughout the interval,
is
'
then so is F(x). In connection with example 3 of 191,2,
we have in the above a fresh proof of the continuity of the function represented by a power series in its interval of convergence. If we use the lim-defmition of continuity 19, Def. 6) instead (v.
of the e- definition, the statement of the
theorem may be put
into the
form:
"m
(
jyn (X))
n=0
appears as a special case of the following elaborate theorem: In this form
it
Theorem
2.
We assume
formly convergent in the open
when x approaches XQ from
the
that the series
interval
interior
lim /(*)
>
18
x
F(x)
much more
= ^fn (x)
and that the . x l 19 , of the interval .
.
is
uni-
n=o
limit,
=*
18 or < #, Whether the series remains convergent at a? X Q may be and indeed whether the functions fn (a;) are defined there at all, is immaterial .
,
for the present theorem.
We
19 are therefore concerned here, as also in the two subsequent statements, with a one-sided limit.
194.
Series of variable terms.
Chapter XI.
340
00
The
exists.
series
n the above
manner,
21 a n
^ en
exists.
Moreover,
if
we write
lim F(a) or,
and\imF(x), when
converges
o
2a n = A
,
x+x
in
we have
A,
otherwise,
n=0
(The
form
expressed shortly by saying: In the case of uniwe may proceed to the limit term by term.) th form of the definiProof. Given e > 0, first choose w (v. 4 and every x in our so that for k 1 n every every ^> x 191)
form
latter
convergence
is
,
,
>
tion
,
interval,
Let us for the 19,
And
Theorem
moment keep n and la,
it
k fixed,
and make
x+x
By
.
follows that
2
>
true for every n n and every & ^> 1. Hence a n is us sums of this denote series Let the A by partial convergent. n and It is easy to see now that F(x) >A. If, for a given e, its sum by A. we not only have w n Q is determined so that, for every n this
is
>
then, for a (fixed)
m>
n
,
,
|F(*)-4|
= !(*.(*) - A J ~(A- ^+rm (x)\^\s m (x) - A m As
for
*-a?
a;
involves 5 m
(a?)
>^4 m ,
we can determine 5
every x belonging to the interval, such that also have a; s, we then
For these
+-5-
\
so that
<
x |
XQ
\
.
f
\F(x)-A\<*, which proves If
(o;n )
is
that we required. chosen arbitrarily in the interval of uniform convergence,
all
it
follows from
rd n, (a?n ) -* (v. 191, 3 form) that the sequences F(xn ) and sn (xn ) will invariably exhibit the same behaviour as regards convergence or divergence, and that if they converge, the limits will coincide. may contrast this with the case
and
We
Passage to the limit term by term
47.
of the series, already seen to be non-umformly convergent, whose partial are s n (x)
we
If here
-r~ir~ir~*
xn
take
F (xn )
we have
,
=
con-
e. it is
i.
0,
sums
J, i. e. it also converges, but with the vergent with the limit 0, whereas sn (xn ) limit J. The two sequences do not have the same behaviour.
Theorem
The
3.
series
and
vergent in the interval/,
= Efn (x)
F(x)
all the functions
^x^
a
over the closed sub-interval /':
is
assumed
F (x)
b, so that
con-
uniformly
fn (x) are supposed is
integrable
also continuous
Then F (x) is also integrable over J' and the integral F (x) over the interval/' may then be obtained by term-by -term integration in that sub-interval.
,
b
b
J
a
precisely:
has for
its
sum
The
s
m
e.
series
[//(
side is also convergent
and
so large that for every n
>m
on the right hand
F (x). determine m
S
7i-0 '-' a
the required integral of
Proof. Given and every x
Since
n
a
(More
of
b
X= f\Vf L =0 n (x)}d J
or
F(x)dx
Jf
i.
s
> 0,
we
in a ... b,
sum
(x) is the
itself integrable
number of
of a finite
By
over/'.
the interval J' into p parts t l9 lation of sm (x) in it, , we have
19, i2 ,
.
.
.
it
integrable functions,
is
theorem 11, we can therefore divide such that, if ov denotes the oscilij> ,
< *=i
Now which
the oscillation of rm (x)
m
is
<2
certainly
was determined. Also the
?7,
1:~S
oscillation of the
ky
>
sum
manner ' n
t ^ie
of two functions
never greater than the sum of the oscillations of the two functions. So i v of the interval a . . . b, we have . for the same subdivision i l9 i 2
is
,
.
.
,
<
iv
where v v denotes the 11)
F(x)
for every
oscillation of .F (x) in
also is integrable over/'.
e,
*"
.
Thus
(again
Furthermore, as
by
theorem
19,
F = s n + rn>
we
have,
n^m, b
b
b
fF(x)dx- afsn (x)dx
=
frn (x)dx <-J
a
the latter by
of functions;
19,
theorem
applying
19,
21.
Now sn (x)
theorem
f
a
22,
we
is
the
sum
of a finite
number
therefore at once obtain
b
b
f F(x)dx- E aff(x)d a ~~
(061)
195.
342
Chapter XI. Series of variable terms. b
however, implies the convergence of
Tliis,
2J
fv (x)dx and the iden-
a tity
its sum with the corresponding integral of F(x). Matters are not so simple in the case of term-by-term differen-
of
tiation.
In
190,
we
7,
saw, for instance, that the series
every #, and so represents a function F(x) defined for every terras of this series are, without exception, continuous and dilfer-
converges for
The
real x.
entiable functions.
If
we
term by term, we obtain the series
differentiate 00
2
cos n x n=l
,
20
for every x. Even which is divergent for every x, as for instance the series
Kxample 5, 191, by term we obtain (cf.
the position
2),
The theorem on of a different stamp.
for
e. g.
a series converges uniformly
better,
since on differentiating term
cos n x
V a series which diverges
no
is
if
~i x=
n Q.
term-by-term differentiation must accordingly be runs as follows:
It
CO
Theorem
196.
Given
4.
21
a
/= a
liable in the interval
.
by
E fn (x) n=0 (a < b); if
1
/;'(*).
series
.
.
n=0 deduced from in /,
one point by
the,
the
terms
are
differen-
series
by differentiating term by term, converges uniformly
it
then so
whose
does
of J.
the
given series, provided
it
converges at least at
if F(x) and (p(x)are the functions represented is differentiable and we have F(x)
Further t
two series,
f
*(*)-?(*). In other words, with the given hypotheses, the series entiated term by term. 20
The formulae established on
p.
357 give, for every x sin
1 jr-
2
21
-f
cos x
+ cos 2 x + '
As regards the convergence
the first instance.
may
={=
2 k
be
differ-
JT,
+!
+ cos n x = of the series, no assumption
is
made
in
47.
to the limit
Passage
term by term.
343
Proof, a) Let c denote a point of / (existent by hypothesis) for which 2fn (c) converges. By the first mean value theorem of the differential calculus ( 19, theorem 8)
2 (/,(*)-
= (*- c)' 2 //(),
(<))
v=n+l
r-n + l
where f denotes a suitable point between x and c. can, by hypothesis, choose n so that for every n and every x in J,
Given
> nQ
Under the same
we
conditions,
we 1,
therefore have
n+fc
< e.
27
r=n
> 0,
'
a
6
e
every k ^>
,
H(
fn (c)), and hence Sfn {x) itself, is uniformly 2(fn (x) convergent in the whole interval / and accordingly represents a de-
This shows that finite
function F(x) in that interval. b)
Now
let
A(*o
These functions are defined to
As above, we may
/.
/ and
XQ be a special point of
+ A)-A.(*o) for every
=
write
gy(/t)>
h^O
for
(,,
= 0,1,2,...).
which XQ
-f-
n + ls
n+fc
27 /;'(* + **) ^ (*) =y=+l r=fl 27
and we
find,
as in
that
a),
w
27 ^n
n=0 converges uniformly for
& belongs
write
all
(o
< * < i)
W
these values of h.
This series represents
the function h
By theorem 2, we may F* ( xo) ex i sts> Wlt^
let
>0 term by
= 270 n=0 A->0 F*(x = q>(x
^(*o) This signifies that
h
)
ft, (*))
),
term,
and we conclude
that
= n=0
as asserted.
Examples and Remarks.
< <
r, the series ^ xj* has the radius f>0 and if Q. By theorem 4, the converges uniformly for every x x given power series accordingly represents a function which is differentiable for XQ \
If
an (x 1
(a?
sfy)*""
|
<
\
<
|
|
<
\
<
Series of variable terms.
Chapter XI.
344
2a
n therefore remains differentiable at every function represented by n (x-~x ) interval r. the of x x open point |
2.
and
its
The
<
|
derived function
^-
-
function represented by
-
is
.
n^
n=l
(Cf,
differentiable
is
Example
5,
191,
for
every
2.)
3. The condition of uniform convergence is certainly sufficient in four theorems. But it remains questionable whether it is also necessary.
a)
In the case of the continuity -theorem
The
1
or
its
x
corollary, this
is
all
cert-
192, 2 and 4 have everywhere-conand represent everywhere-continuous functions themselves. Yet convergence was not uniform. The framing- of necessary and sufficient
ainly not so. tinuous terms their
scries considered
in
is not exactly easy. S. ArzelA (Rendiconti Accad. Bologna, (1), Vol. 19, 1883) was the first to do so in a satisfactory manner. A simplified proof of the main theorem enunciated by him will be found in G. Vivanti (Rendiconti del circ. matem. di Palermo, Vol. 30, p 83. 1910). In the case in which the functions / (x) are positive t it has been shown by V. Dim that uniform convergence is also necessary for the continuity of F(x). Cf. Ex. 158.
conditions
p. 85.
b)
The
convergence
in theorem 195 on term-by-term integration uniform again not a necessary condition may also be verified by various
fact that is
CO
examples.
Taking the
series
n=i
fn
x
( )
discussed on pp. 330-1,
whose
partial
sums are .
and whose sum
is
F(a?)
v=l
=
0,
we
nx
see at once that
00
Thus term-by-tcrm integration
leads to the correct result.
In the case of the
i
series
192,3, however,
in
which we also have J F(x) dx = 0, term-by-term o
integration gives, on the contrary,
In this case, therefore, term-by-term integration
48.
Now
we
is
not allowed.
Tests of uniform convergence.
acquainted with the meaning of the concept convergence, we shall naturally inquire how we can determine whether a given series does or does not converge uniformly in the whole or a part of its interval of However convergence. and we know it often is so to determine difficult it may be the mere convergence of a given series, the difficulties will of course be considerably enhanced when the question of uniform convergence of uniform
that
are
Tests of uniform convergence.
48.
345
is approached. The lest which is the most important for applications, because it is the easiest to handle, is the following:
Weierstrass 9
bounded
2fn
each of the functions fn
//
is
(x)
defined
and 197.
say
2y n
if the series
(of positive terms)
converges, the
converges uniformly in J.
(*t)
Proof.
the
and
J
throughout series
test.
in the interval /,
the
sequence
By 81, 2, the left. By 191,
5 th form,
If
right
is
(#n)
hand
chosen
we have
arbitrarily in /,
*0 when n *oo; hence
side
2 fn (x)
is
therefore
so does
uniformly conver-
gent in /.
1.
In the example 191, 3
Examples. we have already made
use of the substance of
Weierstrass* test. 2.
The harmonic
series Jj?
convergent on the semi-axis fact, for such s's,
,
which converges for x >1,
#>l + 1
where
<5,
<
d
is
uniformly
is
any positive number.
In
1
where 2yn converges. This proves the statement. known as Riemann's The function represented by the harmonic series is therefore certainly continuous for every 2a (a) ^-function and denoted by
x>
1.
3.
Differentiating the harmonic series term by term,
This again
uniformly convergent
is
m#^>l-|-<5>l.
logn ciently large n,
we
n6
-<1
(by 38,4);
the series
In fact, for every suffi-
for these n's and
for every
#>
then have 1
log n
1
n
7T~
is accordingly difFerentiable for every x represented by the series (*).
Riemann's f-function is
we deduce
4. If 27
>
1,
and
its
derivative
a n converges absolutely, the series 2 an cos n x and 2 a n sin n x
are uniformly convergent for every x, since e. g. a n cos n series accordingly define functions continuous everywhere. |
x
\
^ an = yw
.
These
In spite of its great practical importance, Weierstrass* test will necessarily be applicable only to a restricted class of series, since it 22
In
so that x
fact, if
>
1
+
we
8.
consider a special x
>
1,
we can
always assume 8
>
chosen
Series of variable terms.
Chapter XI.
346 in
requires
that
particular
When
absolutely. delicate tests,
the
scries
this is not the case,
investigated
we have
should
converge
make use
to
more
of
43. The which we construct by analogy with those of most powerful means for the purpose is again Abel's partial summation
On
formula.
from
tain
it
we
lines quite similar to those already followed,
first
ob-
the 00
A
Theorem.
198.
series of the
form
2a
n
(
x }'^n(x }
certainly converges
n-^O
in the interval /,
uniformly
if,
in /,
00
2jA v -(bp
1)
b v+1 ) is
uniformly convergent (as a
series)
and
v=0 2 2) (A n -6 n + 1 ) is uniformly convergent (as a sequence) *.
Here
the functions
A n = A n (x)
As formerly
Proof. ities
a v , b v and
first
have
Av
n+k
we have merely
as no longer
sums
denote the partial
to interpret
n+fc
x and k vary
2a
the
numbers, but functions of
2A r -(br -br ^) + (A n+t .bn+1l+1 2*, \ =r=n+l v^n-t-1 Letting
of
quant-
we
a;
-A n .bn+1
}.
any manner with n, we have on the
in
n (x).
left z
sequence of portions
of the series to the series
G^n'^n+i)' (v.
191,
2a b 2A v (bv v
v,
and on the right the corresponding one relative & y+1 ), and a difference-sequence of the sequence
Since by hypothesis the latter sequences always tend to it follows that so does the sequence on the left.
5 th form),
This (again by 191, Exactly as in
proves the statement
5),
43, the above theorem, which
in character, leads to the following
ageable tests
special, but
is still
more
very general easily
man-
:
2
a v (x) b v (x} is uniformly convergent in /, test. converges uniformly in /, if further, for every fixed value 25 and if, for the numbers b n (x) form a real monotone sequence
1.
if
24
more
Abel's
2a v (x)
of x,
A
23 fl , b , n are now always functions of x defined in the interval /; only n n For the notion of the for brevity we often leave the variable x unmentioned.
uniform convergence of a sequence of functions cf. 190, 4. 24 For simplicity's sake, we name these criteria after the corresponding ones for constant terms. 25
Cf. p. 315, footnote 8.
Cf. footnote to 184,
I.
Tests of uniform convergence.
48.
every n and every x in J, the functions and the same number 28 K.
347
b n (x) are less in absolute
vahie than one
Let us denote by an (x) the remainder corresponding to
Proof.
oo
the partial
sum A n (x);
of Abel's partial stitute u for
i.e.
n=o
a v (x]
summation, we
A r and we
= An
may
(x)
+ #(#)
In the formula
(by the supplement
183)
sub-
bv + l )
and
obtain
,
n\-k
2
it
(a n -& n +
1
)
again
<* 1
suffices
show
to
converge uniformly
both
that
2a
v (bv
However, the#n (#/s,
in /.
as remainders
and the bv (xjs of a uniformly convergent series, tend uniformly to in absolute value for every x in /; it follows that (an 'b n + 1 ) remain in /. On the other hand, if we conalso converges uniformly to
sider the portions n\-k
we can
easily
show
that these tend
uniformly to
in /,
thereby
completing the proof of the uniform convergence in / of the series under discussion. In fact, if av denotes the upper bound of a v (x) in /, Thus if f % is the largest of the numbers & n + l9 -*() (v. form 3 a). n+l
\Tn \<
2\b Vv=n+l
v
- 6, +1 ^ vl&.-M -
involves the fact that T"n
|
*0
uniformly in /.
00
2.
Dirichlet's
test.
J
n-O t'/
^^
and
a^
(a;)-
6 n (x) is
uniformly
convergent in /,
2
f
a n (x) are uniformly bounded 26 in of the series in J, the converif the functions b n (x) converge uniformly to partial
sums
gence being monotone for every fixed x.
Proof. The hypotheses and 192, 5 uniform convergence (again to 0) of (A n 'b n +
:L
immediately involve the K' denotes If, further, ).
The b n (x)'s form, for a fixed x a sequence of numbers bQ (x), 6 (#), . . . ; for a fixed n, however, b n (x) is a function of x, defined in /. The above assumption may, then, be expressed as follows: All the sequences, for the various values of x, shall be uniformly bounded with regard to all these values of x\ in other words, each one is bounded and there is a number which is !ft
}
K
simtdtaneously a bound above for them all. Or again: All the functions defined in / for the various values of n shall be uniformly bounded with regard to all these values of n\ i. e. each function is bounded, and a number J\ exists which simultaneously exceeds them all in absolute value.
Chapter XI. Series
348
a number greater than
all
the |^4 n
of variable terms.
n+k 2j bv r=n+l
n+k
b v+l
I
x 9 we have
for every
(a;)|'s
|
K
<*
-\b n + le+1
bn + l
\
way x and & may depend on n, the right hand side will by the hypotheses, hence also the left. This proves the uniform convergence in / of the series under consideration. The monotony of the convergence of bn (x) for fixed x has only been used in each of these tests to enable us to obtain convenient In whatever
tend to
b v + l \. By slightly modifying upper estimations of the portions 2\by the hypotheses with the same end in view, we obtain
Two
3.
The
du Bois-Reymond and
tests of
2a
Dedekinrt.
b v (x) is uniformly convergent in J, if both converge uniformly in J and if, at the same time, the functions b n (x) are uniformly bounded in /.
2a
a)
and
v
2
series bv
\
We
Proof.
n+k
the remainders
every v n ;> m,
|
use the transformation
n+k
As
v (x)
& v+ i
> m,
cc
y (x)
now converge
and every x
say,
in /,
n+k
we have, for Hence for every
uniformly to 0, |
<*(#)!
<
1.
n\-k
^r=n-fl 2\b,-b, +l
\;
x and k are made to depend as n increases, hence so does the expression on the left. That # M -& nfl tends uniformly to in / follows, by 192, 5, from the fact that an (x) does and that the 6 n (oj)'s are uniformly bounded in /.
on the manner any
the expression
on n,
in
b) series
The
S\b
v
series
b v+1
Sa \
v
even
right
now
(x)
if
tends to
b v (x)
is
uniformly
converges uniformly in
J
y
formly bounded partial sums, provided the functions b n in J.
Proof.
From
the hypotheses,
converges uniformly number greater than
(x)
->
uniformly
again follows at once that A n b n Hl more denotes a
Further, if K' once the |^ n ()|'s for every x,
in /. (to 0) all
it
J
in if the the series 27 a v has uni-
convergent
and
n+k
'-Sl&r-W vn+l
whence, on account of our present hypotheses, the uniform conver&y + 1 ) may at once be inferred A v (bv gence in / of the series
2
Tests of uniform convergence.
g 48.
Examples and
Illustrations.
1. Tn applications, one or other of the often reduce to a constant, for every n\
\vill
349
two functions a n it
and
(x)
b n (x)
usually be the former.
will
Now a series of constant terms 2 a v must, if it converges, of couisc be regarded as uniformly convergent in every interval; for, its terms being independent of #, so are its portions, and any upper estimation valid for the latter is valid ipso facto for every x. Similarly the partial sums of a series of constant terms 2a y if bounded, must be accounted uniformly bounded m every interval^ ,
Let (a n ) be a sequence of numbers with -T n convergent, and let for the x bn (x) = x n 1 The series 2a n x n is uniformly convergent in conditions of AbeV test are fulfilled in this interval. In fact, 2a n as remarked in 1., is uniformly convergent; fuither, for every fixed x in the intern zn 1. By the theorem 194 on term-by-terin val, (x ) is monotone and passage to the limit, we may therefore conclude that 2.
< <
.
,
,
|
lim
<
|
(2a n
x")
=
an x n ),
(lim
e.
i.
This gives a fresh proof of Abel's limit theorem 100. 3.
The functions
/ (namely, again we deduce that
in
bn (x)
<
1),
-
=-
also
and monotone
2 an
or
Hence, as above,
x.
(Abel's
limit
theorem
series.)
Let an
cos n x
(x)
v5!
/
N
t
/
N
a n (x).bn (x)
n=l then
every fixed
denotes a convergent series of constant terms.
Dinchlet 4.
x
for
-*..
"
if
form a sequence bounded uniformly
=sinng, and &() =
or
vi cos -
=
nx
M \i sm ---\]
x
_,
or
n-1
n-1
The
>*0.
-,
nn
i
(a
"*
^
>
series
t\\
0)
,
27 5 satisfy the conditions of DiricMet's test in every interval of the form. TT IT. 8, where 8 denotes a positive number
;S 2
In fact,
by 185,5, the
in the interval
partial
/we may take
K
sums
--- -
\
of
< 2a n (x)
and bn
uniformly, because b n does not depend on a:. it follows for the same reason that
(x)
are
^
uniformly bounded
tends monotonely to 0,
If (bn )
denotes any monotone
null sequence,
,2"
bn cos
nx
and
S bn sin n x
are uniformly convergent in the same intervals (cf. 185, 5). All these series 28 for every accordingly represent functions which are defined and continuous 87 Or in intervals obtained from the above by displacement through an integral multiple of 2 jr. 28 Every fixed x ={= 2 k n may indeed be regarded as belonging to an interval of the above form, if 8 is suitably chosen (cf. p. 343, example 1, and p. 345,
footnote).
12*
(G51)
199.
350 x
Chapter XI. Series of variable terms.
Whether the continuity subsists at the excluded points x = 2 k TT we not even in the case of the series 27 b n sin n x although once determine, certainly converges at these points (cf. 216, 4).
=j=
2 k
cannot it
IT.
at
t
Fourier
49. A.
Among
series.
Euler's formulae.
the fields to which
we may apply
the considerations developed
in the preceding sections, one of the most important, and also one of the most interesting in itself, is provided by the theory of Fourier series, and
more
by that of
generally
trigonometrical series, into
29 pose to enter By a trigonometrical series
which we now pro-
.
1
aQ
2 30
is
meant any
series of the
form
2 (an cos n x + bn sin n x), + w=i
a n and b n
If such a series converges in an interval of the form 2Tr, converges, in consequence of the periodicity of the trigonometrical functions, for every real #, and accordingly represents a function defined for all values of x and periodic with the period 2 TT.
with constant
.
c^x
it
We
have already come across trigonometrical series convergent everywhere, for instance, the series, occurring a few lines back,
.fr-~' We
a
^ ~
t
0; B
?r~^~'
have never been in a position, so
of these series for
all
values of x.
far, to
It will
a ^
.,
1;
etc -
determine the
sum
of any
appear very soon, however, that
trigonometrical series are capable of representing the most curious types such as one would not have ventured to call functions of functions at all in Euler's time, as they
may
exhibit discontinuities and irregularities
of the most complicated description, so that they seem rather to represent a patchwork of several functions than to form one individual function. 29 More or less detailed and extensive accounts of the theory are to be found most of the larger text books on the differential calculus (in particular, that referred to on p. 2, by H. v. Mangoldt and K. Knopp, Vol. 3, 8 th ed., Part 8, 1944). For separate accounts, we may refer to H. Lebesgue, Lecons sur les series tngonome'triques, Paris 1906, and to the particularly elementary Introduction to the 152. theory of Fourier's series, by M. Backer, Annals of Math. (2), Vol. 7, pp. 81 1906. A particularly detailed account of the theory is given by E. W. Hobson, The theory of functions of a real variable and the theory of Fourier series, Cambridge, 2 ud ed., Vol. 1, 1921, and Vol. 2, 1926. The comprehensive works of L. Tonelli, Sene trigonometnche, Bologna 1928, and A. Zygmund, Trigonometrical series, Warsaw 1935, are quite modern treatments; the little volume by W. Rogostnski, Fouriersche Reihen, Sammlung Goschen 1930, is particularly attractive and con-
in
tains a wealth of matter. 80
It is
only for reasons of convenience that
aQ
is
written instead of
a
.
Thus we
shall see later
=:0
/
2j
nx
sin
_ -
\
=i
A. Euler's formulae.
Fourier series.
49.
(2
on z x
or for ft
+
I
210 a)
(v.
=
that
e.
= 0,
n ft:T,'(ft
351
g.
2, ...), but
1,
-x !)*-!! for 2 o
2
+
(ft
the function represented by this series thus has a graph of the following type: *
Fig. 7.
we
Similarly,
shall see (v. 209) that =
^ j|
^i
\t->
'*
2w
n^O
for
for 2
j
~r
n=
ft
ft
n,
JT
but
< < (2 :r
ft -f-
1)
ar,
and
+X =-
~
for
thus the function represented by the series has a graph of the type: A
- 2ji
-
.?r
Fig. 8. In either case, the graph of the function consists of separated stretches (unclosed at either end) and of isolated points. However, the circumstance that simple trigonometrical series such
of representing functions which are themand "pieced together", is precisely what discontinuous selves altogether
as the above
was
are capable
chiefly responsible for the
of function,
thorough revision
and thence the whole foundation of
which the concept analysis, came to be
to
We
th shall see that century. subjected at the beginning of the 19 of the so-called are of most series capable representing trigonometrical 31 this a far more in constitute functions" ; respect, they "arbitrary in than series. instrument higher analysis power powerful 81
Of course the concept of an "arbitrary function"
is
not sharply defined.
The term usually denotes a function which cannot be assigned by means a single
closed,
formula
(i.
e.
one avoiding the use of
limiting- processes) in
of
terms
Chapter XI. Series of variable terms.
352
We
mention only incidentally that the range of this instrumeans restricted to pure mathematics Quite the conno by trary: such scries were first obtained in theoretical physics, in the course of investigations on periodic motion, i. e. chiefly in acoustics, optics, electrodynamics, and the theory of heat; Fourier, in his Thorie de la chaleur (1822) instituted the first more thorough study of certain trigonometrical series, although he did not discover any
ment
will
is
of the fundamental results of their theory. What functions can be represented by trigonometrical series and by what means can we obtain the representation of a given function, sup-
posing this to be feasible? In order to lead up to a solution of this question, let us first assume that we have been able to represent a particular function f(x) by a trigonometrical series convergent everywhere:
On f(x)
is
account of the periodicity of the sine and cosine functions, then necessarily periodic with the period 2n, and it is suffi-
cient, therefore, to consider
any
interval, for all that follows, to be
interval of length 2n. < x where 2 n,
We
choose this
one of the end-
<j points inay, moreover, be omitted. The function f(x) is then represented in this interval by a conknow that f(x) may none vergent series of continuous functions.
We
the less be discontinuous,
although it also will be continuous if the For the moment, series in question converges uniformly in the interval we will assume this to be the case. With these hypotheses, we obtain a relationship between f(x) and the coefficients a n and b n which was conjectured by Euler: i. e. in particular, it denotes a elementary functions alone, is apparently built up from separate portions of simple functions of this type, like the functions given as examples in the text, or the following, defined for every real x:
of the so-called
function
which
for irrational
x for rational x
x ,
however, the "arbitrary" function expressed by means of limiting processes on p 329, footnote. Not until it was found that even a perfectly "arbitrary" function such as these could be represented by a single (relatively simple) expression, as for instance by our trigonometrical series or by other did any necessity arise for regarding it as being actually limiting processes, one function, instead of a mere patchwork of several functions. etc.
Cf.,
Theorem
1.
The
200.
series
-0-00+ J;(0w cosn& 6 n=l
is
the
853
A. Euler's formulae.
Fourier series.
49.
+ &w sinna;)
assumed uniformly convergent** in the sum f(x). Then for n 0, 1, 2, ...,
interval
=
0<Ja?
33 .
(Euler or Euler-Fourier formulae)
As is known by elementary considerations, the following hold for every integral p and q (^> 0):
Proof. formulae
34
= a)
cospx-cos qxdx
J
=====
2* cos
/ o
V sin I
c)
px-
sin
qxdx
px
sin
qxdx
rc
=
( \
o
for
JT
=2 =
o
b)
for
for
=7i
[
for
Let us multiply the series for
for
/"(#),
/>
p />
p
+g = > = =
and
=j=
p
=
=
which
is
uniformly convergent
0<^x<^2n, by cospx; by 192,2 the uniformity of the conver gence is not destroyed, and after performing the multiplication we to 2n. may accordingly (v. 195) integrate term by term from
in
We
immediately obtain:
= V6 a 1
.-t
J f(x)cospxdx
f o J o
cospxdx
for
p
=
'
=a
/ cospx'Cospxdx
for
In consequence of the periodicity of cos re and sin uniformly convergent for every x.
32
facto,
a;,
^;
it
is
then, ipso
88
This designation is a purely conventional one; historical remarks are given by H. Lebesgue, loc. cit., p. 23; A. Sachse, Versuch einer Geschichte der trigonometrischen Reihen, Inaug.-Diss., Gottingen 1879; P. du Bois-Reymond, in his answer to the last-named paper; as well as very extensively by H. Burkhardt, Trigonometrische Reihen und Integrale bis etwa 1850 (Enzyklop. d. math. Wiss., Vol. 84
II,
1,
Parts 7 and
8,
191415).
We
have only to transform the product of the two functions in the integrand into a sum in accordance with the known addition theorems, fe. g. cos
p x cos q x =
-jr-
integrate straight away.
[cos
(/>
q)
x + cos (p + q) xn ,
in order to
be able
to
Chapter XI. Series of variable terms,
354 i.
e
in either case jt
f(x)cospxdx;
j
remaining terms give, on integration, the value 0. In the same way, multiplying the assumed expansion of f(x) by sinpx and then integrating, we at once deduce the second of Eider's formulae
for the
2
b = p
Jt
f(x)s'mpxdx.
I
The value of this theorem is diminished by the number of assumptions required to carry out the proof. Also, it gives no indication how to determine whether a given function can be expanded in a trigonometrical series at all, or, if it can, what the values of the coefficients will be.
However, the theorem suggests the following mode of procedure: Let f(x) be an arbitrary function defined in the interval <[ x <J 2 n, and integrable in Riemann's sense in the interval. In that case the 19, integrals in Euler's formulae certainly have a meaning, by
We therefore theorem 22, and give definite values for an and bn note that these numbers, exist, on the single hypothesis that f(x) is .
integrable.
The numbers
by Euler's formulae
will
-^-
a Q a lf a^, ,
2
.
.
and
b 13 b
9
...
thus defined
be called the Fourier constants or Fourier
coefficients of the function f(x). -?r &(\
.
The
series
4- b + n~^ J? #n cos n x ^ ^ f
"
sin
n x)'
may now be
written down, although this implies nothing as regards possible convergence. This series will be called (without reference to its behaviour or to the value of its sum, if existent) the Fourier
its
aeries generated by, or belonging to, f(x)
9
and
this
is
expressed
symbolically by
/()~4This formula accordingly implies no more than that certain constants an> b n > have been deduced from f(x) (assumed only to be integrable) by means of Euler's formulae, and that then the above series has been written
down 35
.
3i The symbol "/-v/" has of course no connection here with the symbol introduced in 4O, Definition 5, for "asymptotically proportional". There is no
fear of confusion.
49.
From theorem we
have,
it
is
true,
Fourier series.
and
1.
some
the
A. Euler's formulae.
manner
in
which
justification for the
converge and have f(x) for
its
this series
hope
355 was derived,
that the series
may
sum.
Unfortunately, this is not the case in general. (Examples will be the contrary, the series may not converge shortly.) On
met with very whole
in the
nor even at any single point; and if it does so, It is impossible to say off-hand necessarily f(x). case the other may occur; it is this circumstance which
interval,
sum is not when the one or
the
prevents the theory of Fourier series from being entirely a simple subject, but which, on the other hand, renders it extraordinarily fascinating; for here entirely new problems arise, and we are faced with a funda-
mental property of functions which appears to be essentially new in the property of producing a Fourier series whose sum is The next task is then to elucidate the function itself. to the equal
character:
connection between
new property and
the old ones, viz. conand so More conon. tinuity, monotony, differentiability, mtegrability, arise as which are therefore follows: the stated, problems cretely this
Is the Fourier series of a given (integrable) function f(x) con-
1.
< x
vergent for some or all values of x in
// it converges, does the Fourier series the value of the generating function ?
2.
sum
3.
the Fourier
If
u^ ^P> x
is
series
converges
the convergence
at
uniform in
n? of f(x)
all points of this interval?
have for the
its
interval
As it is conceivable that a trigonometrical expansion of f(x) might be obtained by other means than that of Euler's formulae, we may also raise the further question at once: 4. 7s it possible for a function which is capable of expansion in a trigonometrical series to possess several such expansions, in particular, can it possess another trigonometrical expansion besides the
possible Fourier expansion provided by Euler's formulae? is
It
not very
easy to find answers to
all
no complete answer to any of them is known would take us too far to treat all four questions
fact
these questions; in at the present day.
in accordance with our attention chiefly to the first two; the third we shall touch on only incidentally, and we shall leave the last almost entirely out of account 36
It
modern knowledge.
We
shall
turn
.
36
It should be noted, however, that the fourth question is answered under extremely general hypotheses by the fact that two trigonometrical series which 2 TT cannot represent the same function in that interval x converge in without being entirely identical. And if/(#), the function represented, is integrable 2 TT, its Fourier coefficients are equal to the coefficients of the trigonoover metrical expansion; cf. G. Cantor (J. f. d. reine u. angew. Math., Vol. 72, p. 139. 1870) and P. du Boit-Reymond (Munch. Abh., Vol. 12, Section I, p. 117. 1870).
^ ^
.
.
.
356
Chapter XI. Series of variable terms.
With the designations introduced above, the content of Theorem
1
be expressed as follows:
may
Theorem 5^ x 5^ 2
TT
(t.
la.
If a
e.
all #), it is the
for
converges uniformly in Fourier series of the function repre-
trigonometrical series
sented by it, and this function 37 admits of no other representation by x metrical series converging uniformly in 2 TT.
^ <
a
trigono-
The
fact that the Fourier scries of an integrable function does not necesconverge will be seen further on; that even when it does converge, it need not have / (
sarily
;
values of x, \\ithout coinciding everywhere (v. The fact that 19, theorem 18). in an interval of convergence the series need not converge uniformly is shown by
the example already used above; for the series
converges everywhere
J
^ ^
8 x and if the convergence were uniform, say in the interval 8, (v. 8 0, it would have to represent a continuous function in that interval, by 193. This is not the case, however, as we mentioned before on p. 351 and will prove later on p. 375.
185,
f>),
These few remarks suffice to show that the questions formulated above are not of a simple nature. In answering them, we shall follow the line adopted by G. Lejeune-Dirichlet, who took the first notable step towards a solution of the above questions, in his paper Sur la convergence des series trigonometriques
38
.
B. Dirichlet's integral.
We
proceed to attack the the question of convergence: If the Fourier series
-
a
2i
first
of the proposed problems, namely,
+ E (a n cos n x + bn sin n x)
generated by
i. e. with coefficients given in terms a given integrable function /(#), XQ its is to converge at the point x of f(x) by Euler's formulae,
=
partial
"
1
must tend
to a limit
whether or no
when n
this is the case,
+b ->
+
oo
.
v
sin v
XQ)
It is often possible to
by expressing
s n (x ) in
determine
the form of a definite
integral as follows:
87
88
,
sums
This function Journ.
f.
is
then (by 193, Corollary) everywhere continuous. angew. Math., Vol. 4, p. 157. 1829.
d. reine u.
49.
B. Dinchlet's integral.
Fourier series.
+b
For i>^l, the function a v cos v XQ
I
sin v
XQ
is
represented by
39
2
2,-t
~
v
357
+
f(f)cosvtdt\cosvx
j
Thus
We
take the important step of replacing the sum of the (n -f- 1) terms 40 for every by a single closed expression. We have indeed 4= %kn, for every a and all positive integral w's,
now
in brackets z
cos (a
+ + ,?)
sin
(
f
os
(
+ 2 2) H-----
1-
+^m+
sin
1
f
-~-J
cos
+ /w-)
(
201.
+ -5-)
2 sin 25
sin HI
39
point
a;
40
~cos
In order to distinguish
--
/ f
,
-j"
Z\ m 4~rr ~)
the parameter of integration from the fixed
we
henceforth denote the former by Proof. If the expression on the left ,
1
t.
is
denoted by
Cmt we have
+ sin *-)
+ sin (a + 2T+T -|-
r=2sinm-^-.cos fa-f
w + I ~J
.
Moreover the above formula continues to hold for z = 2 A JT provided we ute to the ratio on the right hand side the limiting value for z i. e. the value wcosct. ,
attrib-
358
Series of variable terms.
Chapter XI.
from which many analogous formulae may be deduced as XQ> m cases 41 t n, we obtain 0, z Taking a
=
.
+ cos
-5-
XG )
(t
^
~~~
O~
Accordingly
particular
=
=
-{-
-\
x
cos n(i
)
-
-
I
42 ,
'
N
"
'
sin(2n+l)-^5L
/A
(a)
we may
Finally,
transform
this
f(x) need only be defined
over
The
this interval.
latter
modify the value of f(2jr) to
and define f(x)
/*(())
expression
The
somewhat.
property remains unaltered 19,
(cf.
further, for
function
0<^x<,27i and integrable
in the interval
We
theorem 17). every x such i)jz,
we merely
if
will
it
equate
that
=
(A>
2,
1,
by:
41
For subsequent
we mention
use,
the following:
JT
a gives:
-.--{-a substituted for
sin
sin(a-f
jr)
+ sin(4-2*)H
/
g
m
sin
f
a
sin of
=
gives:
cosz-j-
g -
{-sin(r sin
2.
m -f 1
-j-
-x
'
.+
cos2^H
w
T
cosm+ 1
~
mz =
cos
;
z
sin
sin 3.
a =
gives
-5t
:
sin ^ 4- sin 2 s -f
-
-f-
m-
- -
sin
m
-}-
1 -~-
w* =
sin
.
;
Z
sin 4.
= cos
2a;, /
,
(y
-f-
a \
=y ,
x) -f
5.* = 2x, u = S in(y
05,
cos
/
give: ,
(y 4-
o 3
\ a:) -| ,
~+y-x,
(-
cos
/
(y
,
-f-
o 2
m
7
%
1 -a;)
=
sin
wo; -cos r
sin
)
4- ma;) -\
give:
SinWa + ^) + sin(y+ 3 + ... + sin(y + 2^^. g = a!
(y v
x
)
42 For /rrajy, as we observed once before, we should sine-ratio the limiting value for / *a; , here (2w-f 1) .
-
^
+ ma;) .
s
attribute
to the
Fourier series.
49.
Our function f(x)
now
is
defined for
/
(p(f)dt
=
/
cp(t)dt
=
9
/
(p(c
anc*
*)^*
-f-
/
the integrand in
now a
is
(a)
for
any function theorem 19), what-
19,
a
c
As
Now
to
it
and we have
real values of x
all
be periodic with period 2n. arranged with period 2n, we have (by (p (x) periodic ever the values of c and c' may be, for
359
B. Dirichlet's integral.
W<^
=
/
a-f-2?c
function of this type,
we have
8
/?
^J we
sin
the
into
this
integral up and n to 2n, substituting becomes If
to
split
t
7i
T the
pans
relative
for
in the second,
t
- 2w
to
intervals
the latter
t
r
sin(2n+l)-=-
* sin
i.
e.
by the above remark with regard
to
TT 2
J
sin(2n+l) ~ sin
-
o
.
i
and we accordingly obtain
o
Substituting 2
This
for
/
2
we
are ultimately led to the formula
3
by which the partial sums of the Fourier by f(x) may be expressed. We may therefore state,
is Dirichlet's
series generated
t,
integral*
,
important result, the theorem: In order that the Fourier series generated by a junc2. tion f(x), integrable (hence bounded) and periodic with period 2n, may
as our
first
Theorem
43
We
designate
as.
Dinchlet's
integrals
two forms a
a
sinkt
/
,
or
all
integrals
of
either
of the
360
XL
Chapter
converge at a point xQ9
it
Series of variable terms.
is
necessary and sufficient that Dirichlefs
integral
1
, (
sin/
n J should tend
sum
to
a
limit
(finite)
as
n
*
of the Fourier series at the point
+ oo #
TiWs
.
xs
fo'wutf
then the
.
Let us denote this sum by S(XQ ). The second question (p. 355), concerning the sum of the Fourier series, when convergent, may be included in our present considerations and our result may be put in a form still more advantageous in the sequel, by expressing the quantity a Dirichlet integral also. As S(XQ ) in the form of
-H-
+ cos + cos 2 t
1 -\
-----(- cos nt
=
sin (2
*+!)-
we have
or,
effecting
same transformations
the
before
as
with
the
general
integral,
o
44 Multiplying this equation
by
5 (#
),
we
finally obtain,
by subtraction from
202,
-
2O3
Our preceding theorem may now be expressed
Theorem2a.
/w order that the Fourier
t
as follows:
series
generated by a function
f(x), integrable and periodic with period 2ir, should converge s (x ) at the point XQ it is necessary and sufficient that, as n -> 00 ,
rf
+
to ,
the
sum
Dirichlet' $
integral
This equation may also be obtained from 202, by substituting f (x) ~ 1 ; a = 2 and, for every n 7-: 1, a n - b n - 0, i. e. sn (X Q ) = 1 for every n and every x . 44
this gives
Fourier series.
49.
should tend to 0, where for brevity
Dinchlet's integral.
361
we have put
theorem by no means solves questions
this
Although a
13.
1
and 2
such
in
manner
that the answer in given concrete cases lies ready to hand, yet furnishes an entirely new method of attack for their solution. Indeed
it
same may be
the
on
said with regard to the third of the questions proposed at once be modified to the following:
theorem 2 a may
p. 355, for
Theorem to s (x) at
3. On the assumption that the partial sums s n (x) converge every point of the interval a 5^ x 5^ ft, they will converge uniformly
and only
to this limit in the interval, if,
the integral, depending on
if,
J
sin
x9
t
u
as n
tends uniformly to e 0, we car assign
>
value for every
Before
-
>
oo
-
-f
a 5^ x fg
in
that
ft,
N = N so that this integral n > N and every x in a rg x fg (e)
is
is less
to
say
than
if,
given
in absolute
ft.
we make
use of theorem 2 to construct immediate tests of
series, we proceed first to transform and simplify theorem in various ways. For this purpose, we begin by proving the following theorems, which apparently lead us rather off the track, but
convergence for Fourier
this
also claim considerable interest in themselves.
Theorem
4.
... 2
integrable over
is
Iff(x)
and
77,
if (a n )
and
(b n)
are
its
2
Fourier constants, then
71-1
The
Proof.
^ 0,
as its integrand
is
)
converges.
( 1
a cos v
t 'I-
<-
dt
2 ** sin vi\
On
never negative.
the other hand,
2n
'2x
f
bn
n
2
[/(O
i'
is
-f-
2
integral
27t
J
(a n
2
[/(O]
2
dt2[a, J/ + J [27 (a
it is
27i
v
(*)
id t]
cos v
cos v
t
+b
v
2
sin v
2 [b 2
t)]
v
d
J/(f) sin v
td t]
t
o
dt
-2
where each summation expression
is
is
non-negative,
extended from v
we have
=1
to
v
= n.
Since this
362
Chapter XI. Series ot variable terms.
Thus the partial sums of the series (of positive terms) bounded and the series is convergent, as asserted.
The above Theorem
in question are
contains in particular
The Fourier constants (aj and
5.
of
(6 n)
an integrable
function form a null sequence.
From
we may deduce
this,
Theorem
6.
// y>(t) is integrable in the interval b
An /*
/ tp()cosn
=/ a
Proof. the
If
= f(i)
other real
ty(f) sin
tit
+
a<^t^b,
then
-+ 0,
nt dt ->
0.
a and b both belong to one and the we define f(t) t y; 1) n <^ 2 (k
form 2 k n
and
quite simply the further
=
same
interval of
in a <[
t
at the
t,
An
=fy
(t)
cos n tdt
= / f(t) cos ntdt = na n o
a
B
=
n bn similarly n of the function f(f). By
where a n and & M denote the Fourier constants theorem 5, A n and 5 W therefore 0. If a and 6 do not fulfil the above condition, we can split up the interval a < t < 6 into a finite number of portions, each of which satisfies the condition. A n and B n then appear as the sum of a (fixed) finite number as n *oo. Hence A n and B n do of terms, each of which tends to the same 45
and
,
.
This important theorem
will
enable us to simplify the problem of
the convergence of Dirichlet's integral
Supposing 6 chosen
~
(/
.,o)
arbitrarily
46 .
with
< d < -J
,
the function
|[n
45 This important theorem appears intuitively plausible if we imagine the curve y \p (t) cos nt to be drawn for large values of n: We isolate a small interval a . ft in which y (t) has an almost negligible oscillation (is practically constant) and proceed to choose n so large that the number of oscillations of cosn* is fairly large in the interval; in that case, the arc of the curve y = y>(/)cosw/ corresponding to .../? will enclose positive and negative areas in approximately equal numbers and of approximately the same size, so that the integral is almost 0. 46 Of course theorem 6 may be proved quite directly, without first proving theorem 4. The latter is, however, an equally important theorem in the theory, even though, as it happens, we shall not need it again in the sequel. .
.
is
6^t^^.
integrable in
368
B. Dirichlet's integral.
Fourier series.
49.
Hence,
for fixed d,
n ~2
/
c)
The limit as
Dirichlet
w->oo,
value of d
t/>
(0 sin (2 n
and only
>
new
the
t
-
d t -*
theorem 2 a
of
integral
if,
+ 1)
,
will therefore
tend to
as
for a fixed, but in itself arbitrary,
if
integral
2^
o
n
as
tends to
values of f(xQ
Since d
>
Now
increases.
2t) in
0^<<5,
may be assumed
contains at the
same time
the latter integral only involves the 2d. 2 d<^x<*xQ e. of f(x) in#
+
i.
arbitrarily small, this
remarkable result
the following
47 The behaviour of the 204. (Ricinann's theorem. ) X the at of f(x) point Q depends only on the values of XQ the This neighbourhood may be asin of neighbourhood f(x) sumed as small as we please In order to illustrate this peculiar theorem, we may mention the following consequence of it: Consider all possible functions f(x) (inte2n 2 n) which coincide at a point XQ of the interval grable in this of however and in some neighbourhood small, possibly point,
Theorem
Fourier
7.
series
.
.
.
varying with
.
.
the
particular
Then
function.
the
Fourier series
.
.
of all
however much they may differ outside the neighthese functions bourhood in question must, at XQ itself, either all converge or all in the former case they have the same sum S(XQ ) (which and diverge, be to not or may may equal f(x^)). After criterion
theorem
inserting
obtained
these remarks, we proceed to re-formulate the above, which we may henceforth substitute for
2:
Theorem
8.
The necessary and sufficient condition for the Fourier X Q to the sum s(# ), is that for an ar-
series of f(x) to converge at
bitrarily chosen positive 6
<
-5- ,
Dirichlet's integral
3
_;_ /o
i
sin/
should tend 47
to
as
n
increases
i \ A
dt
48 .
liber die Darstellbarkeit einer Funktion durch cine trigonometrische nd ed. p. 227). Reihe, Hab.-Schrift, Gottingen 1854 (Werke, 2 4* As of regards uniformity convergence, we can assert nothing straight away, since we are ignorant as to whether the integral (c) above considered, as n increases, for every fixed a? , will do so uniformly for which tends to every 2 of a specified interval on the a;- axis. Actually this is the case, but we do not propose to enter into the question further.
Chapter Xf. Series of variable terms.
364 There
is
no
in
difficulty
that
showing
denominator
the
In fact the the last integrand may be replaced by t. tween the original integral and the one so obtained, i.
sin
t
in
difference bee.
the integral
6,
because
s
automatically tends to
-
-T-T
Thus we may
Theorem
205*
as
n
by theorem
increases,
continuous and bounded 49 , and hence intcgrable, in
is
finally state:
The necessary and sufficient condition for
9.
series of a function f(x), periodic with the period
Q...271,
< t^ d.
2n and
the Fourier
integrable over
converge to S(XQ ) at the point XQ , is that for an arbitrarily
to
chosen positive d
(
<
-2
the sequence of the values of the integral
,
J
-g-
f J
,
.
X
v
sin (2
n -f2 T-
)
1) t
,^ dt
o
forms a null sequence. Here
>
can assign d
<
N>
? and
0, so that
50
for every n
> N,
6
^L
f
\
ft'
sin (2
n
+
1)
t
C. Conditions of convergence.
Our preliminary first
investigations
two questions of
p.
have
prospered so far
355 may now be attacked
that
the
By
the
directly.
above, these are completely reduced to the following problem: Given a function
"
In fact.1 -T sin/
-
:
tends to 60
t
~
-:
/.sin
/
=
-\
1
in
h-" ;
the interval, and thus
itself
as l-~*0.
The student should make
it quite clear to himself that the second foractually equivalent to the first, although d need only be determined after the value of e has been chosen.
mulation
61
is
For
t
=
0,
we
attribute to
in the integrand the
value A.
should tend
C. Conditions of convergence.
Fourier series.
49.
a limit as k increases, and what, in
to
value of this limit?
Since in
this
that
365
case,
the
is
52
& has a fixed but arbitrarily small value,
integral,
the answer to this question depends only on the behaviour of q>(i) immediately to interval of the form t <Jj (^ (5).
cf.
Riemann's theorem 7
the
of
right
We may
< <
What
0,
an
say in
accordingly inquire the right
properties must cp(f) possess immediately to that the limit in question may exist? order in 0, of large number of sufficient conditions for this have
also:
A
which we
of
shall
them
renders
only
most
for
sufficient
been found, the great generality of which purposes. The first of these was
two,
explain
established by Dirichlet in the above-named paper (v. p. 356) and was the first exact condition of convergence in the theory of Fourier series,
The second
winch Dirichlet's work is altogether fundamental. due to U. Dini and was discovered in 1880.
in
i.
e.
1.
Dirichlet''s rule.
in
an
// cp
interval of the
is
(t)
monotone
< <
form
t
d
(
^
to
<5)
the right
is
206.
of 0,
then the limit in
we have
question exists, and
6
lim
Jh == lirn^
^
^j yw
^
o
where
y>
denotes
the
hand) limiting value lim (p(), which
(right
tainly exists with the assumptions
Proof.
1)
In the
first
made 53
cer-
.
place,
lim
The
existence of this limit,
tegral,
follows simply from
i.
e.
the
the convergence of the improper infact
that,
o
>
values #' and x" both
,
we have
19,
(by
> 0,
and any two
theorem 26)
x"
x" sin
given e
t
, ,
hence x"
u
i
52 There is no simplification in observing that it would suffice tor k to tend to -f oo through odd integral values. 63 In fact, as
Chapter XI. Series of variable
366 as
Now,
we saw on
360, equation
p.
sin f _j
(2
*n=J6
= 0, 1, 2,
are
for
n
On
the other hand, the
.
.
.
,
all
n
(b),
+ 1) A/
Stai
=~
the integrals
, t
**
Therefore
.
terras.
we
also
have
t
w
~>
~
numbers
o
the developments on p. 364) form a null sequence, by theoiem (cf. Accordingly we also have
6.
n
_
.
Since,
this
however
19,
(v.
V
.
Psin (2 n
,
theorem
-f-
1)
*
TT
25),
implies that the above-named limit has the value 2)
By
^-.
a constant K' exists such that
1),
;
'o
for every
a?^0, and
therefore a constant
K(=2K')
such that
exists
b
a
for every a, b 3)
^ a <^
such that
Suppose
>
given
&.
and choose a
positive
<5'
^
,
so that
6,
and we
Writing 2
we
then have
fk
Jk
'
/A sin*
tending to
t
as k
>
+ oo,
by theorem
3G7
C. Conditions of convergence.
Fourier series.
49.
9 can accordingly choose k so large that
|
/
<
'
/
fc
fc
|
-g-
>
for every k
A'.
Further, T
2 f
/
sin
A =-J o For
the
ft *
,
2
.
sin
ft
o
second of these two
quantities,
we have * sin*
-,,
>
and we may accordingly choose &
>
&'
so large that
For / ", the first of the two quantities on the right second mean value theorem of the integral calculus theorem 27), which gives, for a suitable non-negative d" <^ <5',
for every A A of (d), we use the 19,
The
.
"ir^
J
latter
.
fc
integral
=
-^-dt and
therefore remains
< K in
absol-
^(5"
ute value, by
2).
Accordingly \
I
Combining the
=
<1 JT"\ x ^ I
S .
3~A'
.K A
<
e
<^
'
3
three results of this paragraph,
by means of
A = (A-A') + A"-f/r> we
see
given e !> 0,
that,
Thereby the statement 2.
Dtni's rule.
is
we can choose k
so that, for every k
>^
,
completely established.
If lim cp (t)
=
q>
exists,
and
if
for every positive
T
l9'
J
V* ^ >o1
(
T
remain
less
than
^
a fixed
number,
positive
then
lim/
fe
exi
fr-V-f-oo
More
shortly:
has a meaning.
If
the integral
I
^-^
^-^dt
t
which
is
improper
at 0,
368
Chapter XI. Series ot variable terms.
When
Proof.
r decreases to 0,
tonely but remains z
0,
Given
bounded;
which we denote
e
> 0,
the
therefore
it
for brevity
we may choose
by
a positive (ft
above integral increases monotends to a definite limit as
Vo
I
d'
<6
so small that
dt< ^'
J] Writing, as in the previous proof,
the difference (/
ft
k'
so large that
|
/k') tends /k'| < Jk
by theorem
to 0, -|-
for every &
>
before, with a suitable choice of &
&'
we
6,
> A'.
and we may choose Further, as
we saw
also have 8
J for every
i.
e.
k
when
>&
chosen,
d' is suitably
we conclude
cisely as before,
which proves the
We may 3.
Finedly,
-
1
7
" fc
<
also remains |
that, for
every k
>&
.
-q-
Thus,
pre-
,
validity of Dini's rule.
easily
MpscMtz's
deduce from rule.
//
it
the two following conditions.
two positive numbers
A
and a
exist,
such that 55 fn
< ^ /
<5,
Jk
then
+
.
Proof.
65
that lim
The "L^c^'te-condition", rp (i)
=
cp Q
exists.
|
9>
-n
K ^'^
as ^-*0,
itselt
implies
Fourier series.
49.
C. Conditions of convergence.
369
<
d the former integral remains less than so that for every positive t as required. a fixed number and in consequence of Dints rule /,, >
=
exists, then
A* 9V The
Proof.
implies
the
0<^<(5j,
<->+o
existence of
boundedness of the
i.e.
Hence / *
this
ratio
of a
fulfilment
an
in
interval
of
the
with a
Lipschitz- condition
form
= l.
asserted.
fc
to
corollary
these
conditions
immediately ob-
is
tained
Corollary. // cp (f) can be split up into the sum of two or more functions, each of which satisfies the conditions of one of the four rules above, then lim cp (t) (p again exists, and the Dirichlet integrals Jk t->+o of the function (p(t) tend to
=
.
The above rules may at once be transferred to the Fourier series an of integrable function f(x), which we assume from the first to be 2 n and to be extended to all other real values of x in given <^ x
<
by the equation In
order
to a
the Fourier series
thai
sum s(#
)
at the point
XQ
generated by f (x) should converge the integrals
,
,5
T
/H
=
2 f -jJ
f
.
N
v(*;*o)
~ dt
S
sin(2tt-f I)/
..
o
must, by theorem 9 (&05), form a null sequence, where, as before,
v (<; *)
= \ [/K + 2 + /(* - 2
- s K)
f)]
.
This form of the criterion shows, over and above Riemanris theorem 2O4, that neither the behaviour of f(x) immediately to the right of a? nor that immediately to the left of # , have in themselves any influence whatever on the behaviour of the Fourier series of f(x) at XQ What
,
.
is
is
that
the
behaviour of f(x)
important stand in a certain relation
to
to
the
that on the left of
x
should such that namely,
right ,
X
of
the function )
56
~ il/K + 20 +ffo> -
It suffices that cp' (0) should exist as the rv*ht 19, Def. 10), as in fact the possible values of 9 (t) for t
2
ft]
- S(XQ
)
hand differential coefficient (v. do not come into account. ~^*
370
Chapter XI. Series ot variable terms.
should possess the necessary and
sufficient
of the limit of Dirichlet's integrals It
is
not
Jk
properties
(206)
known what
57
for the existence
relative to q)(t). are. The four conditions
these properties convergence of Dirichlet's
given above for the integrals furnish us, however, with the same number of sufficient conditions for the convergence, at a special point XQ , of the Fourier series of a function f(x).
Each
of these conditions requires, in the
should tend to a limit
which we are about
A common
.
to set
Km i [f (*
(8)
-
must the
exist.
The
is
up
first
+ 2*) -/(*,,--
value of this
l/(*o
considered as a function of
At the same theorem 2 a, be 0.
above.
2
by theorem
limit,
of the Fourier scries of f(x) at X Q> This convergence is ensured if the function
=
for
assumption
all
accordingly the following:
sum
o)
instance, that the function
if
+ 2 + f(*9 ~ f)
the
rules
The limit
<)]
will
2,
then
also
be
the latter converges.
2
<)]
- s (* ),
one of the four conditions given in those conditions must,
fulfils
time, the value
We
conditions are satisfied:
207.
1 st
assumption. The function f(x) is defined and integrable (hence x 2 n and its definition is extended to all bounded] in the interval
^ <
real values of
x by means
2 nJ assumption.
of the relation
*1,
ln[f(x
+ 2t) + f(x
-2()],
where XQ denotes an arbitrary real number, but is kept fixed throughout, and its value is denoted by s (# ), so that the function
9 (0
=9
= | [/(* + = 0. lim 9
(': *o)
has a right hand limit
2,...
The limit
2
+ /(* -
2
I)]
exists 58,
- i (*o)
(f)
With
these joint assumptions, we have the following four criteria for the convergence of the Fourier series off(x) at the point x : 87
Define
e. g.
/ (x)
as entirely arbitrary to the right of x x 8 8) and, in x
an interval of the form x
~
<
<
+
x (but integrable x < x 0> let f(x)
- <
1 /( 2 *o x) say. The Fourier series off(x) at x is convergent with the (Proof, for instance, by means of Dirichlet's rule 208, 1 below.) 68 The two-sided limit then necessarily also exists.
sum
in
= 2
>
Fourier series.
49.
IMrichlct's rule.
1.
< < t
form is
C. Conditions of convergence.
monotone
//
an
in
the Fourier series of f(x) converges at
<5j,
371
interval
XQ and
its
of the
sum
59
equal to S(XQ ).
Dint's
2.
rule.
// for a fixed (otherwise arbitrary) positive
num-
ber d the integrals
p
dt
< < d,
T remain less than a fixed number for every r such that Fourier series of f(x) converges at x and its sum is S(XQ ).
The same
Lipschitz's rule. the integrals should
3.
that
numbers
A
and a should
be
exist,
is
true,
bounded, we
such
that, for
if
instead
stipulate
every
t
that
of
two
such that
the
requiring positive
4 th rule. The same is true, if instead of the Lipschitz- condition we require that
The application of these rules following corollaries:
is
made
considerably easier by the
Corollary 1. The function f(x) also fulfils the assumptions 1 and 2 Fourier series converges at X Q to the sum s(# ), if f(x) can
and
its
be split up into the sum of two or any fixed number of functions, each of which satisfies these two joint assumptions (for a suitable s) and in some neighbourhood of x fulfils the conditions of one of the above rules.
Corollary
2.
Similarly,
tion 2 that each of the
lim f(x t->+0
should
exist,
it
suffices to stipulate in place of
two (one-sided)
+ 2 = f(x + 0) f)
and
assump-
limits
lim f(x
- 2 = f(x - 0) 1)
f-v+0
and
^=/t*o +
that the
2 ')-/'(*o
two functions and
+
=
-
-
-
2 t) f(x (t) f(xQ 0) should each, individually, satisfy the conditions of one of the four rules. The Fourier series of f(x) is then convergent at XQ and has the sum * (*o)
= ![/(*(> + 0) + /X*o -<>)] One
or two special in applications, corollaries:
portance
cases,
which, however, are of particular im-
may be mentioned
in the following further
69 In case it converges at XQJ the Fourier series of a function f(x) satisfying the assumptions 207 accordingly has the sum /*(#) if, and only if, the limit s(a? ), whose existence is stipulated in the second assumption, =f(x ). Similarly in the case of the following rules.
208.
Chapter XJ. Series of variable terms.
872
Corollary 3. If f(x) satisfies the first assumption and is monotone both to the right and to the left of x , the limits mentioned in the preceding corollary exist, and the Fourier series of f(x) converges at XQ to
sum s(z
the
)
= y [f(x + 0) Q
-f-
f(x
Hence,
0)].
more
still
particularly:
The Fourier
Corollary 4. first
will
assumption
series of a function f(x) which satisfies the at the point x and its sum will be the
converge
value f(x) of the function at that point, monotone on either side of XQ
if
f(x) is continuous at
X Q and
.
Corollary limits
If
5.
f(xQ
further,
if,
o-o
Um
the first assumption, and the two both the (one-sided) limits
satisfies
f(x)
0) exist;
and
then the Fourier series of f(x)
exist;
have the sum
s (XQ )
will
converge at XQ and
will
= ~ [f(x + 0) + f(x - 0)]. Q
Corollary 6. The Fourier series of a function f(x) which satisfies the assumption will converge, and will have as its sum the value of the function, at any point XQ at which f(x) is differentiate.
first
50.
Applications of the theory of Fourier series.
As we see from
the
rules
of
convergence
developed
above,
extremely general classes of functions are represented by their Fourier This we propose to illustrate by a number of examples. series.
The
function f(x) to be expanded must always be given in the 2 n and must possess the period 2 n: f(x 2 n) <^ x f(x). series is in Fourier obtained in the then, general, corresponding
The
=
<
interval
form
In particular cases, the sine- or cosine-terms if
f(x)
(the
x
is
graph of f(x)
= k n,
= 0, (k
is
1,
symmetrical with 2,
.
.
n is
evident
partial
In fact,
.),
if
integrals.
-
bn
respect to the straight lines
and therefore
2n
as
may be absent
an even function,
n
2n
= / f(x) sin n x d x = f + / =
we replace x by 2 TT The Fourier series
,
n x in the second of these two
of f(x) thus reduces to a pure
cosine-series.
on the other hand, f(x)
If,
graph of f(x)
(the
k
= 0,
is
2,
1,
373
Applications of the theory of Fourier series.
50.
.
.
an odd function,
is
with respect to the points x
symmetrical
= kn,
and therefore
.),
271
n
.
an
= / f(x) cos n x dx = 0, o
Thus here the Fourier
series of f(x) reduces series. sine pure There are accordingly three different ways in which an arbitrary given function F(x), which is defined and integrablc in a <^ x <^ b, may be prepared for the generation of a Fourier series.
as
equally evident.
is
to a
t
l
method.
a ^> 2 n, a portion of length 2
b
If
n
cut out of
is
say cc^,x
transferred the interval (a, 6), a defined in we 2 71: thus obtain function the to point a; <^ x f(x) It is then defined for the whole #-axis 60 by means of the condition of periodicity f (x -^- 2 stant (b) in
=F
= /(#).
TT)
If 6
b^x<.a+27T
#
<2
<
define /(#) to be con-
?r,
and proceed
is
as before
61 .
^
method. Precisely as above, define a function f (x) in 5^ x TT / (2 77 (not 2 77) by means of /? (#), put / (#) #) in TT fg #
3 rd method. Define/(#) 0,
but put
f(x)~
for all further x's
The
/(2
TT
as
< x < put/(0) = /(TT) = < < 2 then again define /(#)
above for
^) in
?r
by the condition of
TT,
jc
TT;
periodicity.
functions which aie obtained by these methods from a given function F(#), and which are now suitable for the generation of a Fourier series, we shall distinguish as f^(x) t f%(x}, f9 (x)- Whereas /*j
three
certainly give a pure cosine scries and f% (x) a pure sinewill lead, as a rule, to a Fourier series of the general (x)
will
(#)
f
series,
form (unless, in fact, f (x) is itself already an odd or an even function). Since our rules of convergence enable us to recognize the convergence only at points XQ for which lim it
exists,
will
~^ [f(x + 2t) + f(x -
be advisable
to
modify
our
2
t)]
functions
further at
the
=F
>
60 If b 2 TT, a portion of the curve y a (x) is left out of the representation altogether. If we wish to avoid this, we need only alter the unit of measurement on the x-axis so that the interval of definition of (x) has the length 2 TT;
F
i.
e.
we 01
substitute
Or
a
+
-
^
- x for
x.
77
else give the interval of definition of fying the unit of measurement on the *-axis.
13
F (x)
the exact length 2
TT
by modi-
(G51)
Chapter XI. Series of variable terms,
374 junctions 2 k
by writing
TT
/(0)=/(2A7r)--:lim x ->
whenever
this
limit exists.
provides not exist,
the
condition
our resources
as with
we
=
209.
1.
now go on
3
(2 k
E=
77)
= 0.)
If
/3
(x),
limit
this
and does
value f(2kir) does not come into account, cannot discover whether the Fourier series con-
to concrete
examples.
Here
F(x) ss a =f 0.
Example. fi(x]
=/ (0)
For corresponding reasons we have already put
verges there or not. ^ above. /a 00
We
certainly the case for
is
(This
/3
functional
the
"
I
/g (a;)
EE a, while for
we have
to put
x
=
n
< < n, <x<2
and x
= n>
x
a
a
7i.
Dirichlet's conditions are evidently fulfilled at every point (inclusive of the junctions), for each of the three functions. The expansions obtained
must accordingly converge everywhere and must represent the functions For f^(x] and f%(x), however, they are trivial, as they themselves. reduce to the constant term 2
=
aQ
= a.
For fs
\f3 (x)sm
nxdx =
we
however,
obtain:
2n
*
JT
(x) f
~ \smnxdx =
\s\nnxdx
\
smnxdx
t
n i.
e.
The expansion accordingly f
/
x
4 a
r
for
even values of n,
for
odd values
of n.
is
. ,
sin 3
x
,
sin 5
a;
5
or 3t
.
~T in sin 3 a?
.
sin 5 a?
at
and at
JT,
-- in
This establishes the second provides the
sum
of
the
examples given on p. 351, and whose convergence we were
of this curious series, of
already aware
(v.
185,5)
series, with the first of
connection
(v.
62
For x
.
which we
= ~,
,
m
are familiar
1 -
13
Example.
F(x)
----
_L r
~
J
L_
_|_
ax, in
an ax
in
Here
0<#<27r, and
at
<J
7i
B)
a;
in
at 2
<^
?r
in
<x<
at
and
rc,
rc,
2
at TT,
n
in
first
of the
210.
,
,
I
which establishes the
gives:
sin4ap
sin Hac i
rc,
rrc,
After an easy calculation, the expansion of -L.
'
2'
(a 4= 0).
o;
* 3
-4
a2jr x
=
13
a (2
s' u *
. . .
'
17
~ - JL
-
ax
\
special
entirely different
n
11
(
an
obtain
1,11, _
'
we
^,
122):
^ 4-
2.
375
Applications of the theory of Fourier series.
50.
x
n
ftn
T*-
in
<
at
and
examples of
at
p. 351.
Similarly, the
expansion of f9 (x) gives .
x
(b)
.
sin oc
sin 2 '
+
sin 3
x
sin 4
a?
,
T" in at rr, a?
Or, more shortly,
_f*in at
ft
-ir-
in
ft
<
< 05
<[ 2 rr
<
TT,
TT.
62 This and the following examples are already found, for the most part, in Euler's writings. Many others have been given by Fourier; Legendre, Cauchy, Frullani, Dirichlet and others. They are collected together, in a convenient form for refer-
H. Burkhardt, Trigonometnsche Reihen und Integrate bis etwa 1850, Enzyklopadie d. math. Wiss., Vol. II A, pp. 902920.
ence, in
376
The
Chapter XI. Series of variable terms. function f2
cos
(#),
a: ,
-Tz-
however, provides the expansion:
x
cos 3 53
"I
cos 5
,
x
_
,
h
H3
1
8 |
inn S The
=
of these expansions gives for x
first
x -=
third, for
known
the
known
the series, also previously
0, gives 1
+ V + 5^ + 7 +
'
'
=
'
from which we may immediately deduce the
series for
^;
the
to us (137),
8"'
relation
i+i+i+i+...=v 83 previously established (136, 156 and 189) in an entirely different way . On comparing the two results, we obtain the remarkable fact that in
<x^
TT
the function x
capable of the two Fourier expansions
is
sin 3
_
~~
4
With
cos
rcos_x "
L"!
lr
,
7
3y
x
cos 5 y ,
.
and
_.
.
"1 '
"^
'
32
a view to penetrating
6
'
J
further into the significance of these / (x) and a few of
still
graphs of the function
results, it is well to sketch the
the corresponding curves of approximation. This we must leave to the reader, and we shall only draw attention to the following phenomenon:
The convergence that of the scries 63
A
fifth proof,
a
and
sin
uniformly convergent in
term over that
this is
117 b.
interval.
we
tt
^-
t
uniform
is
sums
for all #'s;
not so
are discontinuous, the
The expansion 123 is made in 123, together
see that the expansion
f*
^
and may therefore be integrated term-by-
process, or 7r
a
"8
note 38 to 156.)
c
Now
shewn by a recurrence Hence at once
This method was
210
b, since their
quite different again, is as follows: f x 5 1, by the stipulations
uniformly convergent in with 199, 2. Putting x
is
of the series
210
essentially given
by writing cos 1
=
_
w== o(2w~+ l)
by
Euler.
a
t
=
z and using Example
*
(Cf. the note referred to in the foot-
first at 0,
377
Applications of the theory of Fourier series.
50.
the second at
In the former case, the approximation curves
TT.
the zigzag line representing the limiting curve along the lie its whole of length, whilst in (a) and (b) the corresponding state of affairs cannot occur (cf. 216, 4). and does not close to
=
cos a x (a arbitrary 64 but 4= 0, 1, Example. F (x) We first form the function /2 (x), and accordingly define
3.
a)
ax
cos
f
2,
,
.
.).
0^#lS7r
in
| cos a (2
.
TT
in
x)
TT
^x<2
TT;
function continuous everywhere, which by Dirichlefs rule will also generate a Fourier series continuous everywhere, which represents the function, and is necessarily a pure cosine-series. Here we have thus
/2
TT
an
(#) is a
n
JT
2 J cos a x cos w
x
jc c/
J [cos (a
+ w) ^ + cos (a
w)
A;]
d x\
hence, as a was assumed not to be an integer, 7Tdn
Therefore the function /2 cos OLX in
For ^ ^^
TT
^
we
TT,
from entirely
^+
(x) in is
TT,
f
i \
(^
l;
sma TT
n 2a
g'lT^a
^ ^2
or in other words the function represented by the series Jt:
TT,
:
obtain from this the expansion 117, previously deduced
different sources:
cos 77
^?
_
----
(X.TT
sin a
=
COt
7T
a 7T
TT
2 a = - + -=a -+a a 1
r=2
2
-^ 2
'
I
-
2 a
.
.
.
*
^,2 -('
2
We
24. thus enter the sphere of the developments of Of course the other series expansions there deduced may also be obtained directly from
Thus 212
our new source.
_TT_ ~___!__ a sin a
TT
~*~ - __ 1
2j*
a2
x
gives for
_
,
2 a __ ~__"^__ 3
2 a
a 2~- 2 2
2
Subtracting the cotangent expansion obtained just before,
^
1
~_
cos arr
~sirTa
~"
w mn
aw
_ ""
~2"
4a
"" a2
^T
4a
"~
2
a2
^
2
a2
-
32
~
we
further obtain
4a
""
"
*
"
a 2 ~^~5 2
and so on. b) If we now similarly construct an cos a x, we have cos a x in
/3
= (#)
[ 84
<# <
J
J
odd function /3
at
and
cos a (2
TT
at
(#)
from
TT,
TT,
je)
in
TT
<x<2
Because otherwise the cosine-expansion would become
TT.
trivial.
F (x)
=
378
Chapter XI. Series of variable terms.
Here a may is
The
easily
<x 213.
worked
integral values without reducing the result to
assume
also
a trivial one.
coefficients b n are obtained
from
integrals
whose value
out, and they lead to the following expansions, valid
in
a) for
0,
-,
T .-
j8)
for a
. .
.
2,
1,
tin
*
\ [12
:rp
4 r
2
*
sin
^
.
sin 2
2
all
+ 32 ~^ sin 3 * + x
4
+4
-rr 2
,
5*
sin
.
.
p
2
sin 4
x
'
'
']>
+
if /> is .-
,
, '
'
it
'
.,
.
p r
is
innumerable numerical series
the above series,
odd.
may
be
taking particular values of x and a.
4.
The
5.
If the function
F (x)
treatment of
cos
x+
sin a
F (x) =
tion of a pure cosine series,
214.
+ g,
/>
p*
deduced by
-.-inSa
= i = integer
cospx =
From
+
x leads to quite similar expansions.
log (2 sin ^ J
we
arranged for the genera-
is
<x<
obtain the expansion, valid in
^. + ^^ +... = -
log
(2
sin
J)
TT,
.
however, to be shewn by a special investigation that the result
It has,
holds in spite of the fact that the function is unbounded in the neighand 2 ?r, and therefore is not (properly) intcbourhood of the points V where this will follow quite simply in another below, 55, grable. (Cf.
way.) 6.
F (x) = e yx +
Example.
We
cosine series.
~ ax ,
a 4= 0,
is
be expanded in a
to
have therefore to take /
f
\
t
F (*)
^*^* in ^x<2
in
TT
J*W~~\F(2ir-x)
IT.
After working out the extremely easy integrals giving the coefficients a nt we obtain JT
which t
go*
is
for 2 a
+-
g-qv
valid in TT
_ -
a
_1
TT 5=1
x
for simplicity,
to the relation,
^+ we
TT.
If
we
^
substitute
e. g.
,
x
=
TT
and write
are led, after a few simple transformations,
valid for every
IT _I
a
,
t
4=
_ 1 _ Al
^2 Z
.
1-
i.
to an "expansion in partial fractions" of this remarkable function;
e.
its
379
Applications of the theory of Fourier series.
50.
we
expansion in power series function
the
-~
was considered,
t
can at once deduce from
-JT?
the latter by multiplying
by
t
2
for
and adding
24, 4,
where
our function reduces to
1.
Various remarks. The very fact that trigonometrical series are capable of representing extremely general types of functions renders the question as to the limits of this capacity doubly interesting. As was already remarked, necessary and sufficient conditions for a function to
new
be representable by
of
kind, for
its
all
Fourier series are not known.
attempts to build
damental properties (continuity,
On
the con-
it
up
directly
differentiability,
by means of the other funetc.) have so far
mtegrability,
We
failed. all
its
we find ourselves obliged to consider this as a fundamental property of functions,
trary,
must deny ourselves the satisfaction of supporting this statement by working out relevant examples, but we should nevertheless like
details
put forward a few of the
in
to
facts in this connection.
of the conjectures which will naturally be made at first sight is that This is not the all continuous functions are representable by their Fourier series. case, as du Bois-Reymond was the first to show by an example (Gott. Nachr. 1873, 1.
p. 571)
2.
tinuous
One
.
On is
the other hand, to assume the function differentia ble as well as con66 is necessary, as is shown by Weierstrass* example of a uni-
more than
formly convergent trigonometrical
a n cos(b n irx) n=l
which accordingly a function that
series, viz.
/
00
is
(0 \
<
a
<
1,
b a positive integer,
is the Fourier series of its sum (v. 200, continuous but nowhere differentiable.
1 a),
ab >
3 1 4-
\
^ n), /
but which represents
We
65 now have simpler examples than that mentioned above. E. g. L. Fejer has given a very clear and beautiful example (J. f. d. reine u. angew. Math., Vol.
137, p. 06
1875.)
1.
1909).
Abhandlungen zur Funktionenlehre, Werke, Vol.
2, p. 223.
(First published
380
Chapter XI. Series of variable terms.
Whether continuous functions exist whose Fourier series are everywhere is not at present known. 4. A specially remarkable phenomenon is that known as Gibbs* phenomenon 67 which was first discovered (by J. W. Gibbs) in connection with the series 2 10 a: 3.
divergent
,
The
curves of approximation y neighbourhood of * = 0. More
maximum
greatest
ordmate.
Then
n
6a
of
->
y
=
17808
-r\
overshoot the mark, so to speak, in the by (n the abscissa of the between and IT and let yn be the corresponding
precisely, let us denote
sn (x)
but
;
=^ s n (x)
n does not
Thus
value g equal to
(1
which the curves y
= sn (x) approximate
210
a (p. 351,
fig.
length exceeds the
mation curve portion
is
is
7), a
.
.).
.
-*-
it
-, as
we
should expect, but tends to a
appears that the limiting configuration to
contains, besides the graph of the function stretch of the >--axis, between the ordinates g, whose
2
"jump" of the function by nearly
drawn
for n
=
9
m
the interval
.
In
-.
.
.
?r,
th
the
fig. 9,
and for n
=
approxi-
44 the
initial
given.
51.
Products with variable terms.
Given a product of the form
//a =i whose terms
are functions of x,
+/.(*)).
we
shall define (in
complete analogy with
the theory of series) as an interval of convergence of the product, an interval at every point of which all the functions fn (x) are defined and the product
/
itself is convergent.
Thus
e. g.
the products
/(:
5). //(-
+
).
//('
+ <-
*).(.
are convergent for every real x t and the same is true of any product of the form 77(1 + an x), if 2 an is either absolutely convergent (v. 127, theorem 7) or a con2 ditionally convergent series for which 27 a n converges absolutely (127, theorem 9). }
For every x in f> the product then has a specific value and therefore defines a determinate function F(x) in /. again say: the product
We
represents the function
The main
in /.
F (x)
question
in /, or: is
F (x)
as before:
expanded in the given product how far do the fundamental prois
(of continuity, differentiability, etc.) belonging to the terms fn (x) hold for the function (x) represented by the product? Here again the
perties still
F
67
J.
Gronwall, 68
first
W. Gibbs, Nature, Vol. 59 (London 1898-99), p. 606. Ober die G&fasche Erscheinung, Math. Annalen, Vol.
The maxima
in the interval occur at
being the greatest
x
=
maximum. The minima occur
*
~-.,
.,
at
x
=
2
*
-
f
.
.
4
77
-
n
Cf. also T. II. 72, p. 228, 1912.
IT
n
, . .
.
.
.
,
the
Products with variable terms.
51.
381
answer will be that
this is the ease in the widest measure, as long as the are uniformly convergent. considered products What the definition of uniform convergence of a product is to be is almost obvious if we refer to the corresponding definition for series,
we
since in either case
are essentially concerned with sequences of functions down the definition corresponding
However, we shall set (cf. 190, 4). to the 4 th form (191, 4) for series: Definition
The product II
.
convergent in an interval
depending only on
/,
e,
//,
+/(*))
(1
given e
> 0, a
not on x, can
is
said to be uniformly
single number
N=N
17
(z)
be chosen so that
x in /. for every n ~> N, every k > 1 and every It is not difficult to show that with this definition as basis the theorems 47 hold substantially for infinite products 71 We will, however, leave the details to the student, while we prove a few theorems which are less
of
.
which
far-reaching, but
will
amply
suffice for all
which have the advantage of providing us
our applications, and
same time with
at the
for the uniformity of the convergence of a product.
Theorem and
+ fn (x))
The product //(I
1.
We
first
criteria
have
converges uniformly in
/218.
a continuous function in that interval, if the functions fn (x) continuous in / and the series 2 fn (x) converges uniformly in /.
represents
are all
\
\
Proof.
If
2
converges in /, so does the
product // ( 1 (- /(#)), by 127, theorem 7; indeed, it converges absolutely. Let F (x) denote the function it represents. Let us choose m so large that 1
fn (x)
l/m+l (*) for
every x
in
J
I
|
+
I/H 2
(*)
and every
I
+
+
k^l;
Ifm+K (*)
this
is
|
<
possible,
1
by hypothesis.
Consider the product
// (!+/(*)),
n=m+l 69
in
The symbol
4648;
cf. p.
o in this section again holds only with the 327, footnote 1.
same
restrictions as
70
This definition includes that of convergence. If the latter be assumed, the "remainder" rn (x) and (1 n+ (#))(! H fn +* 0*0) define uniform convergence as follows: //(I n (x)) is said to converge uniformly in/, if for every (xn ) in/, however chosen, rn (x) -> 1.
+/
we may speak of
i
+/
Writing //(I
+/(*))
= Pm (x)
and 77
i/=l
quite easily deduce
fv
(x) there,
i>
e.
g.
the continuity of
by means of the
(1
= rofl
F (x)
at
x
+/(*))
= Fw (*),
we may
from that of the functions
relation
F (x) - F (*) - P m (*) Fm (*) - Pm (*) Fm (*) = [Pm (x) - P m (*)] Fm (x) + [Fm (*) - F m (*)] P m (*). 13
(051)
382
Chapter XI. Series of variable terms.
and denote
its
partial products this product.
represented by Fm,
by p n
We
~ Pm+l + (Pm+2 ~~ Pm-\ + = Pm+l + Pm\ */w+2 + Pmj l)
1
i.
Fm (x)
e.
from In
30.
fact, for
2
>
n
(x),
have
Fm (x) be the function
m. Let 190, 4)
(cf.
+ (Pn */w+3 +
+ + Pn-l '/ +
Pn-l)
as is indeed evident by an infinite series Now (by 192, theorem 3) this series converges uniformly in J. m, we have every n
is
also expressible
>
and
!/(*)
I
n*=M-\-l
of
uniformly convergent in /, by hypothesis. Accordingly the sequence its partial sums, i. e. the sequence of functions p n (x), tends uniformly
to
Fm in/, so
is
CO
that the product n
and
in /,
//
+/n (x))
(1
-m+l
seen to converge uniformly
when we
this property is not affected
Fm (x)
is
prefix the first
m
factors.
necessarily continuous in /, since the terms of the By 193, The same series which represents it are all continuous in that interval. is
is
then true of the function
F (x) =
A
Theorem 2\fn
.
.
// the functions
2.
but
(x) |, also differentiable in /.
of J
+ /i (*))
.
+ fm (*)) Fm (x),
(1
q. e. d.
similar proof holds for
if not only is
(1
where
F (x)
is
4=
fn
'
\fn (x) Moreover its \
given by
(x) are all differentiable in
converges uniformly in /, then
J and F (x)
differential coefficient at every point
72
Proof. The
proof may be put in a form analogous to that of the previous theorem; however, in order to make other methods of attack familiar, we will conduct the proof by means of the logarithmic function, as follows.
72
8
-
For
-
is
If
g
m
Let us choose
so large that
(x) is difTerentiable at a special point
called the logarithmic differential coefficient
(x)
= gi (x)
-
a
(x)
.
gk
(x),
we
SLM _ Si' W g(x)
provided that the functions
g, (x)
,
have, as
*/(*>
"*"
g,(x)
x and g
ofg
is
(x),
(x) 4=
because
,
log
|
g
(x)
|.
well known,
"*--*- *(*> ,
there, the ratio it is
,
& ffi
A (x) are all differentiable at
'
the point
x
in question.
51.
> m,
for every x in /, so that, in particular, for every n I
By 127, theorem
/(*)!
8, the series
J? log fi=m + l is
(!+/(*)) The
then absolutely convergent in /.
series obtained
from
it
by
differen-
__
term by term,
tiating
383
Products with variable terms.
!+/(*)'
is
l/n (*)
I
<\
from
that of
as before,
every
for
< 2,
if,
> m
n
y
1 |
+ fn (x) > |
and
\
therefore
so that the uniform convergence of the last series follows '
Z\fn
(x)
|.
Accordingly (by 196)
we put
//a +/(*)) =
n=m + l i.
For since
indeed also uniformly (and absolutely) convergent in J.
*",,
e.
^ m
n
log (1
+/(*))
= log Fw (*).
i-1
Since finally
and the
F (x)
last factor
on the right has been seen to be differentiate
itself is different! able in /.
If,
leads at once to the required result, tioned in the preceding footnote.
further,
F (x)
by the rule of
in
J
9
the last relation
4= 0,
men-
differentiation
Applications. 1.
The
219,
product
Fm (*)- II (l-S)' n=mH X is
(
uniformly convergent in every bounded interval, since, with
fn
(x)
w>0 x
=
>
1 ,
evidently a uniformly convergent series in that interval. The product accordingly defines a function m (x) continuous everywhere, which, in particular, is never is
F
zero in * |
|
<m
-f 1.
This function
is
also differentiate, for
S \fn
=
'
(x)
|
2 |
x
\
S-
1
384 is
Chapter XT. Series of variable terms.
uniformly convergent in every bounded interval.
Hence
for
|
x
\
<m
Fm^(x) _ 1 + f 2*_ *(*) * n JZ+i*-* ,
By 117,
this
however implies
*'<*>-- w ff cot.. "w
G m (#)
where
- G
**
J
J7 _ *-i- ^i.
*v,(*)
'<*>
G TO (*)'
,,-i
denotes the function sin
x
TT
=
1, . . . , ih w, to its limit (obviinterpreting this expression as equal, for x 0, ously existent and -J- 0) as jc tends to these values. (The corresponding convention is made for the middle term in the relation immediately preced ing.) If however two functions (x) and (x) have their logarithmic derivatives equal in an interval,
G
F
two functions never vanish, it follows that they can only constant factor (4= 0). Hence, in x < m -f- 1, in \vhich the
|
sin
where
c
is
last relation
side ->
have,
c,
first
=
x
TT
x
c
To
a suitable constant. by x and let x -> 0.
differ
//
(l
\
<
a
determine
The
hand
left
2
its value, we need only divide the side then -> ?r, while the right hand
because the product is continuous at x 0. Accordingly c for x m 1, but hence, as m was arbitrary, for all xt |
by
\
TT
and we
+
sin
= 7i x
Tx
.
M---1
m
2 and J:, as well as the remarkable product This product, and those discussed below 257, 9, and many other fundamental expansions in products, are due to Eider. 2. For cos IT x we now find, without further calculation,
COS7I* =
The
3.
sin 2 --
7T
x
n
"X
=
sine-product for special values of x leads to important numerical
product expansions.
E. g. for
==
A;
^>
/(2>-i)(2>i " As
g
i
j.
2
|
->
1,
we may
clearly
omit the brackets, and
we
+
accordingly write
ft
77
=
2
'
2
'
4 4 6 1 6-! 8 '
'
'
8
2~~ l-3-3-6-6-7-7 ; 9...
(Wales' Product). Since
it
follows from this that
/2\2
/4\ 2
246 i" 3 '5 78
/
2& 2~v-
2k -
1
rv*~"
Arithmetica infinitorum, Oxford 1656.
V7r
(Cf. pp.
218
9,
footnote
1.)
385
Exercises on Chapter XI.
we
same tune the remarkable asymptotic
obtain at the
2. 4. 6..
.2*
relation
2*
middle coefficient in the binomial expansion of the (2 w) th 2n of all the coefficients of this expansion or for the cothe sum 2
for the ratio of the
power
to
efficient of
xn
in the
expansion of
.
x
yl
The sequence
04.
(v.
128
4)
as 77(1
-h
of functions
cannot be immediately replaced by a product of the form 77(1 for
diverges
J
a; =J=
0-
this
However,
divergence
is
of
-f /*,(#))
such a
kind that
By 1S8,
2 and 42,
3, this
implies that
= tends, as
only
if
x
n *-QO, -\-
0,
a specific
to
2,
1,
.
.
.
.
and
finite
limit,
4"
(*)
0;
the latter, of course,
Accordingly
a definite number for every x 4= 0, 1, 2, .... The function of x so dea tn ma-function (T-junction). It was introduced into analysis by Euler (see above) and, next to the elementary functions, is one of the most important in analysis. Further investigation of its properties lies outside the scope of this book. (Cf., however, pp. 439 440 and p. 630.) is
fined is called the
G
Exercises on Chapter XI. I.
Arbitrary series
154* Let (nx) denote the to x,
or the value 4-77,
if
two consecutive integers. aj's.
The
nx
function represented
difference between
lies
The
~^-
series
by
ff
nx and
the integer nearest
exactly in the middle of the interval between
it,
however,
(p, q integers), while it is continuous for all irrational values of x.
155.
of variable terms.
*
is
for all
s
uniformly convergent for discontinuous for x
converges uniformly for
all x's
/ sin
^~2 q
^
other rational values of x and
a n ->0,
2^
all
n x \*
*(- .IT) Does this remain
true for an
~
1 ?
XL
Exercises on Chapter
336 156. The products
c)
77-l + sin-,
converge uniformly
The
157*
l
//
d)
158.
whose
series
sums have the values
partial
it
it
=
27l -
Draw
every interval?
in
of continuous positive functions certainly converges represents a continuous function F(x). (Cf. p. 344, Rem. 3.)
---
J>]
^=
in
converge uniformly
gr
every interval?
Is the
represents continuous?
a situation of the following kind occurred:
160. In the proof of 111, expression of the form
An
sn (x)
A series 2 fn (x) if
159. Does function
sin
every bounded interval.
in
converges for every x. Is this convergence uniform the curves of approximation. uniformly
+(_!)"
F (n) = considered, in which, as n increases. At the May we infer that
(n)
+ a, (n) +
.
+
for every fixed k
is
same
time, the
lira
F(ri)
fc
(n)
+
.
.
.
+ a fn (n)
the term a^ (n) tcnJs to a limit a^ of terms increases, p n *> QO
t
number
=
2,"
.
fcf
/i=J
n-><x>
provided the series on the right converges? missible if, for every h and every n
Show
that this
is
certainly per-
t
|
ak
(n)
remains \
and
2,"
fc
yk
Formulate the corresponding theorem for infinite products. converges. Exercise 15, where such term-by term passages to the limit were not allowed.
(Cf.
>
161. The two series
x4
x9
<x<
are both convergent for
x6
x9
x>
Q 1
and have the same sum
-
-
log 2 for x -
-f-
1
.
Li
What
is their behaviour when x * vergent for x >! 1 ,
for
0?
Examine
12112
the
two
in
< x <J 1
series,
con-
a;->4-H-0. 162. The
is
1
its
sum?
series
Is its
J?
^
convergence uniform?
-^
converges
What
Kxercises on Chapter XI.
163. Show
that, for
x ->
164. Show
that, for
#->!
a)
2
~
I
0,
a
whose
sums have the values
partial
sn (x)
over an interval with endpoint 0.
not be integrated term by term the curves of approximation.
may
Draw
Fourier scries.
II.
166. May we deduce from a)
*
a:a"*""T'
i
n=i series
2
w *n
n-i 1)
lo
*
*
n ~\ Vf (l-*)'Jt/ (-
165. The
,
rrr^r -" 4~
n=l
M C)
+
1
387
the series
210 a by y
integration term
by term:
^-^~-~
fi=i
'2
_ In
_ _
y
which intervals are these relations valid?
167.
* cos -
What would be
(Cf. 297.)
same way, deduce from 210 c the
In the
(2
relations
n-
the results of further integrations? In which intervals are these
expansions valid?
168. From 209, 210, and the relations in the two preceding exercises, deduce the following further expansions and determine their exact intervals of validity: .
a)
cos*
--
cos 3 x
3 cos 3 x
+ ,
,
-- + ...=
cos 5 x g
,
n
_,
cos 5 x
b) v
C)
.
x n-- 843 +-^5-+ sin
sin
a;
,
,
169. From 215, deduce
...
further
or by differentiating term by term. the new series so obtained?
Is
= nx/n*
x*\
_(___),
etc.
expansions by substituting n the latter operation allowed?
x
for
What
x
are
the
Series ot complex terms.
Chapter XII.
388
ax ? What is 170. What are the sine-series and the cosine-series for e sma; ? Show that the latter is of the form complete Founer expansion of e a 2
H- 6j sin
where a v and 171
x
-
a 1 cos 2 x
6 3 sin 3
x -\- a 4 cos 4 x
-f-
b.
sin 5
X
h 4-
-
b v are positive.
If
a;
and y are positive and
fl
~-^
if * *
sinwscosny
-2-
Determine the values of the integrals
n
If ,r
,
ax
Jsinx (The former
=
1
,
and
f J
sin
a;
,
ax.
I
x
'37498..., the latter
=
1-8519
)
173. For every x and every n, sin 2
smx-i
^
#
f J
r--
sin
a;
T-"-
o
where the bound on the right hand
side cannot be diminished (cf. the preceding
exercise).
(Further
exercises
on special Fourier series will be given
in
the next
chapter.)
Chapter Series of 52. After
XII.
complex terms.
Complex numbers and sequences.
we have
discussed in detail, as in Chapter
I,
the
modes
of
the concepts essential for building up the system of real numbers, no new difficulties are raised by the introduction of further
formation of
all
types of numbers
algebra are
known
Since the (ordinary) complex numbers and their to the reader, we may accordingly be content
with briefly mentioning one or two main points here. 4 that the system of real numbers 1- I* was sho wn
m
is
in-
any further extension, and is, moreover, the only system capable of symbols satisfying the conditions which we laid down for a number system. Yet the system of complex numbers is a system of bymbols to which the name of number system is applied. This apparent contraof
52.
389
Complex numbers and sequences.
removed. For our definition of the number concept sense an arbitrary one, as we emphasized on p. 12, footnote 16: A series of properties which appealed to us essential in the case of rational numbers was raised to the rank of characteristic properties of numbers in general, and the result justified our doing this, in all essenin so far as we were able actually to construct a system which possessed all these properties. tials, a single one, diction
was
is easily in a certain
If
we
desire to attribute to other systems the character of a system we must therefore of necessity diminish the list of char-
of numbers,
The question arises acteristic properties which we set up in 4, 1 4. which of these properties may be dispensed with first of all; i. e. which of them may be missing from a system of symbols without its becoming impossible to legard the latter as a
number system.
4
of a system of symbols, the first with Among which we may dispense, without fear of the system losing the character of a number system entirely, are the laws of order and monotony.
the properties
2
These are based, by 4, 1, on the fact that of two different numbers of the system, the one can always be called less than the other, and the latter greater than the former. If we drop this distinction and in 4 replace both the symbols < and ;> by 4=> ^ appears that the modified conditions
4
of symbols,
the
no other system
are
satisfied
by another more general system
system of ordinary complex numbers, but thai substantially different from the latter can satisfy them
namely
the system of (ordinary) complex numbers is a which, as is known, may be assumed to be of the where and y are real numbers, and i is a symbol whose form x x yi, 9 is for regulated by the single condition t 1, manipulation which the fundamental laws of arithmetic 2 remain valid without ex3.
Accordingly,
system of symbols
+
=
<
>
and are suitably replaced throughout ception, provided the symbols In short: the last-named for restriction, we may work by =4=. Except formally with complex numbers
exactly as with real numbers.
4. In a known manner (cf. p. 8), complex numbers may be brought into (1,1) correspondence with the points of a plane and may thus be represented by these: with the complex number x --J- yi we associate the point (x, y) of an ay-plane. Every calculation may then
be interpreted geometrically. Instead of representing the number x-\-yi by the point (x, y), it is often more convenient to represent it by a directed line (vector) coincident in magnitude and direction with the line
from
(0, 0) to (x, y).
will be denoted in the sequel by a single and unless the contrary is expressly mentioned or follows without ambiguity from the context, such letters will in5.
letter:
Complex numbers
z, f,
a, b,
.
.
.;
variably denote complex numbers.
390
Chapter XII. Series of complex terms. 6.
x+yi,
the absolute value (or modulus) meant the non-negative real value
By is
(am z, z=^tf), we mean the angle sin cp 1
= -py
we may
for
\
9
I/a;
+y
2 '*
complex numbei
by
* ts
which both cos(p
amplitude
=
and
~.
p
calculate with absolute values, the rules 3,
4 hold unchanged, while Since
the
When we
.
of the
z |
5.
loses
all
II,
meaning.
accordingly operate, broadly speaking, in precisely as with real numbers, by far the greater
same ways with complex
part of our previous investigations may be carried out in an entirely analogous manner in the realm of complex numbers, or transferred to the latter, as the case may be. The only considerations which will have to be omitted or suitably modified are those in which the numbers themselves (not merely their absolute values) are connected by the
symbol
<
In
or
>.
order to
avoid
repetitions,
which
this
parallel
course would
to all definitions and otherwise involve, we have prefixed the sign II valid word for word which remain onwards, theorems, from Chapter
when arbitrary
numbers are replaced by complex numbers,
real
We
the whole
of our preceding developments
what modification complex numbers. the
somewhat
required when we
is
A
few words
will
(this
with a few small alterations need only glance rapidly over
validity extending equally to the proofs, which will be explained immediately).
and
transfer
also
indicate at each place them to the realm of
be said on the subject of
different geometrical representation.
Definition 23 remains unaltered A sequence of numbers will now be represented by a sequence of points (each counted once or more than once) in the plane. If it is bounded (24, 1), none of its points with origin at 0. lie outside a ciicle of (suitably chosen) radius
K
Definition
27, and
28
25,
of a null
that
sequence,
and the theorems 26,
relating to such null sequences remain entirely unaltered.
The sequences
with
(zn)
i
-
(_;)n
are examples of null sequences whose terms are not all real. The student should form an exact idea of the position of the corresponding sets of points and prove that the sequences are actually null sequences.
The
powers in the general sense, and based on the laws of order for real numbers. They cannot, therefore, be transferred to the realm of complex numbers in that form (cf. 55 below). The fundamental notions of the convergence and divergence of a sequence of numbers (39 and 40, l) still remain unaltered, definitions in
of logarithms
were
7 of roots, of
essentially
Complex numbers and sequences.
52.
although the representation of zn -> a circle of arbitrary (positive) radius e
391
now becomes is
the following described about the point
l :
If
as
centre, we can always assign a (positive) number n Q such that all terms of the sequence (z n ) with index n n Q lie within the given circle. The remark 39, 6 (1 st half) therefore holds word for word, provided we in-
>
a complex number
terpret the ^-neighbourhood of
as being the circle mentioned
above.
In
setting
up the
definitions
40,
the symbols
2, 3,
<
and
>
essential part; they cannot, therefore, be retained unaltered. although it would not be difficult to transfer their main content
played an
And to
the
in
the
complex realm, we complex realm we
will
shall
drop them call
entirely,
and accordingly
every non-convergent sequence
'
3
divergent
.
Theorems 41,
1 to 12,
and the important group of theorems 43, 3, remain word for word the same,
with the exception of theorem together with all the proofs.
The most important of these theorems were the Cauchy-Toeplitz limit-theorems 43, 4 and 5, and since we have in the meantime gained complete familiarity with infinite series, we shall formulate them once more
in
this
place,
with
the extension indicated in
44,
10,
and
for
complex numbers.
Theorem
1.
The
of the
221.
matrix
20*
(A)
are
coefficients
assumed (a)
the terms in each
n
fixed
to satisfy the
two conditions:
column form
a null sequence, i.e. for every
0, as
1
<x>.
For complex numbers and sequences, we preferably use in the sequel
the letters
We
*,(,,....
+
-> * OO , that (zn ) is definitely say, in the case the limit CO, or tends or diverges (or even converges) to oo. That would be quite a consistent definition, such as is indeed constantly made in the theory of functions. However, it evidently involves a small inconsist2
might
|
|
divergent with
ency relative of numbers (
to the use of the terms in the real domain, that e. g. the sequence n l) n should be called definitely or indefinitely convergent,
according as it is considered in the complex or in the real domain. though, with a little attention, this may not give us any trouble, to avoid the definition here.
And even we prefer
392
Series of complex terms.
Chapter XII.
K
(b) there exists a constant
values of i. e. 9
any number
sum of row remains
the absolute
such that the
of terms in any one
than K,
less
for every fixed k ^> 0, and any n:
+ KlH
Kol Under these numbers
conditions,
*'
when (*,
= a kO *0 + a
a kn\<
\~\
*i>
-
an y nu ^ sequence,
zs
)
K
the
00
kl
Z\
= 2 a kn Z
H
n=o
9 form a null sequence Theorem 2. The coefficients a kn of the matrix (A), besides satisfying the two conditions (a) and (b), are assumed to satisfy the further
also
.
condition
3 or>
27 a kn
(c)
In
this
--=
n-O case, if z n
+,
A k ->
we have
V = **0*0 +
fl
fcl
k ->
as
1
oo.
also
=
*1+'''
iXn*,i -*f
n=o
(For applications of this theorem, see more especially 233, as well as 60, 62 and 63.) Unfortunately, we lose the first of the two main criteria of 9,
which was the more useful of the two. Moreover, the proof of the second main criterion cannot be transferred to the case of complex numbers, as it makes use of theorems of order throughout. In spite of this, we shall at once see that the second main criterion itself remains valid for complex numbers. The proof in all its forms in two different ways: either we reduce the new be conducted may (complex) theorem to the old (real) one, or we construct fresh foundations for the proof of the new theorem, by extending the developments of 10 to complex numbers. Both ways are equally simple
and may be indicated 1.
easily
The reduction accomplished
briefly: of
by
complex sequences splitting
=
to
real
sequences
up the terms into xn -\- iyn and f f
their
is
real
most and
=
imaginary parts. If we write zn -(- irj, we have the following theorem, which completely reduces the question of the convergence or divergence of complex sequences to the corresponding real
problem:
Theorem
322. if,
and only
yn converge 3
if,
to
The sequence
1.
s=
(xn
+ iyn
xn converge
to
)
converges to
and
the
=
-f- i
v\
imaginary parts
rj.
In consequence of
fore, as the
(z n )
the real parts
(b),
A k = Za n kn
is
zn 's are bounded, by 41, Theorem
absolutely convergent.
absolutely convergent and there2,
the series
zn ^ai cn n
=i!sk
1S
a l 80
Complex numbers and sequences.
52.
393
are null #n -*f and yn +rj, (xn f) and (y n 77) of same is the true of sequences. By 26,1, i(yn rj) and, by 28,1,
Proof,
(
(xn
zn
If
b)
-
a) If
xn
**,
-")
~~
e
l-
y}>
n
f
-
(
zn
)
a null sequence; since 4
is
|2OT
and (yn
f)
+ i(y I
are also null sequences, by 26,
rf)
2,
i.
we have
e.
both
The theorem
established.
is
The theorem Theorem conditions
namely,
sufficient, to
assign n
for every
For the convergence of a complex sequence (zw), the second main criterion 47 are again necessary and
2.
the
of
which we are aiming follows immediately:
at
n
> 0,
every choice of e
that, for
we should
be able
so that
>>
n Q and every
n'
>
nQ
.
Proof, a) If (2 n) converges, so do (xn ) and (yw ) by the preceding As these are real sequences, we may apply 47, and, theorem 0, we may choose n l and n 3 so that given e
>
I
xn'
xn
^
yn
I
<
o
f
r
ever y n
>n
n
>w
i
and every
n'
>n
t
,
and I
|
< TT
for every
9
and every
>
n'
Taking n Q greater than n x and n2 , we have accordingly, and every w' > n
n^
for every
n
> nQ
,
The
conditions of our theorem are therefore necessary. b)
conversely,
If,
given e only that n i.
e.
and
n' (by
> 0,
fulfils
n' are both last
>n
since
We
have
in
,
of the
conditions
so that
zn
we have
'
zn
general
and
-
theorem,
\<
e,
also, for the
footnote)
*n'-*< e *
the
|
and our
(z n )
we can determine n
<'
provided
same n
394
Chapter XII. Series of complex terms.
By 47,
and (yj are convergent, so that (zn ) must the preceding theorem; the conditions of our theorem
this implies that (x ] n
also converge,
by
are therefore also sufficient. 2. Direct treatment of complex sequences. In the treatment of real constituted our most frequent resource. intervals nests sequences, of In the complex domain, nests of squares will render us the same services:
223.
Let
Definition. for simplicity be
Q
,
Qlf @3
assumed
,
to the
parallel
and
contained in the preceding
is entirely
we
form a null sequence,
the sides
denote squares, whose sides will coordinate-axes. If each square
...
a nest. For nests of squares,
Theorem.
we have
if
the lengths
I
0>
l
lt
shall say that the squares
...
of
form
the
There exists one and only one point belonging to all the (Principle of the innermost
squares of a given nest of squares. Point.)
hand bottom corner of Qn be denoted by hand A point right upper corner by b n -}-ib*. 5 if to the and , square Qn if, only y belongs
Proof.
+ =x
n z
ta
the
-f- i
in
Now, a;-
Let the
anc*
left
me
consequence of our hypotheses, the
and
axis,
similarly the intervals
form a nest of intervals. There and exactly one point
the #-axis
is
iv\
intervals of the corresponding nest.
= +
intervals
= /n
Jn
= an
.
.
.
bn
exactly one point f on on the y-axis belonging to all the But this means that there is also therefore
all the squares f exactly one point f 117, belonging to Qn are now in a position to transfer definition 52 and theorem
We
to the
224.
on
n*...&n* on the y-axis,
.
54
complex domain:
is said to be a // (z n ) is an arbitrary sequence, accumulation or the point of limiting point of sequence if, given an
Definition.
arbitrary
6
> 0,
the relation
k-l< is
satisfied
one n
225.
>
for
an infinity
of values
of
n
(in particular,
for at least
any given nQ ).
Theorem. Every bounded sequence possesses at least one limiting (Bolzano- Weierstrass Theorem.) Proof. Suppose \z n K and draw the square QQ whose sides & an d *% AH them's on the parallels to the axes through
point.
\
lie
<
5 This statement at the same time expresses, in pure arithmetical language, the relations of magnitude framed in geometrical form in the theorem
and
definition
228.
52.
Complex numbers and sequences.
395
it, i. e. certainly an infinity of z n 's. (> is divided by One at least of the four the cooi dinate axes into four equal squares if there were only a must contain an infinity of zn 's. fcict, (In
are contained in
each, there would also be only a finite number the case) Let Q denote the first quarter' not @ which has this property. This we again proceed to divide into four equal squares, denoting by @3 the first quarter which contains an infinity of finite
in
number
in
which
is
,
1
The sequence Q , Q , Q2 , ... forms a nest of points z n , and so on. lies within the preceding and the lengths of the since each Q n squares, form a
sides
null
innermost point of For if e is given than
nest
this
>
and
m
7 ;
is
denote
f
the
f is a point of accumulation of (z n ). chosen so that the side of Qm is less
the whole of the square
-|-,
Let
namely (2K-^-J.
sequence,
Qm
lies
within the
6-
neighbourhood
an infinite number of points zn also lie in this it, Therefore f is a point of accumulation of (zn ) 9 and the existence of such a point is established. of
,
and,
with
neighbourhood.
The
second main criterion for the complex domain, theorem 222, 2, formulated above may now be established once more, but without any appeal to the "real" theorems, on the same lines as in 47. i.
e.
validity of the
the
of
Proof, a) If zn determine n Q so that
>,
i.e.
-C<
f)
(zn
and
is
a null
b) In fact,
\.
e.
If,
is
m>
nQ and n
Zn every M with "
Taking
K
*-* We
*
to
n
[see
part a) of
accordingly necessary.
conversely, the e- condition
if
>
For these ns and w"s, we therefore also have
the proof of 47].
condition
we can
*'-f<-
provided only that n and n' are simultaneously
The
sequence,
be
n
">
> m,
m
larger
lies
than
in all
is fulfilled, (z n )is
the
the
circle
m
of
certainly
radius
numbers
e
\z^\ 9
bounded.
round z |*3
|,
.
.
WC
6 regard the four quarters as numbered in the order in which the four quadrants of the xy -plane are habitually taken. 7 The process of obtaining this point corresponds exactly to the method of successive bisection so often applied in the real domain.
396
Chapter XII. Series of complex terms!
By our preceding theorem, point
limiting
+ f,
'
follows
it
there
Supposing
.
exists
tas at least one a second limiting poi&J that (zj
/
choose 6 e
_JL|r_ 3 K
t ;
< | l
which is positive. By 224, the definition of limiting point, we can choose n Q as large as we please and yet have an n n for which ' Thus zn n for which zn e and also an ri f above any number w , however large, there exist a pair of indices n and w' for which 8
>
|
<
|
>
>
|
|
<
Iv -*! > This
only a
finite
and outside
number
the
.
have
The
condition of the theorem
|<e
\z n
a series
2 an
sequence
of
As
our theory of
infinite
226.
Theorem. only
if,
2 $ (flj
of their
two series have s
A
the series
= + s'
is ,
there-
and consequently zn
*.
.
complex terms.
sums, the basis for the extension of been provided by the above.
partial
222,
series
2 9t (an
1,
2 an )
we have of
first
the
complex terms
is
convergent
of the real parts of its terms
sums
and
s'
s" respectively,
the
and
sum
if,
and
the series
Further
imaginary parts converge separately. the
we
chosen,
suitably
series has already
to
Corresponding
n
>n
complex terms must obviously be interpreted
of
its
If
n
therefore sufficient also 9
is
Series of
53.
the
circle
of points z n for every
fore
as
Accordingly f must be the unique of radius e round f there is
our hypothesis.
contradicts
limiting point,
>
of
if
these
2 an
is
; 5 ".
2 the second principal criterion (81) for remains unaltered in all its forms, and, at the same time, the theorems 83 deduced from it, on the algebra of convergent series, also retain their full validity. In accordance with
222,
the convergence of infinite series
85
we
also remains unchanged, Since, in the same way, theorem as before, distinguish between absolute and non-absolute con
shall,
vergence of series of complex terms (Def. 86).
hence
9 Hence we may also say: (zn ) converges if, and only and possesses only one point of accumulation. This is then
the limit of the sequence.
if,
at
it
the
is
bounded
same time
Here again we have Theorem. The vergent if, and only
the
series if,
397
Series of complex terms.
53.
2 an
both
of
complex terms
the series
2^i(an )
is
and
absolutely
2%(a n )
con-
are ab-
solutely convergent.
The proof
z
= x-\-iy
results
simply from the fact that every complex number
satisfies the inequalities (cf. p. 393, footnote 4)
In consequence of this simple theorem, it is at once clear that, with series of complex terms as with real series, the order of the terms is immaterial if the series converges absolutely (Theorem 88,1).
however, 2an is not absolutely convergent, either 2$l(a n ) or must be conditionally convergent. By a suitable rearrangement of the terms, the convergence of the series 2a n may therefore be desIf,
^3(0
troyed in any case, as in the proof of theorem 89, 2, that is: In the of series of complex terms also, the convergence, when it is not
case
depends essentially on the order of succession of the terms. (Regarding the extension to series of complex terms of Riemanns absolute,
rearrangement theorem
The
44,
remarks on the following page.) 4, as also the main rearrangement absolutely convergent series, still remain cf.
the
next theorems, 89, 3 and
theorem 90, which
relate to
valid, without modification or addition, for series of complex terms. Since the determination of the absolute convergence of a series
a question relating to series of positive terms, the whole theory of is again enlisted for the study of series of terms: that was Everything proved for absolutely convergent complex is
series of positive terms
of real terms
may be utilized for absolutely convergent series terms complex If we omit power series from consideration for the present, we 18 observe, on looking over the later sections of Part II ( 27), that series
of
the
developments of Chapter
X
are the
first
for
which there
is
any
question of transference to series of complex terms. Abel's partial summation 182, being of a purely formal nature,
and
its corollary 183, of course hold also for complex numbers, and so does the convergence- test 184 which was based directly on them.
The
may also all be retained, provided we on in 220, 5, in accordance with which all sequences assumed to be monotone are real. In the case of du BoisReymond's and Dedekind's tests, even this precaution becomes unnecessary: they hold word for word and without any restriction for arbitrary series of the form 2 a n b n with complex a n and b n Riemanris rearrangement theorem ( 44) is, on the contrary, essen-
keep
special forms of this test to the convention agreed
,
.
227,
Chapter XII. Series of complex terms.
398
a "real" theorem.
tially
In fact,
if
a series
2an
of complex terms
is
not absolutely convergent, so is one at least of the two series JJjR^iJ and 2%(a n), by 227. By a suitable rearrangement, we can therefore, in
accordance with Riemanns theorem, produce in one of these two
series a prescribed type of convergence or divergence. But the other one of the two series will be rearranged in precisely the same manner, and there is no immediate means of foreseeing what the effect of the It has recently rearrangement on this series or on 2an itself will be. been shown, however, that if 2a n is not absolutely convergent, it may be transformed by a suitable rearrangement into a series, again convergent, whose sum may be prescribed to have either any value in the whole complex plane or any value on a particular straight line in
this plane,
according to the circumstances of the case
The theorems 188 and 189
of
10 .
Mertens and Abel on
plication of series ( 45) again remain valid word for word, with the proofs. For the second of these theorems we must,
multi-
together it
is true,
rely on the second proof (Cesaro's) alone, as we have provisionally skipped the consideration of power series (cf. later 232). At this point we are in possession of the whole machinery required for the mastery of series of complex terms and we can at once proceed to the most important of its applications. Before doing so, however, we shall first deduce the following
extremely far-reaching
criterion.
Weierstrass* criterion u
228.
A n bounded,
with
where
00
.
<x
2a
A
series
is
12 complex and arbitrary, and
n of
complex terms, for which
A>1,
We thus have the following very elegant theorem, which in a certain sense completes the solution of the rearrangement problem: The "range of summation" i. e. the set of values which may be obtained of a series Z a n of complex terms is either a definite point, or a as sums of convergent rearrangements of Ea n definite straight line, or the entire plane. Other cases cannot occur. A proof is given by P. L&vy (Nouv. Annales (4), Vol. 5, p. 506, 1905), but an unexceptionable statement of the proof is not found earlier than in E. Steinits (Bedmgt kon10
vergente Reihen und konvexe Systeme, J. f. d. reine u. angew. Math., Vol. 143, 1913; Vol. 144, 1914; Vol. 146, 1915). For the (more restricted) result that every conditionally convergent series Sa n ** s can be rearranged to give another convergent series Sa n f =* s' with s' =t= s 9 W. Threlfall has given a fairly short proof (Bedingt konvergente Reihen, Math. Zschr., Vol. 24, p. 212, 1926). 11 J. f. d. reine u. angew. Math., Vol. 51, p. 29, 1866; Werke I, p. 185. 12 An equality of this kind may of course always be assumed ; we need only
An
=
"
as a definition. What is essential in the condition a n /) footnote to 166), that when a and A are suitably chosen as is here, previously (cf. It is substantially the same thing to assume that the An* should be bounded. + B n /n* wjth A > 1 and Bn bounded. ! + //!
write
H*
(
\
1
n
'
Series of complex terms.
53.
is
and only if, SR (a) > 1 If 0<8fl(a)<^l, invariably divergent.
absolutely convergent is
series
if,
399 .
For
JR (a)
<
the
both the series
n=o are convergent.
Proof.
1.
In that case,
and
it
if
Let |
An
cc
\
= (t-\-iy < K,
and
we
say,
follows at once that,
if
is
/?'
Now
suppose SH()
= /?^1-
for sufficiently large values of n, is
2\a n
\
3
a.
JL
divergent. If, on the other hand,
9ft
it
(a)
first
assume /?==
1
<
test,
the series -^|an
is |
In that case, since
follows from Gauss's test
=
> 1.
$
n
\
By Raabes
for every sufficiendy large n. therefore convergent. 2.
us
any number such that
^^.
an
let
write, as is permissible,
ft
172
that
< 0, our last inequality shows that
then
2
a n must Therefore 3b. If $K(a)
now
= = 0,
diverge. i. e .
>
f
it
is
we
easy to verify that
where
H>
1
and
is
the
^'s are again bounded.
then have
smaller
of
the
Accordingly,
if
two numbers 2 and c
Jl,
and
denotes a suitable constant,
2
13 As regards the series an itself, it was shown by Weierstrass, I.e., that this is The proof is somewhat troublesome. divergent whenever Ot (a) fS 1. 5^ 0? (a) fg 1 further more exact investigation of the series 27 a n itself in the case is given by A. Pringsheim (Archiv d. Math, und Phys. (3), Vol. 4, pp. 1 19, in 17. particular pp. 13 1902), J. A. Gmeiner, Monatshefte f. Math. u. Phys., Vol.
A
also
19, pp.
149103.
1908.
400
Chapter XII. Series of complex terms.
n
for every
^>
w,
It
say.
follows by multiplication that
n-1
>n
"n
Hence
an
|
so that 4.
>
\
2a n
Cm
am
-\
n>m,
for every
,
\
i---
again diverges
(cf.
17O,
finally,
If,
we
,
and
w
cannot tend
to 0,
1).
have to show that both
the
series
are convergent. I
Now
as in
a* + 1,
<1
.
an
|
so that |
an
1.
we
have, for every sufficiently large n,
8'
n
diminishes monotonely
|
fore tends to a definite limit ^> 0. a)
the series J"(|an all
moreover,
its
from some stage on, and there Accordingly,
|# n + i|)
|
f
with
,
is
convergent, by 131, and has,
terms positive for sufficiently large
n's.
Now
1-
= since the fraction
on the
when n
that
*-f-c
right
on the
than a suitable constant A.
converges with2(|
n
hand side tends
left is,
By 70,
~~
to the positive limit
for every sufficiently large 2, this means that 2\a n
We can
|0 w +i|)-
|
V
show more
n, less w hl .
precisely,
|
how-
ever, that
b)
an
+ 0.
For
again follows, by multiplication, from
it
(n
^
that
The (cf.
hand
right
17O,
1)
side
(by
we must have
126, 2) tends fl >0. Now n
to
as
n
hence
the series
*=o
n+1 ) and therefore converges absolutely, since a *0 with a n we may omit the also, by a); n +1 n This proves the conbrackets, by 83, supplement to theorem 2. n of a vergence ( n l) This theorem enables us to deduce easily the following further theorem, which will be of use to us shortly: is
a sub series of
2(an
\
|
2
.
+
|
,
1
Power
54.
Theorem.
i
_
|
|
>1
2a
*3i
b) converge conditionally, 1 single point z
^
diverge, if 9R (a)
arbitrarv > ^
I
n*>
I
(4) bounded,
< 1,
>1 0< 3t(a)<^l,
()
if
=+
divergent for every
=1
z
,
the series will
\
\
,
for the
except possibly
.
Since
Proof.
statements
the
a
*.
for the points of the circumference
a) converge absolutely, if
c)
_
n n z is absolutely convergent for \z\
and
,
_!i
~n
~~a"7
z
229.
as in the preceding theorem,
//,
_?i. the series
401
Analytic functions.
series.
relative
to
are
\z\^\
=1
verified.
immediately
For
the statement a) is an immediate consequence of the conz , of vergence -2|0 n ensured by the preceding theorem. Similarly c) is an immediate consequence ,of the fact established above, that in this |
|
|
case |
an
remains greater than a certain positive number for
\
sufficiently large n. if
Finally,
< 5R(a) ^ 1
follows from Dedekind's test
theorem
sums
that n
2z
of
ference
2\a n
=
z |
|
a n ^. 1
\
and
z 4*
-f-
1, the
convergence of
3. For we proved converges and a n >0;
184,
in
every
2a n z n
the preceding the partial
that
are bounded, for every (fixed) 24>-|-l on the circum1 , follows simply from the fact that for every n
II-*!
54.
Power
The term "power of
form
the
2a n zn
,
where now both the be complex.
series.
Analytic functions.
series" is again
used here
to
denote
2a
more
generally, of the form coefficients a n and the quantities z
or,
a
n (z
series n
z^
9
and ZQ may
The theory of these series developed in 18 to 21 remains vab'd without any essential modification. In transferring the considerations of those sections, we may therefore be quite brief. Since the theorems 98, 1 and 2 remain entirely unaltered in the
new domain,
the
on the behaviour
same of
metrical interpretation 14
true of the
fundamental theorem 93 itself, domain. Only the geo-
series in the real
power is somewhat
different:
The power
scries
we take into account Pnngsheim's result mentioned in we may state here, more definitely except for * = -fl.
If
footnote,
is
2a n z*
the preceding*
Chapter XII. Series of complex terms.
40 2
for every z interior to the circle of
indeed absolutely
converges
radius r round
the origin 0, while it diverges for all points outside that circle. This circle is called the circle of convergence of the power
and the name radius applied
series
for the
first
number
to the Its
time, completely intelligible.
magnitude
r thus is
becomes,
given as before
by the Cauchy-Hadamard theorem 94. Regarding convergence on the circumference of the circle of convergence, we can no more give a general verdict than we could regarding the behaviour in the case of real
show
tely will
at the
power
endpoints of the interval of convergence (The examples which follow immedia-
series.
may be
behaviour
that this
The remaining theorems
230.
of the
most diverse
nature.)
18 also retain their validity unaltered.
of
Examples. 1.
2zn
r=l.
]
sum
with the
zn
vergent, as
does not -*
5-
f
;
=
the boundary points zn 8.
5]
e.
=
z
for
the series 1
\
|
,
it
is
is
convergent,
everywhere di-
1
series l5
remains (absolutely) convergent at
all
.
\
r=l.
;
=
z |
i.
circle,
there.
This
1.
The
series
is
not
certainly
convergent for
all
the
is
also
ft
= l gives
points, for*
boundary
the boundary,
Z
zn Jf?
On
.
L
2.
In the interior of the unit
the divergent series
.
However,
it
1 gives a convergent series. In all these points, since z = theorem 229 of the preceding section shows, more precisely, that the series must converge conditionally at all points of the circumference |jr| = l for we have here different from + 1
not divergent for
fact,
;
result may also be deduced directly from Dinchlefs test 184, 2, since has bounded partial sums for z =f= -f 1 and s = 1 (cf. the last formula of
The same
Z zn
|
the preceding section)
and
n
tends monotonely
|
to 0.
the convergence can, however, only be conditional 4.
and
5?
t,
-.
4n
;
r
=
1
.
2an zn
zn
16 .
This series diverges at the four boundary points
and converges conditionally
16 If
As
at
every other point of the boundary.
most of the subsequent examples) the real power series 2an x n 16 These facts regarding convergence may also be deduced from 185, 5, by splitting up the series into its real and imaginary parts. Conversely, however, the above mode of reasoning provides a new proof of the convergence of these two real series.
this
power
has real
coefficients (as in
series of course has the
same radius as
.
Power
54.
zn
For
5.
nowhere but
at
The
6.
r
,
series.
= -f-OO.
403
Analytic functions.
For 2'n!* n ,
thus this series converges
r=0;
z=0.
series
J
(-!)-.
and
^(_l)*-^__
are everywhere
convergent.
A
7.
power
series of the general
2an (z
form
z ) n converges absolutely and diverges outside
at all interior points of the circle of radius r round zot n this circle, where r denotes the radius of nz .
2a
more
Before proceeding to examine the properties of power series in detail, we may insert one or two remarks on
Functions of a complex variable. to
If
domain 17
we
every point z within a circle $ (or more generally, a a value w is made to correspond in any particular manner, )
say that a junction
w
= f(z)
of the
complex variable z
is
given in
The correspondence may be brought about
this circle (or domain).
in
a great number of ways (cf. the corresponding remark on the concept of a real function, 19, Def. l) in all that follows, however, the functional value will almost always be capable of expression by an explicit formula in terms of z, or else will be the sum of a convergent series ;
whose terms are for
shortly;
the
Numerous examples
explicitly given.
moment we may
each point z within the
which
at
power
series represents the
The concepts
sum
will
think of the value w, circle
occur very
for instance,
convergence of a given
of
of the series at that point.
of the limit, the continuity,
and the
differentiability of
a
function are those which chiefly interest us in this connection, and their definitions, in substance, follow precisely the same lines as in the real
domain 1.
:
Definition of limit.
If the function
every z in a neighbourhood of the fixed point
_
iim f(z]
=
w=f(z) ,
we
is
defined
18
for
say that
co
or
17
sequel,
by a
A
we
finite
strict
shall
f(z)-+a>
for
*-*,
of the word "domain" is not needed here. In the always be concerned with the interior of plane areas bounded
definition
number
of straight lines or arcs of circles, in particular with circles
and half-planes 18 f(z) need not be defined at the point satisfy the condition
of course be assumed
0< |*
|
The d
but only for all z's which above definition must then
itself,
of the
231.
Chapter XII. Series of complex terms.
404 if,
given an arbitrary e
>
we can
,
assign d
d
>
(e)
so that
\f(z)-a>\<e
<
for every 2 satisfying the condition 19
to exactly the same thing whose terms lie in the to ,
coincide with to
verge
\
< 6;
which comes
or
converging
and do not
wn =
corresponding functional values
the
,
z |
for every sequence (zn ) given neighbourhood of if
f(zn ) con-
co.
we
consider the values of f (z) , not at all the points of a neighbut only at those which lie, for instance, on a parti, cular arc of a curve ending at , or in an angle with its vertex at , or, more generally, which belong to a set of points M, for which If
bourhood of
=
we say that limf(z) co or f(z)*-co a point of accumulation, in that set M, if the as z+> along that arc, or within that angle, or
is
above conditions are
come
M which
at least for all points z of the set
Definition of continuity.
2.
in
fulfilled,
into consideration in the process.
a neighbourhood of
at the point
,
and
If
at
the function
we
itself,
w
= f(z)
say that f(z)
is
is
defined
continuous
if
lim f(z) exists
and
equal to the value of the function at
is
i. e. if , f(z) */"() when z is restricted to an define the continuity of f(z) at arc of a curve containing the point , or an angle with its vertex at , or any other set of points that contains and of which is a limiting
We may also
M
point; 3.
the definitions are obvious from 1.
Definition of differentiability.
fined in a
entiate
neighbourhood
at
,
if
of
and
at
If
the function
itself,
f(z)
is
w
= f(z)
said to
be
is
de-
differ-
the limit
lim '
exists in
accordance with
1.
Its
value
is
called the differential coeffi-
and is denoted by /"(). (Here again the cient of f(z) at variation of z may be subjected to restrictions.)
We
must be content with these few
definitions
mode
concerning
of
the
general functions of a complex variable. The study of these functions in detail constitutes the object of the so-called theory of functions, one of the most extensive domains of modern mathematics, into which we 30 of course cannot enter further in this place 19
Same
80
A
.
proof as in the real domain. rapid view of the most important fundamental facts of the theory of functions may be obtained from two short tracts by the author Funktionen:
Power
54.
The above
series.
405
Analytic functions.
explanations are abundantly sufficient to enable us
to
most important of the developments of 20 and 21 to power series with complex terms. In fact, those developments remain valid without exception for our present case, if we suitably change the words "interval of convergence' to "circle of convergence" throughout Theorem 5 (99) is the only one to which we can form no analogue, since the concept of integral has not been introduced for functions of a complex argument. All this is so simple that the reader will have no trouble, on looking through these two sections again, to interpret them as if they had been transfer the
intended from the first to relate to power series with complex terms. At the most, a few remarks may be necessary in connection with Abel's limit theorem 100 and theorem 107 on the reversion of
a power
y
+ As
In the case of the latter, the convergence of the series 1 ---^ hence of the series an( > H y -f- b^ y -\---- which satis-
series. 2
y'
,
theorem, were only proved for real values of y. however, as we have thereby proved that this power
fied the conditions of the
This
is
clearly sufficient,
a positive radius of convergence, which is all that is required. regards Abel's limit theorem, we may even corresponding
series has
As
degree of freedom of the variable point z prove more this reason we will go into the matter once more: n a n z to be a given power series, not everywhere convergent, but with a positive radius of convergence. We first observe to the greater
than before, and for Let us suppose
2
exactly as before, we may ducing any substantial restriction that,
assume
On
this
the
radius
=1
=
z of convergence, 1, we assume that at least one which the series continues to converge. Here again |
\
+
1.
In fact,
"n*0*
2a n
the series
'
z
n
if
z
4
s
+
exists
point z
we may assume we need only put
at
that Z Q is the special point
without intro
circumference of the circle
1>
=*,!>
also has the radius 1
and converges
at the point -j- 1.
proof on^mally given, where everything may now be interpreted as "complex", then establishes the n Theorem. // the power series 2' a n z has the radius 1 and remains
The
convergent at the point we also have
if
z approaches 21
the
-f-
point
1 of the unit circle,
+
1
along
the
and
positive
if
real
2 an = s
axis
t
from
then
the
0.
origin
th Grundlagen der allgemeinen Theorie, 4 ed., Leipzig 1930; II. th und der ed., Leipzig Teil, Anwendungen allgemeinen Theone, 4 Weiterfiihrung 1931 (Sammlung Goschen, Nos. 668 and 703). 21 We are therefore dealing with a limit of the kind mentioned above in
theorie,
231,
I.
Teil,
1.
14
(051)
406
Chapter XII. Series of complex terms.
We
233.
can now
easily
prove more than
Extension of AbeVs theorem.
this:
With the conditions of the preceding
theorem, the relation
remains true
by
mode
if the
the condition
+1
is restricted only approach of z to remain within the unit circle and in the angle between two arbi-
of
z should
that
trary (fixed) rays which penetrate into the interior of the
unit circle
+
point
from
starting
,
1
the
(see Fig. 10).
The proof
will
be con-
ducted quite independently of previous considerations, so that
we
thus
shall
a third
obtain
proof of Abel's theorem.
Let Z Q ,
any
sequence
limit
-\-
Fig. 10.
portion
We
have
show
to
k
of
the
of
be
.
points
the
in
1
,
of
described unit
circle
that
ftoif,
as
we
before,
2an z
write
n
= f(z).
choose for a ftn
and apply the theorem
to the
sn
= "o +
- **)<* =
i
(b)
and
sum
of (c)
that the
of the & th
fulfilled.
it
with
its
=z
follows immediately that
]
n=0
This proves the statement, provided
(a)
is
now A k
satisfy
clearly fulfilled,
=
the
conditions
as * fc
1,
(a),
and the
CO
^z =
1, so that (c) is k n=o such that requires the existence of a constant is
(l
z k]
K
Finally (b)
for all points z
of
row
It
chosen numbers a kn
Now
221.
sums
!->
H s.
partial
(
tends to s as k increases.
we can show
theorem
Toeplitz
sequence of
which, by hypothesis, converges to
also
In
the value
k
vertex at
1 in the angle (or
+1).
It
any sector-shaped portion only remains, therefore, to establish
Power
54.
This reduces
the existence of such a constant.
=l
the following statement: If z
< Q
and
only on
and Q O
,
407
Analytic functions.
series.
+ sn a constant A = A Q (cos
*
i
Fig. 10) to proving
(v. 9?) ,
(<^
w#A |
^
|
) exists,
^
<^
depending
such that
,
^^
A
for every z of the type described. In the proof of this statement, it is sufficient to assume QQ COS
=
A
=
statement then runs:
- yi - 2 Q cos + ea
1
cos
or 2 Q cos
<
for
cos
Q
left
tainly suffices to
+
9?
2
cos
and \
show
By
9?
+
2 Q* cos
replacing
latter is
by
99
,
and
increased; therefore
2 it
by cer-
that
q cos
9
^
q cos
+ ^
2
9
,
This extension of Abel's theorem to "comwhich is obviously true. of modes or plex approach" "approach within an angle" is due to
-
O. Stolz 22 This completes the extension to the case of complex numbers of all the theorems of 20 and 21 with the single exception of the theorem on intenot which we have defined in the present connection. In particular, gration, .
thereby established that a power series in the interior of its circle of convergence defines a function of a complex variable, which is continuous it is
and
the latter "term by term" and as often as we please and domain, accordingly possesses the two properties which
different iable
in that
above all others are required, in the case of a function, for all purposes of practical application. For this reason, and on account of their great in further developments of the theory, a special name has importance been reserved for functions representable in the neighbourhood of a point 28 In recent years the Zeitschrift f. Math. u. Phys., Vol. 20, p. 369, 1875. question of the converse of Abel's theorem has been the object of numerous investii. e. the question, under what (minimum of) assumptions relating to gations, the coefficients a n the existence of the limit of J (z) as z > 1 (within the angle) an An exhaustive survey of the present state of research entails the convergence of in this respect is given in papers by G. Hardy and J. E. Ltttlewood, Abel's theorem and its converse, Proc. Lond. Math. Soc. (2), I. Vol. 18, pp. 205 235, 1920; II. Vol. 22, pp. 254269, 1923; III. Vol. 26, pp. 219 23G, 1926. Cf. also theorems ,
.
H
.
278 and 287.
Chapter XII. Series of complex terms.
408
zQ) n They are said to be analytic or regular z by a power series E a n (z at ZQ By 99, such a function is then analytic at every other interior point .
.
of the circle of convergence; it is therefore said simply to be analytic or 23 In particular, a series everywhere convergent reregular in this circle .
which
presents a function regular in the whole plane,
therefore shortly
is
called an integral function.
which we have proved about functions expressed about analytic functions. Only the two foltheorems by power of are which special importance in the sequel, need be expressly lowing, formulated again. All the theorems series are
234.
1-
If two functions are analytic in one and the same
21) their
(by
For the
and
their difference,
sum,
quotient the corresponding statement
the function in the denominator
105, 4) only
if
of the
and provided,
circle,
circle ,
is
is primarily true (by not zero at the centre
necessary, that this circle
if
then so are
their product.
is
replaced by a
smaller one.
If two functions, analytic in one and the same circle, coincide in a neighbourhood, however small, of its centre (or indeed at all points of a set having this centre as point of accumulation), the two functions are completely 2.
theorem for power
identical in the circle (Identity
series 97).
Besides stating these two theorems, which are new only in form, we shall prove the following important theorem, which gives us some information on the connection between the moduli of the coefficients of
a power series and the modulus of the function
it
represents
:
00
Theorem.
235.
= Z an (z
If f(z)
=
zQ) n
o
\^\^
< < r and M ~ M (Q) if Z = the circumference z Q
\
Proof 24 for
We
.
which however
G
= 0,
(P
P
a number which
is
ZQ
\
f(z)
\
\
<
for
3= 1
then
r,
1, 2,
.
.
.),
never exceeds along
Q.
consider the function
*(*) 28
M
z \
(Cauchy's inequality.) choose a complex number 77, of modulus
\
first
1 -rf
converges for
any integral
=
(*-
25
exponent q
^
0.
=
1,
Now we
k
*o)
A
function is accordingly said to be "analytic" or "regular" in a circle ,ft can be represented by a power series which converges in this circle. 24 The following very elegant proof is due to Weierstrass (Werke II, p. 224) and dates as far back as 1841. Cauchy (Me"moire lithogr., Turin 1831) proved the formula indirectly by means of his expression for / (z) in the form of an integral. that f(z) never exceeds on z z The existence of a constant 2 is practically
when
it
M
M
|
2
an
\
|
\
n
clearly has this property. This also such that a n Q n M. But the above theorem states that every never exceeds has the property that a n Q n is always 5^ M.
obvious, of course, since
\
Q
\
^
\
|
=
|
25
if*
Such numbers
cos (q a
TT)
+
i
17
TT)
;
this is
M
|
\
\
of course exist, for
sin (q a
Mis obviously that / (z)
never
1
if if
cos (a
t\
a
is
chosen
TT)
+
i
sin (a
irrational.
TT),
then
Power
54.
409
Analytic functions.
series.
^
for a specific integral value of the exponent k constant coefficient a. If we denote by gQ , g l9
function for z
this
= Z + QQ
v
v
,
rj
= 0,
1
,
2, fc
.
.
.
and an
g
>2
,
...
the
we have
,
arbitrary
values of for
n
^1
kn
1
*i
hence I
The
expression on the right hand side contains only constants, besides it therefore follows that the arithmetic mean
the denominator n;
~
\-gn-l
as
n
increases.
In the case k
0,
we
should be concerned with the
identically constant function g(z) s= a, for
which
since the ratio is equal to a for every n, in this case. more general function
If
we
consider
the rather
^^^
^)-t? + where metic
I
and
m
are
fixed
integers
and now form the
^> 0,
arith-
mean -----hgn-l
=
g (z (where, as before, gv by the two cases just treated. for every z of the
certain constant
-fIf,
v
Q
rj
),
further,
circumference
z |
K, we have gp-f
ffi
= 0, it is
ZQ
\
1,
.
.
known
= Q,
is
+ b , this clearly that the function g(z)>
.),
never greater than a
also
+ " + gn-l W
v
^ n A" ~~
^
7J
and therefore also
With these preliminary remarks, the proof of the theorem is now As 2j\ a n \Q* con quite simple: Let p be a specific integer ^0. we can determine so that e > 0, verges, given q p
>
Chapter XII. Series of complex terms.
410
A
we
fortiori,
then have for
values of z such that
all
z
ZQ
|
=Q
\
,
B
l< + nq!(*-*o) I
l
and
same
therefore, for the
values of z 9
n=0 if
M has
ference
the
ZQ
The The
|
=Q
Accordingly, on the circum-
in the text.
meaning given
Z |
,
modulus signs is of the kind just considered. there obtained now becomes
function between the inequality
b \
|
^K
,
I
and, as e
was
arbitrary
and
**p
> 0,
K^tf p
*
===
1
we
I
I
have, in
fact,
footnote to 41,
(cf.
l)
<
q. e. d.
The elementary analytic functions.
55.
I.
1.
The
++
1
1
l-z
I-ZQ -(Z-ZO
this
1
)
:
=
is
-
11 1~
S
1
power
/
series
..xi.
<
i. e. for z z z 1 converges for every z than -f- 1; in other words, the circle of convergence of the circle with centre ZQ passing through the point +1. [
the series is
expressible as a
I~
series
nearer to z
The
w
rational function
for every centre ZQ
and
Rational functions.
-
function
is
1
With reference
to
|
1
\
thus analytic at every point different from this
example,
we may
+1.
draw
attention to the of fundamental importance in the theory n whose circle of convergence is the
following phenomenon, which becomes of functions: If the geometric series 2z
briefly
,
unit circle, is expanded by Taylor's theorem about a new centre z within the unit circle, we could assert with certainty, by that theorem, that the new series converges at least in the circle of centre zl which touches the unit circle on now see that the circle of convergence of the new series may the inside.
We
very possibly extend beyond the boundary of the old. This will always, be the case, in fact, when zt is not real and positive. If zt is real and negative, the new circle will indeed include the old one entirely. (Cf. footnote to 99, p 176.")
55.
The elementary analytic
functions.
I.
Kational functions.
411
Since a rational integral function
2.
<*o
may be regarded
+a z+a
as
i
*
z*-\-----1" am z m
a power series, convergent everywhere, such
Hence
functions are analytic in the whole plane.
the rational functions
of general type
are analytic at all points of the plane at which the denominator is i. e. not 0, everywhere, with the exception of a finite number of Their expansion in power series at a point z , at which the points.
4s 0> s obtained as follows: If z is replaced by in the numerator and denominator of such a function, both zo *o) * ), the function takes these being then rearranged in powers of (z denominator
+
the
is
*
(z
form
We
'
where, on account ot our assumption, & 4. 0. may now carry out the division in accordance with 105, 4 and expand the quotient n in the required power series 26 of the form Scn (s #o) *
II.
The
is
The exponential
function.
series
a power series converging everywhere, and therefore defines a func-
tion regular in the whole plane, i. e. an integral function. point z of the complex plane there corresponds a definite
To
every
number w,
sum
of the above series. This function, which for real values of z has the value e z as defined in 33, may be used to define powers of the base e (and then
the
further those of 80
An
any positive base) for
all
complex exponents:
method consists in first splitting up the function into Leaving out of account any part which represents a rational integral function, we are then concerned with the sum of a finite number of fractions of the form alternative
1
partial fractions.
A
A
i
1
\tf
each of which we may, by 1, expand separately in a power series of the form This method enables us to see, moreover, that 2c n (g z )*, provided z 4= a. the radius of the resulting* expansion will be equal to the distance of z from the nearest point at which the denominator of the given function vanishes.
Chapter XII. Series
412
236.
For
Definition. attributed
the
to
all real or
power
e* is
complex terms.
ol
complex exponents,
the
meaning
defined,
be
to
without ambiguity, by
the
relation
And
p is any positive number, p without ambiguity, by the formula
z
if
shall denote the value determined,
where log/> is the (real) natural logarithm of p as defined 21 in 36. z (For a non-positive base b, the power b can no longer be uniquely defined; cf., however, 244.) As there was no meaning attached per se to the idea of powers with complex exponents, we may interpret ihom in any manner we please. Reasons of suitability and convenience can alone determine the choice That the definition just given is a th )rof a particular interpretation. from formula 91, example 3 (leaving suitable results one, oughly out of account the obvious requirement that the new definition must coincide with the old one for real values of the exponent 28); this formula was proved by means of a multiplication of series, the validity of which
holds equally for real and complex variables and the formula must accordingly also hold for any complex exponent; it is
237.
ei e z whence
*
= e i+*
also
This important fundamental law for the algebra of powers therefore certainly remains true. At the same time it provides us with the key to the further study of the function e z
238.
1.
Calculation of
= cos y 27 It
definition
may be lt
xk
is
-f- i
noted
ez
sin
how
For
.
y
real y's,
we have
.
far
removed
the product of k factors
all
l value belongs e. g. to 2 the definition. determined above by quely
no knowing" what 28
.
this definition is
equal to x". ;
from the elementary
At
yet this value
is
first sight,
in
there
any case
is
uni-
there can exist no other function than the function a2 just regular in the neighbourhood of the origin and coincides on
By 234, 2,
defined which the real axis z
is
=x
with
the function e x defined
may indeed say that every necessarily be unsuitable,
definition
of e
z
by 33. For this reason we differing from the above would
The elementary analytic
55.
Hence
functions.
follows that, for z
it
e* = e* + iv
-==
By means of this lormula complex z's.
~x+
ex
20
i
e iv
The exponential
11.
function.
413
y>
e
r
(cos
the value of e
j/
z
may
easily
be determined
for all
This formula enables us, besides, to obtain
complete manner an idea
points of the complex note the following facts.
at the various
We
values}.
We
2.
have \e z
= **<*> =
plane
(in short, of
2
\
s
y
|
x
iy
-
\
\
assumes stock of
e
In fact
e*.
= (cosy -f isiny Vcos y -j-sin = e x e = e because e x > and hence e = Similarly, = y, am = 3 z
z
its
\
\e**\
|
a convenient and
in
which the function
of the values
\
,
\
= 1,
the second factor
1.
ez
also from the formula 3. e z
238,
For
we
if
just
2kni,
has the periods e*
1
(z)
used. that
= e *+**\ = e
increase z by 2
71 i
its
is z
for
to say, 2 *w
*-
all
(k
<,
values of z,
^0,
integral).
imaginary part y increases by 2ji, and by 1. and 24, 2, this leaves Every value which e z is able
real part remains unaltered, the value of the function unchanged. \\
hile its
assume accordingly occurs in the n < 3 (z) y
to
=
strip
by a strip
Every such parallel translation. called a period-strip; Pig. 11
is
represents the first- named of these strips. 4. e z
has no other period,
O
in-
between two deed, more precisely: and numbers z special z^ we have if
the relation Fig. 11.
this necessarily implies
For we
first
= then = e x + iv = ^(cos y
infer that e z*~~ z *
ez ae
that
Euler: Intr. in Analysin 14*
1,
inf.
Vol.
T,
we -f-
note that
isiny)
if
= 1,
138. 1748.
(G51)
Chapter XII. Series ot complex terms.
414
we must by
have
2.
cos y i.
e.
cos y
hence y
5.
= 2kn
=
1
=w
= R^ (cos ^ +
e zi
By
3.,
the
*
sin
=w = 6 \wRi e i'^ = R^
t
,
(
# >
with
&J
Now
k
number
0, the
t
as
cos
numbers
= 0,
1,
2,...)
equation, and by 4. no other solutions be chosen, in one and only one way, always
same
are also solutions of the exist.
has one and only
^=^0,
(ft
can
also have
owc0 awd only once in the period
4=
for given lt
3 certainly a solution of e
is
,
w
e* assumes every value
w^
we
sin
-f- i
Thus, as asserted,
.
z strip; or: the equation e one solution in that strip.
If
= 0. Further, y= 1 sin y = 0,
hence x
*=!,
may
so that
<
n 6.
The value
is
3 (^
-f 2 k
n i}
<^
never assumed by e z \ e z.e- z
+ n, for,
q. e. d.
by 237,
= \,
so that e z can never be 0. 7.
The
derivative (e z )' of e z
is
entiating term-by-term the power 8.
From 238,
we
1,
also
The
III.
e z y as follows at
again
once by
differ-
z series that defines e .
deduce the special values
functions cos z and sin z.
In the case of the trigonometrical functions, we can again use the
239*
expansions in power series convergent everywhere to define the functions for complex values of the variable.
Definition. 1 ""
The sum of z2 z* 2
is
!
+ il
+
the
power
by sin z,
^ O
'
I
!
FT J !I
convergent everywhere,
z2k
+
(-
*)*
+
'
'
'
'
(2 k)\
denoted by cos z, that of the power 1~! 1 !
series,
r
for every complex z.
+
series, also
(
1,
convergent everywhere^
55.
The elementary analytic
For
and
real z
= x,
We
functions.
III.
The
functions cos z and sin
certainly gives us the
this
have only
z.
415
former functions cos x
as before, whether these definitions are suitable ones, in the sense that the functions defined, which sin x.
to verify,
are analytic in the whole
possess the plane, i.e. integral functions, 30 cos x and sin x. properties as the real functions again the case, to the fullest extent, is shown by the
same fundamental Thai
this
is
following statement of their main properties:
For every complex
1.
cos
we have
z,
the
240.
formulae
#+ /sin # = 0*%
-+ -
whence further
e
**
-~e
-**
.
81112
,
(Eider's formulae).
The proof
follows immediately by replacing the functions power series which define them.
sides by the
on both
The addition theorems remain valid for complex values of
2.
cos sin
(z l
(^
This follows from
and the
+ 2 = cos z 2 = cos 2 -f-
3)
t
2)
i
since by
1.,
cos
z:
sin z 1 sin z^,
z.}
+ sm
sm ^3
%i
cos -?3 f
^
sin z -f sin 2
237
latter involves
cosfo
+* + isin(* +* 9)
==
9)
1
(cos 2j
= (cos Z
A
+
i
sin
zj
cos z 2
(cos z a
sin
-f- i
sin za )
^ sin ^) -f-
i
(cos
2:,
cos *,)
.
and z^ for ^j and 2 a and taking into account the an even, sinz an odd function, we obtain a similar formula, which differs from the last only in that i appears to be changed Addition and subtraction of the two relations i on either side. to Substituting
z
fact that cos z
is
,
give us the required addition formulae.
The
3.
fact that the addition
tions are formally the
of the
same
real variable x,
these functions by cos,?
theorems for our two integral funca; and sin a;
as those for the functions cos
not only sufficiently justifies our designating and sins, but shows, at the same time, that
the entire formal machinery of the so-called goniometryt since it is In particular, evolved from the addition theorems, remains unaltered.
90
Here again a remark analogous
be made.
to that
on
p. 412,
footnote 28,
may
Chapter XII. Series ot complex terms.
416
we have
the formulae
=
cos 3 z
-f-
sin 2 z
= 2 sin z cos
valid without
sin 2 z
cos 2 z
1,
= cos
9
sin
z
z9
etc.
z,
z.
change for every complex
The period-properties of the functions are also retained in complex domain. For it follows from the addition theorems that 4.
cos
(z -f-
sin (2
cos 2 sin 2
n JT
+
sin z sin 2
rc
sin 2 cos 2
JT
= cos 2, = sin 2 .
cos z aw<2 sin z possess no other zeros in the comknown in the real domain* 1 In fact, iz e~ iz or necessarily involves, by 1., e
plex domain
i.
JT)
= cos z = cos 2
77*0 functions
5.
cos z
-f-
2 n) 2
the
=
besides those already
.
=
e.
By 238,
Similarly, sinz
or z
= kn, 6.
=
z^
-j-
=
iz implies e
=n
~ e" iz
,
or e* iz
=
i.
e.
=
=
cos 22
z,
that either
sin zt
5.,
=
t
+
- JL
fl
-Q
2
sm -~^
2
r
S>~ &
i
by
= 2 k jit,
=
cos z
-
2
sin - 1 -,,
that either
must
=
3-
u
>
5.,
2 iz
e.
cos z2 is satisfied if, and only if, under the same condition as in the real sin z^ if, and only if z -f- 2 k n or Similarly sm z^ z^ It follows in fact from 2 k n. z
cos
by
i.
1,
q. e. d.
The relation z ^ -. 2 kn,
domain. zg
can only occur when
this
4,
ft
JT;
= it
similarly
= = 2 cos g sin -^^ = ^ or ^-^ = (2 A +
follows from
g
L
sin z%
,
-^-Q---
,
-*-
ft
1)
.
7. 77z functions cos,? aw^ sin^r assume every complex value w n $1 (z)
= w^ 81
Or
and only
<
=
but only one,
l,
in other
if,
z
if
ze>
=
1.
words: The sum of the power series
has one of the values (2 k
.similarly for the sine series.
-f 1)
A
,
k
=
0,
-
1
-|
1
,
=
if,
2, ...;
and
...
;
the elementary analytic
g 55.
iV. lhe functions cot z and tan
functions.
Proof. In order to have cos z ~- w, we must have e iz*
or
=w
Vww* V 2
-f-
1.
V Yr (cos
(Here
r
r/;
+
two numbers, for instance 72(0)8-?*
of the
the quantity under the radical
Since in
any case
sin
g?)
is
e~ iz
2
10
defined as one
+isin-~V whose
square
is
c
ign.)
w + Vw 2
32
i
e iz -\-
z.
1
=}=
0,
complex num- n< < +n, 3(Y) w -\-V w* 1,
there certainly exists a
ber
/
foi
which e z
such that '
=
=
'
by 238,5. have n
=w+ By
6.,
we 12
w. This at
in the
penod-stnp. however, a second solution,
from
it,
z),
if,
and only
if,
Fig. 12.
+
1n, i. e. w reason in precisely the
and
We
exists
(viz.
the period -strip
z =H
=
certainly has
therefore
one solution
z,
and
ji
<^
or cos z
different in
9R (2)
'
equation least
<
z
i
Writing
=f=
equation sin z -- w.
In
this
same manner with regard
we
case,
to
can also easily convince
the our-
always one and only one solution of the equation in the portion of the period-strip left unshaded in Fig 12, if we include the parts of the rim indicated in black, but omit the parts represented by the dotted lines (see VI below). selves that there
8.
For the
is
we have
derivatives,
(cos z)'
The
IV. 1.
will also TT
cos z.
(sin z)'
functions cot# and
tan^gr.
Since cos z and sin z are analytic in the whole plane, the functions cot
k
as in the real case,
sin 3,
z
= ^? sin z
and
tan
z
=
-cos
z
be regular in the whole plane, with the exception of the points
for the former
and
(2
k
+
1)
for the latter,
which are the zeros of
z and cos z respectively. Their expansions in power series may be obtained by carrying out the division of the cosine and sine series. Since this operation is of a purely formal nature, the result must be the same as it was in the real domain. 24, 4, where the result Accordingly, by sin
of this division was obtained by a special
k
-
__ -Z
t n ~ tan
32
In
fact, since
w*
1
artifice,
=+=
/
(-
1 \fc-l
1)
w8 Vw2 ,
1 4=
i
?.
we have
418
Chapter XII. Series
complex terms.
oi
On account of 94 and 136, we are now also in a position to de termine the exact radius of convergence of these series. The absolute value of the coefficient of z 2k in the first series, by 136, is
Its
if
(2A)
th
root
is
denotes the
So&K t
every k
=
1, 2, ...
sum
5? *^
>
every other i, but
The
^r. VK
is
it
(for
ft
-*'
tan-series is
12*)
=
cot-series
found to be
and
1,
for
less than this for
is
therefore
1);
and the radius of the
=
when k
-^
between 1 and 2
latter lies
1 "^ IT'
i
n, by 94.
Similarly that of the
.
-^
For cos 3 and sin^r ftott 2. cot z and tan* Aatte tf/te period n. change in sign alone when z is increased by n. Here again we may show, more precisely, that cot *a
= cot
a
*,
In fact,
it
and
jgr
involve
tan
a
(*-0,
*,
COt &.2
=
cos
cos z9
z.
:
:
sin
2r
sin 22
t
In the "period-strip",
wi=
equation cot 2
For each II,
5)
for
e.
By
in
the
w
there
which e*
z 2.
To
are never assumed. 10
1,...).
i,
*%*
.
a
is
accordingly
=f
sin fo,
zj ==
^ 0,
sin z^ i.
w
*
=f=
this,
z
= kn.
SR (0)
+~
/ws^
once;
za
e.
^<
strip
see
*.)X .
:
sin
/Ae
*==. The
2
write
,
then becomes
definite
exists
For z
~< i.
i. e.
=
and tan 2 asswme every complex value
2r
values
a
= + **
that in the case of the first equation sin (za Similarly in the case of the second.
cot
jgr
follows from COt Z,1
3.
= tan
jgr
=
SR (*)
a
/
*'-^, u
<;
y
complex number such
we and
n
that
4
s
<.
%
an(* (by (/) <^ n>
then have cot
jar
= w,
is a solution of the latter equation in the there can be no other solution in this strip.
prescribed
The
strip.
impossibility
g 55.
The elementary
V.
analytic functions.
of a solution for cot z
=
results
*
Ihe logarithmic
from the
series.
419
fact that these equations
both involve
+
=
sin a which cannot be satisfied by any value of z, as cos a z 1. For tans the procedure is quite similar. 4. The expansion in partial fractions deduced in 24, 5 for the the real domain remains in the same valid in form for every cotangent 1* it 2,... (and similarly for the excomplex z different from 0,
pansions of tans, there
may be
single
word 33
.
Indeed the complete reasoning given
etc.).
interpreted in the "complex" sense, without altering a In particular, for every z satisfying the above con-
dition,
Now
it
follows,
if
we
substitute z for
^_
2
_
2inz,
that
,
hence we obtain the expansion i
valid for every
tension to the
i
1
(&0,
complex z=$=2kni
complex on
fractions
obtained
between
this
integer).
This
is
the ex-
variable z of the remarkable expansion in partial connection p. 378, and it exhibits the true
expansion and that of cots, which previously seemed
rather fortuitous.
The logarithmic
V. In
25,
we saw
series.
that the series
<
x 1 the inverse function of the exponential represents for every v i. e. substituting for y in function e 1; |
\
3! 33 It was precisely for this purpose that at the time we framed some of our estimates in a form somewhat different from that required for the real domain 207 to which footnote 26 refers). (e. g. those on pp. 200
Chapter XII. Series of complex terms.
420
the above series and rearranging (as is certainly allowed) in powers of because it is purely x, we reduce the new series simply to x. This fact formal in character remains when necessarily complex quantities are considered.
Hence, for every e"
if
w
denotes the
sum
1
\
=z
ew
AT (l)
W
{L.)
= + z, 1
71
"1
vn Z
.
adopt for the complex domain the
A
Definition.
242.
1,
or
of the series
if\
We now
<
z |
number a
said to be
is
a
natural logarithm of
c,
in
symbols,
a
=
if e
cf
c.
In accordance with
number
c
lies
part
=- log
II,
we may then
5,
assert that every
complex
possesses one, and only one, logarithm whose imaginary between TT exclusive and ^ inclusive (to the number 0, 4=
+
however, by
no logarithm can be assigned
II, 6,
more
defined value will be
at all).
This uniquely
especially referred to as the principal value
of the natural logarithm of c. Besides this value, there is an infinity of c we have also ea + 2k other logarithms of c, since with e a c\ thus
=
=
if
a
is
the principal value of the logarithm of
+2k
a
must k
=*=
also
be called logarithms of
0) are called
logarithms of
c.
numbers
TT i
(k
^ 0,
integer)
These values of the logarithm (for no further
c.
.
We
(log*)
= log
c |
logarithm of the positive number that, taken as a whole, all
3
,
|
(log c)
= amc,
denotes the (single-valued) real \c\, and the second is interpreted as the values of the one side are equal to
in the first of these relations log
meaning all
the
M By 238, 4 there can be subsidiary values have, for each of its values,
its
ffl
if
c,
c
|
\
the values of the other.
With these
definitions,
we may
(L) provides a logarithm of (1
any case that the above series But we mav at once prove more,
assert in
+ #)
namely the
43.
Theorem. The
logarithmic series (L) gives, at each point of the unit with the exception of the point 1), the principal
circle (including its rim,
value of log (1
+ #).
34 If c is real and positive, the principal value of log c coincides with the (real) natural logarithm as formerly defined (36, Def.).
The
55.
VI.
elementary analytic functions
The
421
inverse sine series.
Proof. z |
|
^
For
That the series converges for each z =t= - 1 for which was shown in 230, 3. (We have only to put % for z there.) has that value for which -|z, am(l precisely #)
1
this
iff
7T
/
< *
2
Hence we have,
<+ -.
I
TT
2-
for the imaginary part of the
(3)
3(w)
sum w
of the series (L),
=
with integral k. Now w is a continuous function of z in z <. 1, and assumes the value 1 for xr 0. Hence 3 (w) too is a continuous function # in in 1. the Therefore, equation (^)> ^ must have the same value \\e have clearly to take k for all these z. But for 0; hence this from the % \ve learn 1. is its value in the whole of application Finally of Abel's limit theorem that the sum of our series is still equal to the prinz 1 for which 1. cipal value of log (1 z) at the points 2 =|= |
|
|
\
<
JST
j
<
\
=
+
|
VI.
We ~
^
saw
1,
the strip
The
inverse sine series.
in III, 7 that the equation sin
has TT
exactly
<
j)f
(77)
two solutions,
^+
for
The two
TT.
metrical, either with respect to
\
+
\ or
=
w xr
#,
-
for a given
-[ : 1
exactly
complex
solutions (by III, 6) arc
accordingly,
^;
we may
^- z, for an arbitrary given precisely that the equation sin w has one and one in the strip solution clusive of only 1),
more
-I^
SB
(w)
^+f
in
one,
symassert
z
(in-
,
the lower portions of its rim, from the real axis downwards, arc omitted of the rim not counted with the strip are (cf. Fig. 12, where the parts if
drawn line).
in dotted lines, and the others are marked by a continuous black This value of the solution of the equation sin w z, which is thus
=
uniquely defined for every complex function
^r,
is
sin" 1 z.
tv
All the remaining values arc contained,
sin" TT
called the principal value of the
1
z
sin"
-\-
1
z
by
III, 6,
2k TT,
+
2
k
TT,
and may be called subsidiary values of the function.
in the
two formulae
422
Chapter XII. Series of complex terms.
For
x such that
real values of
? ==a;
|as|
the series 123,
+ -2--3- + 274Tr + 1-8 * B
x9
1
.
,
,
represents the inverse series of the sine power series
Exactly the same considerations as in V. for the case of the logarithmic series now show that, for complex values of z such that z <J 1
,
\
|
the series
"' + T-T+2-4-5- + "is
the inverse
sine
.
1-3 * B
,
w
series
power
= -J,
The inverse tangent
VII.
The equation
i
This
i
It
gives at any rate one of the values of sin" z. That this the principal value, may be seen from the fact that, for
a condition which the principal value alone
=|=
.
-|
is
== sin" 1 <: sin"" 1 1 |z|
z
--w* -^7 1
therefore actually
of the
series
z9
1
.
tant0
=
z, as
series.
we know from
one and only one solution
fulfils.
IV, 3, has for every given
in the strip
-- < -
91
(w) <^
-f" -^-
.
called the principal value of the function
is
the other values of which (by IV, 2) are then obtained from the formula The equations tan z i have no solutions whatever. k 7t
tan" 1 z
+
=
.
Almost word
show
that,
z
for |
word
for
\
,
the
same
considerations as above again
the series
= 2 _.+._ +...
(A)
z. To show that this is actually gives one of the solutions of tznw the principal value of tan"" 1 ,?, we have to show that the real part of
the
sum
of the series lies
between
This remains true for every z
and
is
4
s
--5i
(exclusive) * =*
on |
|
1> as
and
-f- -5-
we^
(inclusive).
as for
ar |
|
< 1,
proved as follows:
The sum w
of the series (A), as
may be seen by
log-series, is
w
= -^j-log + iz) (1
-^-log
(1
iz)
substituting the
The elementary
55.
for every \z\ <[ 1, z
analytic functions.
4
s
i>
VIII.
The binomial
423
series.
where principal values are taken
for both
Accordingly,
logarithms.
SRW = y3log(l + ^)~|3lo g (l-^); by 243, both terms hence
being excluded
between
difference lie
~
between
(w) lies
fR
of the
in either case.
and
Thus
-f- ~rr
>
me
,
extreme values
two
the series (A) certainly represents i> q- e. d. |z|<^l and z
+
the principal value of tan" 1 ,?, provided
VIII.
+~
and
-
The binomial
series.
To
complete our present treatment of the special power series investigated in the real domain, we have only to consider the binomial series
where the
in the case
quantities occurring there assume complex values.
i.
The name
Definition.
of
a where 244. principal value of the power b ,
a and b denote any complex numbers, given to the number
tion, is
when log log b 9
we
the exponent a start with the
e.
We
as well as the variable x
uniquely
with b
4
as the only condi-
s
defined by the formula
b is given its principal value. By choosing other values of obtain further \alues of the power, which may be called its
subsidiary values.
All these values are contained in the formula
ta
u
6^a[\ogb+2kai],
each value being represented exactly once, if log b cipal value and k takes all integral values ^0.
is
given
its
prin-
Remarks and Examples. 1.
A
power
6
fl
accordingly has an infinite number of values in general,
but possesses one and only one principal value. 2.
The symbol
**,
for instance, denotes the infinity of
numbers
(all
real
numbers, moreover) ,
(
ft=
,
1.
2, .. )
a of
which 3.
e
2
is
the principal value of the
The only case
of values
is
that in
in
which a power
power
**.
b a will not
have an
infinite
number
which *"
(ft-0,
1,2.
...)
424
Chapter XII. Series
number
complex terms.
of
ka
values; this will occur if, and only if, only a finite number of essentially different Here two numbers arc described (just for the moment) ,is essentially
gives only a assumes, for values.
finite
=
0,
of
2, ...,
1,
and only if, they do not differ merely by a (real) integer. i\ow and only if, a is a real rational number, as may be seen if, at once; and the number of "essentially different" values which may in this case be assumed by k-a is given by the smallest positive denominator with which a may be written in fractional form.
different this
is
if,
case
the
J.
4.
follows that b m
Tt
m
a
if,
where
m
a positive integer, has exactly
is
quite definitely distinguished as the prin-
is
of different values of b a will reduce to one, by 3. and 4., number of denominator I, i. e. a real integer. For
Tne number
">.
and only
= yb,
one of which
different values, cipal value.
a rational
is
exponents (but for these alone), the power thus remains
real integral
now
if,
all
as before
a single-valued symbol.
power
b
If
6.
ba
is
positive and a real, the value formerly defined (v. the principal value of this power.
and p z
z
23G
for e Similarly, the values defined in precisely, the principal values of these powers bols would represent, for complex values of z, an 7.
more
cordance with our convention that defined
z ,
last
and
Nevertheless,
z
p
for
any
we
=
sym-
infinity of vakus, in acshall keep in future to the
p
positive
are now,
t
shall
represent
value
the
the principal value only
e
i
definition.
generally
(>0),
,
In themselves, these
The following theorems will show that it is consistent when fll(a);>0. The value attributed to the po\\er
8.
for b
e
236,
by
as the
33)
now
is
to define b a also in
th.it
case
is
(uniquely).
making these preliminary preparations, we proceed
After
to
prove
the following far-reaching
Theorem 35 For any complex exponent a and any complex
z
.
z |
|
<
1
,
converges
in
binomial series
the
and has
sum
for
the
principal
value
of
the
as
in
power
(1+1)-. The convergence
Proof of real zs
and
statement as
and
's
(v
follows
pp 200 -210), so
word that
sum of the series. Now we may substitute
to the
real a's,
T.
f
d
remr
n
anjrrw
Math, V
for
word
we haxr
for real x's
1,
p.
311
the case
only to prove the
such
1826.
that \x
\
<
1,
The elementary analytic
55.
for
in the
y
after
for
log u
proceed
+ *)
= (l
= ! + + --]
in this
manner, purely formally
complex and writing
z for x;
and rearrange ence as yet
]\ n
= a .2 -^n 'x*
w
( {
1
z
nl
if
series
assuming
n
iti
e^^Z^-r n=l nl
without powers of z. We necessarily obtain the series any question of convergence
to
be gi
(*+*)
= (l
refer-
a
(where
-f- z)
taken for the logarithm and hence for the power could show that the rearrangement carried out was per-
the principal value
we
Now
missible.
Let us
.
in
to
series
substitute in
whose sum would therefore be proved also),
("\x
^
n
power
in the first instance,
we
e.
i.
obtain,
n
binomial series
the
e.
i.
the
425
series.
and so
)/
of x fallowed by 1O4),
powers
-f ar),
The binomial
VJII.
V"
exponential series
learranging in s
functions.
is
1O4
by
this
is
in
certainly so;
everywhere and the series
converges
fact
cc
when a and
for
z <. 1 convergent replaced by their absolute values. |
|
the
exponential
-- z n
-
remains
senes are
all the terms of This proves the theorem in
the
full
its
extent.
+ z) a
and imaginary parts, we obtain complicated in appearance, but which for that very reason shows how far-reac hing a result is contained in the preceding theorem, and from which we also obtain a means for If
we
split
up
(1
into its real
a formula due to Abel, which
evaluating
a
the
= + iy, /3
is
a
power
(1 -f- z)
,
r/>,
/?,
Writing z
.
/
all
real,
+ = R (cos + z
= r (cos
cp -f- i
sin
and
cp)
and writing i sin
0)
,
we have
R = Vl
+ 2ycos//? +
With these values of
= Rfi
.
e -y
R
= principal
2
y'
,
value
3G
of tan- 1
-
and 0, we thus obtain
* [cos
(00
+ y log R) +
i
sin (ji&
+ y log R)]
.
For the case |^|<1, theorem 245 and the remark just made completely answer the question as to the sum of the binomial series. 1 We have now only to consider the points of the circumference z
=
|
.
\
From Abel's theorem, together with the continuity of the principal value 1 in z <[ 1 and the continuity of the of log (1 -f- z) for every 2 =}= deduce the we at once exponential function, |
has accordingly
to
\
be chosen between
+ -5-
and
-H~-
Chapter XII. Series of complex terms.
426
246.
At every
Theorem. at
2 (!
point of the rim
which the binomial series continues
=
+
1,
=1
of the unit circle,
converge, except possibly for as previously the principal value of
sum remains now
its
|2|
to
*)-.
The binomial
whether, and for what values of a and continues to converge on ike rim of the unit
determination series
made
presents no difficulties after the preparations
00
Theorem.
The binomial
series
is
the
circle
in this respect
53. The theorem we have chiefly for this purpose) in which sums up the entire question once more:
247.
z,
(and
the following,
i
n
J^( }z n=o n '
reduces, for real integ-
>
ral values of
tf^O,
a finite sum, and has then the (ipso facto
to
=
a
in particular for a it has the value 1 (also # does not have one these when z values, the series conof // 1) 1 z and 1 , while it exhibits diverges for \z\ verges absolutely for
unique) value (\-\-z)
=
\
.
\
\
>
a) if 91 (a)
,
<
>
on the circumference
the following behaviour it
=1
z \
:
\
converges absolutely at all points on the circum-
ference',
b)
^
if 91 (a)
ditionally
of the series a
-\- z)
Proof.
it
diverges at z
=
1
and converges con-
at every other point of the circumference.
The sum value of (I
< 91 (a)
1
c) if
1, it diverges at all these points;
;
when
it
in particular, w
Writing
l)
(
(")
converges
its
=
value is
n
+1,
f
invariably the principal
in the case z
=
1.
we have
*-(+!) a
is
.
__
\
U-i/ 229 may
be applied, and the validity of a), b) and c) follows immediately. Only the case of the point z 1, i. e. the convergence of the series
hence theorem
requires special investigation.
=
Now
a (a
- 1) (a - 2)
The elementary
55.
and
in general, as
the
partial
sums
may
at
of our series
same index n,
are
equal to the paitial products,
of the product
/
cc
//fl
\ )
The behaviour
immediately evident. In fact /?>0, choose ft* such that
with
of this
is
product 1.
427
series.
once be verified by induction:
00
the
The binomial
IV.
analytic functions.
=
If
sufficiently large n, say
for
every
n^tm,
hence
By 12t>, 2, it follows at once that the partial products, and hence the sums of our series, tend to 0. The series therefore converges 37
partial to
the 2.
sum If,
0.
however,
R (a) =
< 0,
ft
we have
a n
whence
it
and hence
again follows by multiplication that
that the left
diverges in this case. 3.
If,
finally,
$ (a)
hand
~ 0, a =
sum
of our scries
The
fact that this value tends to
most speedily
The
side tends to oo. i y,
with y
say,
^
series
therefore
the w th partial
0,
is
no
limit as
solute convergence of the series
J?(
J
,
we
+ On
ma y De proved account of the ab-
have, by
29, theorem 10.
n
in the present connection as follows:
+
w> +
oo, the right hand side evidently tends to no limit; on Letting the contrary, the points which it represents for successive values of n circulate incessantly round the circumference of the unit circle in a constant sense, the interval between successive points becoming smaller 37
we
The mere
convergence of
(!)*(
J
follows already from
see that the convergence is absolute when SR(a)>0. It which requires the artifice employed above for sum being
228
and
the fact of the its detection.
is
428
and smaller at each the same is therefore
J(
Series of complex terms.
Chapter XII.
w l)
also (
true
diverges
J
view
In
turn.
of
the
when
of
s
Ji(a)
the
asymptotic
hand
left
= 0.
relationship, series
Hence our
side.
Thus theorem
247
is
parts, the behaviour of the binomial series is determined for every value of z and of a, and its sum for all points of convergence is given by means of a "closed expression".
established in
all
its
Uniform convergence. theorem on double series.
Series of variable terms.
56.
The fundamental remarks on
series of variable terms
n=0 are substantially the same for the complex as for the real domain 46); but instead of the common interval of definition we must (v.
now assume this is also
be a
a
A
region
p.
circle
403, footnote 17). z
\z
\
which for simplicity we shall suppose accordingly assume that
of definition,
most purposes
quite sufficient for
circle (cf. 1.
common
exists,
We
in which the
functions fn
(z)
to
are
all defined. 2.
For every individual z in
z
the circle \
ZQ
\
< r,
the series
n=0 is
convergent. The scries
2
fn (z) then has, for every z in the sum, whose value therefore defines a function of z the definition
on
p.
We
403).
2fn(*)
The same problems
= F(z).
as those discussed in
anse
a definite
the sense of
accordingly write
00
=0
circle,
(in
46 and 47
for the
connection with the functions represented variable terms. In the real domain, however, series of by complex is of the greatest importance, both for the it theory and its applithe of use of in its most general make function to cations, concept case of real variables
in
form, while in the complex domain this has not been fou:id profitable. The usual restriction, which is sufficiently wide for all ordinary purtherefore assume poses, is to consider analytic functions only.
We
further that
The functions fn (z) are all analytic in the circle z 2 with series z as centre and radius not expressible by power than a fixed number r. 3.
\
i.
e.
|
< r, less
We
then speak
429
Series of variable terms.
$ 56.
of series of analytic functions**; a series is the following: Is the such concerning which it ?> function F(z) represents analytic in the circle \z or not? Precisely as in the real domain, it may be shown by examples that without further assumptions this need not be the case. On the
for
brevity
the chief problem
|
<
other hand, the desired behaviour of F(z) may be ensured by stipul47, first paragraph) that the series converges uniformly. (cf.
ating
The
definition for this
almost word for word a repetition of 191:
is
Definition (2 nd form defined in the circle z \
which converges in
this
circle
e
for
if,
N>
every
).
A
series
za
<
r
39
\
circle,
> 0,
is
is
it
(independent, therefore, of
>N
n
for every
and every
2
fn (z), all of whose terms are Z Q <^ r, and in the circle z said to converge uniformly in this or
\
choose a single
to
possible
\
number
such that
z)
z in the circle considered.
Remarks. 1. Uniformity of convergence is here considered relative to all the points of Of course other types of region or indeed arcs an open or closed circle* of curves or any other set $)t of points, not merely finite in number, may be taken as a basis for the definition. The definition remains the same in subTn applications, we shall usually be concerned with the case in stance. which the terms fn (z) are defined, and the series 2fn (z) converges, at every z |<; r (or a domain 0)), but the convergence point interior to a circle |* z ZQ is uniform only in a smaller circle < r, (or in a smaller subg, uhcre with, its boundary, belongs to the interior of (i)) domain j, which, together 2 If the power series ~a n (z z ) n has the radius r, and 0<0<>, the series the (closed) circle \Z is uniformly convergent in ZQ\
\
\
<
,
|
\
<
.
equivalent to the following: 88
Here again we may remark
(cf.
190,
4)
that there is
no substantial
difference between the treatment of series of variable terms and that of sequences A series of functions n (z) is equivalent to the sequence of its partial sums and a sequence of functions s n (z) is equivalent to the series s (z), ., Sj (z),
2f
.
s fl
(*)+(,
.
-* (*))H For simplicity, we shall hereafter formulate all and theorems for series alone; the student will easily be able to
(*)
definitions
enunciate them for sequences. 89 This definition corresponds to the former 2 nd form. The 1 st form 191 be omitted, as it did not appear essential for the application of here may the concept of uniform convergence, but only for its introduction. 43 The set of points of a circle (or, for short, the citcle itself) is said to be closed or open according as the points of the circumference are regarded as included in the set or not.
248.
Chapter XII. Series of complex terms.
430
S fn (z)
3 rd form.
is said to be
uniformly
convergent in
*
*
\
|
(or in the
for every choice of points zn belonging to this circle (or set), corresponding remainders rn (zn ) always form a null sequence. The 4 th and 5 th forms of the definition (p. 335) also remain entirely altered and we may dispense with a special statement of them here. the
set
9ft),
if,
un
On the other hand, it is impossible to give as impressive a geometrical representation of uniform and non-uniform convergence of a series as in the real domain.
We
now
are
a position to formulate and prove the theorem
in
announced.
Weierstrass 9 theorem on double series
249.
41 .
We suppose
given a
series
each of whose terms fk the
analytic at least for \z
(z)
is
at
least for
z \
<
r,
so that
42
expansions
all exist
and converge
that the series
2f^(z) converges
zQ
\z
\
uniformly
Further, in the circle z
we assume < z
\
,
\
< r,
so that the series converges, in particular, everywhere the z Q \
for
every Q It
there. 1.
then be shown that.
may
The
coefficients in a vertical
column form a convergent ,
2.
3.
^j A n n=0
For
z |
is
(z
n
z
ZQ
)
|
converges for
< r,
z \
z \
= 0,
series:
1,2,...)-
.
the function
again analytic, with
n
The proof
dates from the year 1841. an M, indicates the place occupied in the given series by the corresponding function, while the lower index relatec to the position, in the expansion of this function, of the term to which th ?o 41
42
Werke, Vol. 1, p. The upper index,
70.
in the coefficient
Series of variable terms.
56.
z
For
4.
z
< r and for every (fixed) v
|
|
431
1, 2,
.
.
.
,
k-o
F
the successive derived functions of (z) may be obtained by term-by-term the series and each , given of the new series converges uniformly differentiation of i.
e.
in every circle
#
% \
|
5^
with g
,
<
r.
Remarks. we
our attention primarily to expansions in power series, the theorem simply states that with the assumptions detailed above, an infinite number of power series "may" be added term by term. If on the other hand we look rather at the analytic character of the various functions, we have the following If
1.
direct
Theorem.
<
r and the s If each of the functions fk (z) is regular for z z g, for every g 27/fc (z) converges uniformly in z r, then this series z ZQ r. The succes(z), regular in the circle represents an analytic function sive derived functions JFX") (z) of 1, are represented^ in that circle\ (z) for every v by the serie*
series
F F
Each of
in succession.
with Q
<
2. is
|
< \
\
<
^
t
these series converges uniformly in every circle
\
z
ar |
5^ g,
r.
The assumption
satisfied, for instance,
vergence
^
\
\
that 27/7c (z) converges in
by every power
It is also satisfied e. g.
r.
The
5C g for every g < r # z Q ) k with radius of con-
z
|
|
series 27 c k (z
by the
J
series *^ _ 1
-
1
~~~
-,
s
for r
=
1
;
cf.
58, C.
of our four statements shows that the present theorem cannot be proved simply as an application of Markoff's transformation of series; for the here this is deduced from the latter assumes the convergence of the columns, other hypotheses. 3.
first
<
>
be chosen Proof. 1 Let an index m, a positive g r and an e be kept fixed throughout. By hypothesis, we can determine a kQ such ZQ <^ g, that, throughout z .
to
|
for every
\
k such that
k'
**
Now
the function
coeiBcient
(a)
we
write
= ** (*) =/o (*) + sk (z) is
s k > (z)
inequality 235,
a
...+/* (*)
definite
power
series,
whose
th
i7t
we
therefore have
series
*W
+<> + A m be
convergent, by 81. Let the first of our statements is
if
,
is
By Cauchy's
Hence the
>k>A
is
+ its
+
sum. As
=
A-0
m
thus established.
could be chosen arbitrarily,
Chapter XII. Series of complex terms.
432
Now
2.
let
M'
be the
ZQ
=
We
z cumference ference
|
** (*)
I
I
|
^
y.
**.+ 1 (*)
I
of
s fco ^
|
(z)
i
+
I
** (*)
I
we
- %+
i
(*)
\
>k
have then for every k
Again, using Cauchy's inequality,
whatever the value of
43
maximum
along the circumon the same cir-
\M' + = M. t'
obtain, for every n
-
0, 1, 2,
.
.
.
,
Hence
k.
M 00
and 27 A n (z n^O restriction on
z
for
z |
|
zQ ) n therefore converges g
was that
<
r.
(In
if
< < Q < r.)
z
is
\
|
<
should be
it
fact,
z
z
for r,
<
Since the only
&.
the series must even converge
any determinate point satisfying the
always possible to assume Q to be chosen Let us for the moment denote by F l (z) the A n (z #o) n ft s thus, by its definifunction represented by the series ^ z < r. tion, an analytic function in
ZQ
z
inequality so that z
|
ZQ
|
|
r,
\
is
it
"
^
|
3.
We
now
have
to
|
show
F
that
l
(z)
= F (z),
F (z)
so that
itself
is
<
ZQ r. For this purpose, we choose, an analytic function regular in z r a positive Q in Q' as in the first part of our proof, a positive (/ # r, xr and an e 0, fixed. We can determine k so that, for all z in z fj \
|
<
< <
y
>
> k > & By Cauchy's > > k and for every n ^ 0,
for every k such that k' k before that, for k'
&'
Making
->
+
,
I
Now
|
|
An
we
-
.
infer that, for every
+
(
the expression between the
<*>
k
inequality,
>k
it
and every n
,
follows as
S 0,
+ an*) ^
+
I
modulus signs
is
the w th coefficient in
k
the expansion of
F
l
(z)
Zf
v
(z) in
powers of (z
z
).
Hence we have,
i>=o
for
< g: - If/, (*) \*' F, (z) ZQ
z |
|
|
v=-0
The
right
hand
side
is,
z
for |
z
\
<^
Q',
43 is a continuous function of am z Sk +i ( z ) 9 along the circumference in question and (9 being real) attains a definite maximum on this circumference. I
I
56. Series of variable terms.
>
and Thus, when e can determine k so that
>
for every k
Q'
k Q and every \z ^
ZQ
\
433
have been chosen
<^
(/.
we
arbitrarily,
This implies, however, that for
these values of z
r=0
The numbers
< Q' <
<
Q
were subjected
and hence
@' v\
for every z interior to the circle
We
4.
=
V +2 1
^ ()
sum
where the
z
\
"
-
0) ./
i
+
A!
+ 2 AS
(1)
(z
(1)
2
s
z
+3a
)
|
z
ZQ
Hence
\
equation holds
(z-
z
)
+ iA
s
(z
-z
9 )
+
(*
- *o) +
(z
-2
9
2 )
+
' ' '
.
.,
of the coefficients in any one column converges to the
begin our evaluations
^e
a
- *o) + 3 V (*
value written immediately below them. to
the
< r.
z
|
other than
restriction
that
write
/o' (*)
fi'
no
to
follows
it
above)
(as
with
and every
z F'
for those values of
= k >
e'
y
(z)
( 7e
Just as in 3. Q
Q
Y
e)
(we have only
we deduce
for
that
,
= 2fk A-O*
'
(z).
Indeed, by the same
<
ZQ r, for reasoning as before, this series converges uniformly in z for the of series If we r. write down the <. g every corresponding system v th derived functions, we obtain, in the same manner: |
F<"> (z)
= 27/fcW (z) k
(v
=
\
1, 2,
.
.
.
,
fixed)
o
z # |<:r; i.e. the scries Zf^(z) obtained by differterm by term, v times in succession, converges in the whole circle entiating z 5^ Q z ZQ
for
every
|
<
\
|
\
|
Remarks.
A
few examples of particular importance
will be discussed in detail in the next section but one. 2. The fact of assuming the convergence uniform in a circular domain is immaterial for the most essential part of the theorem: If is a domain of arbitrary 44 and if every point Z Q of the domain is the centre of a circle z ar shape g (for some Q) which belongs entirely to the domain, is such that each term of the series Zfj. (z) is analytic there, and is a circle of uniform convergence of the given series, then this series also represents a function (sr) analytic in the domain in question, whose derived functions may be obtained by differentiation term by term. Examples of this will also be given in 58. 1.
G
|
F
"
Cf. p. 403, footnote 17.
|
g
Chapter XI 1. Series of complex terms.
434
Products with complex terms.
57.
The developments
rems 1 remain
,
in
such a way
products with "arbitrary"
to
relating
we admit complex
terms hold without alteration when factors.
were conducted
of Chapter VII
and theorems
that all definitions
values for the
In particular the definition of convergence 125 and the theo2 and 5 connected with it, as well as the proofs of the latter,
entirely
unchanged. There
is
also nothing to
modify in 127, the
definition of absolute convergence, and the related theorems 6 and 7. On the other hand, some doubt might arise as to the literal transference of theorem 8 to the complex domain. Here again, however, everything may be interpreted as "complex", provided we agree to
+ aJ
to mean the principal value of the logarithm, for every The reasoning requires care, and we shall therefore n. sufficiently large the out carry proof in full:
take log (1
250.
+
Theorem. series,
The product 77(1 0J converges a suitable index m, with starting
whose terms are the principal values of log
sum of
the
we
this series,
77(1
-|-
an )
n-i
Proof,
a)
O
n-m+i vergent,
its
*
partial
the
If
converges.
are
)
.
.
.
+ am
(1
)
Lm
is
*'-.
For
sufficient.
if
the
tne principal values of the logarithms,
sums
since
consequently,
conditions
wm
),
the
if,
have, moreover,
= (1 + a,) (1 + * 2
The
+ an
(1
and only
if,
s n , (n
> m),
exponential
series is
con-
tend to a definite limit L, and is continous at every
function
point,
certainly tends to a vergent in accordance with
i.
e.
it
value the
+ 0.
Hence
the product is conhas the value
125 and
definition
stated.
The conditions are necessary. For, a given positive e, which we may assume b)
if
the product converges,
< 1,
we can
determine n
so that (a)
for I
an
|
every |
(1
n ^> nQ and every fcj>l. We then have, for every n > n and the inequality an -g-
<~<
,
|
certainly fulfilled
now show
in particular,
<
--
is
We
thus
greater than a certain index m. may that for the same values of n and k hisine' the
for
further
every n
\
435
Products with complex terms.
67.
45 principal values of the logarithms)
n
I
E
(b)
and therefore the
27
series
k
<e
\
+ an
log (1
) is
convergent. In fact, as
|
av
< ^*
\
Ji-mi-l
for every v
>
we
0>
have
also
(c)
by
likewise,
for these values of v,
,
+
log (!
I
and
46
,)
<e,
|
(a),
log [(1
I
+
re+1)
.
.
.
+
(1
+*)]
I
<*
and every k ^ 1. Accordingly, for some suitable integer 47 q we certainly have an+k ) a n+ J a n ^) . . log (1 log(l log(l 2qni\<s, and it only remains to show that q may in every case be taken 0. Now if we take 1, by (c). any particular n ^ # this is certainly true for k It follows that it is true for k For in the expression 2. for every
w
^w
y
+
+
|
+
+
+
+
.
+
= =
,
= a + log (1 n+1 + log (1 + a n+2 + 2q*i the modulus of either of the two terms < by modulus of the whole expression has to be < e; as e < )
)
first
e,
(c), 1,
and by
(d) the
q cannot, there-
be an integer different from 0. For corresponding reasons, it also 3 the integer q must be 0, and this is then easily seen follows that for k by induction to be true for every k. This establishes the theorem. fore,
=
The
part of theorem 127, 8 relating to absolute convergence may viz. be immediately transferred to the complex domain,
also
00
no
the series n
2 =
m+ 1
+ an
log (1
)
and
the product
//
n
-m
\
(1
+ an
)
\
simultaneously absolutely or non-absolutely convergent, in every case. In fact, it 11 of 29 and 30 remain valid. Similarly the theorems 9
are
remains true for complex a n
45
The
y
<-
modulus
s of
that in
logarithms are always taken to have their principal values in what
follows. 46
In
log (i
I
47
the this
sum
fact, for
+
*)
|
^
|
|
*
a |
|
+
<
~,
L*
|a
+
.
*
!
+
i
*p
+
.
.
.
=
-jJ* Lj is
of the principal values of the logarithms of the factors, but
sum by
we
|
For the principal value of the logarithm of a product a multiple of 2
TT i.
Thus log
if
^
. .
(i
e. g.
*""
take principal values throughout.
log
=
i
=
k>
log 1
,
but
= 0,
<
2 1
z
|
.
not necessarily
may
differ
from
436
Chapter Xll.
the quantities
Series ot complex terms.
are bounded,
?^
n
when
since
z |
\
<
while the expression in square brackets clearly has for those
4
.
its
<
modulus
1
z's.
Finally, the remarks on the general connection between series and products also hold without alteration, since they were purely formal
in character.
251*
Examples. *
by
IL
+
(1
is
"-)
theorem
29,
For
Divergent.
10, the partial
2
\
2
an
|
==
s
is
convergent, so that
products
the right hand
expression represents, for successive values of points on the circumference of the unit circle, which circulate incessantly round this circumference at shorter and shorter intervals. pn therefore tends to no limiting value.
4278.)
(Cf. pp.
2.
21
n
f
For
8.
,
z |
of this product
|<
=s '" 1 '
n~"'i
7/
1,
(1
-f-
* 2 ")
In fact
=
.
:p
obvious by 127, 7 and
is
>
its
the wth P artial Product
In fact the absolute)
n
th
is
at
once
convergence
partial product multiplied
by
is
(1-*)
which tends
The complex
to
1.
consideration
of
products
whose terms are functions
of a
variable,
n=l like that of series of variable
terms in the preceding section,
be restricted to the simplest, but also the most important which the functions fn (z) are all analytic in one and the same
will
in z |
_ ZQ
<; r |
that
circle) the circle.
the
circle,
case, circle
e possess an expansion in power series com crgent in in which the product also converges everywhere in The product then represents a definite function F(z) in .
(i.
and
which
is
said,
conversely,
to
be expanded in
the
given
product.
We F(z) |2
next enquire under what convenient conditions the function
by ihe product is also analytic in the circle For the great majority of applications, the following
represented 2
|
theorem
is
sufficient:
Products with complex terms.
57.
Theorem. // the functions fl xr at least in the (fixed) circle z
(xr),
\
/2
|
.
(xr),
if,
;
.
.
fn
,
437 .
(xr),
.
.
are all analytic
252.
further, the series
n-l
^
z z converges uniformly in the smaller circle g, for every positive xr r r\ then the product //(I n (#)) converges everywhere in z (j and represents a function (z) which is itself analytic in that circle. \
\
<
+/
\
<
|
F
The proof
same
the
follows
of
line
as
argument
the
of
that
continuity theorem 218, 1 almost word for word. To establish the convergence and analytic character of the product at a particular point z l z z r and prove the two facts in the circle r, we choose a g for every
<
|
|
first
% of the
<
ZQ
z
circle |
uniformly in the whole of
#
z |
|
/m+l (*)
>m
for every n
|
Pn
(z)\
=
on
converges uniformly in
ZQ
z |
Fm (z)
F(*)
=-
(*)
ZQ
z |
p.
+ (Pm+*
/Wl
a function
/,+ 2
I
and every
It follows precisely as
analytic in
+
Choose
\
Pm+l)
z r,
II (I
+/
^Q
I
fn (*)
then for
;
m |
all
so large that
<1 these w's and #*s,
^ e I/M+1 Wl + - + !/<*) <
3.
I
(ar))
ZQ
\
+
I
+ (Pn
^ Q.
As
the series
#
analytic in
w-1
\
382 that the series
|
<
\
+
+
I
+/whl (*)) ... (1 +/n
(1
|
I
fn (z) converges
\
^ g, so that the product //(I +/n (z))
certainly converges verges there (indeed absolutely).
I
2
series
\
ZQ
|
(*))
all
itself, \
< g.
^n-l)
+
the terms of this series are
by 249, therefore represents
Hence
= (1 +A (*)) ... (1 +/, (*)) Fm (z)
an analytic function, regular in that circle. the above considerations, we may deduce two further theorems, which provide an analogue to Weierstrass theorem on double series: is
also
From
1
Theorem pansion in
power
1.
More
term by term.
With
the assumptions of the preceding theorem, the
ix-253.
F (z)
may be obtained by expanding the product of know that the (finite) product we precisely,
series
P*
(*)
=
//(!+/,(*))
P=l
may z
|
be ZQ
expanded |
15
<
r,
in
a power
series
of centre ZQ
which
since this is the case with each of the functions
converges for
flt /2
,
(051)
.
.
.
.
Series of complex terms.
Chapter XII.
438
Let the expansion be
= A +A?\z-z + A! Z -z + -. + A = 0, 2, the limit for each (fixed] n = Xn lim A >
P]c (
z -)
Then
)
}*
i ''\
1,
.
.
.,
&->+ exists,
and *(*)
= JT(i + 4 0) = 1 ^ - *)"
Proof. I
^
z
z
I
used
in the
same
By
46,
(*
n=
A;=l
theorem
2,
uniform
the
convergence,
in
of the scries
,
preceding proof, implies the uniform convergence
in the
circle of the series 48
PI
W + [P. W - PX ()] +
+ [P W - P*-, k
(*)]
+
theorem on double series to this series, we Applying obtain precisely the theorem stated. Finally we prove a theorem about the derived function of F(z], W^f^ys^rass'
quite similar to
Theorem
218,2: For every z in
2.
-
\z
ZQ
\
for which
F(z)*^Q,
we have
*.
.
of z
//t^
series
and
gives
on the right hand side converges for all these values the ratio on the left hand side, the logarithmic dif-
ferential coefficient of F(z).
Proof.
We
saw
that the
expansion
= PI(Z) + (P F(z)
9 (z)
was uniformly convergent
in
\z
z
- P,
(z))
+
\^e
F'W = P/W + (P,'W-P
'
1 (*))
...
By 249,
+ -...
which implies that
PiW-^F'W at
If at a particular point circle. F(z) each n, and hence by 41, 11,
every point in the
have
PB (z)
=^
for
=J- 0,
we
48 For the remainders of the latter scries only differ from those of the former in that they contain the common factor m (*), which is a con* and hence is bounded ttnuous function for every z in the circle \z SS in this closed circle.
P
I
i
57.
*V(*)
Since, however,
*
~_
f,!(*)_
~il
what our theorem
this is precisely
439
Products with complex terms.
+ M*)'
asserts.
Examples. 1.
If
2 a*
is
any
absolutely
convergent series of constant terms, the product
77(1 represents a function regular in the whole plane, by 252. pansion, in power series, which is convergent everywhere,
By 253,
its
ex-
is
with
Here the indices
A2 ., A/c independently take for their values all the natural Aj <^&- The existence of numbers, subject only to the condition A t
,
.
,
<
,
of
.
.
.
most remarkable formulae. 2. We have
where the product on the right hand side converges in the whole plane The is word for word the same as that given in 219, 1 for a real variable. 3. Taking z = i in the above sine product, we obtain
proof
XI t
or
(Cf.
however the extremely easy evaluation 4.
The sequence
of
J[
\\
-J
of functions
n\n 2 converges for every
by 127, theorem
z in the
whole plane.
In fact
10,
*9
Introductio
60
Fundamenta nova, Kbnigsberg
in
analysin
inf.
Vol.
1,
1829.
Chap.
15.
1748.
in
128,
6).
254.
440
Chapter XII. Series
by 128,
also,
to Euler's
numbers
the
2,
yn
=
(1
--
-|-
complex terms.
ot
--
H
-f-
log n tend, as
j
constant C, so that the right hand expression,
when divided by n z
which
n
>
-f
OO
,
is
+
tends to a definite limit as n * OO. This proves the statement. Further, the limit K(*) say, becomes only for z = Q, 1, 2, .... Excluding these values, we have, for all other values of z, ,
y
=
lim
H
lim
~
This function of a complex variables (restricted only to be ={= 0, ~1 , 2, .) s the so-called a WHHI-function F(z) which we have already defined on p. 385 for real values of the argument. proceed to show that K(z) is analytic in the whole plane (i. e. an For this, it suffices to show that the series integral function). .
.
G
We
K to = ft to + fea W - ft - gn-l
to
^
also a constant
every
|
z
\
exists
and
<J g,
every circle
in
converges uniformly
B1
Let
z |
|
<
. .
=i+
l+i
.
z \
(l)
> w > 2^
^-.
+
.
- gn - 1
[(l
A
. and for every v -= 1, 2, 3, write (see p. 283 and p. 442, footnote 54) t
,
than some constant
i
(<*)
Now
.
g (z)\<
|
we may
less
and n
+ where log
such that
H-----h te
|
W = *l
further,
where |#n (*)| remains 51
<*))
.
B
for
=
every n
.
2, 3, ...
and
Then
.
:
As
(cf. p. 435)
we have
r, v
|
\
<
|,|
<
ff
b = ,4 and the last factor in the preceding expression therefore remains << e 3 for every |^|<^ and every w;> w. Similarly the last factor but one (see p. 295), As the remaining factor is also also remains less than a fixed number A 9 always less than a fixed number A for every \z\ fj #> it follows that On the other * (?) ^i'^a'^8 f r a ^ these values of * and every n >> m also remain bounded ^OT U) hand, the first m functions g.2 (z) g 1 (z) ., ,
.
I
I
^
.
<
|
\
,
\
|
\
ot
which s=^0,
From that |
this
we
,-;-
|
;
1,
infer
< A'
in
2, ... and
in ft,
z |
\
.
,
,
n
=
1
|
|
,
in the text
is
in the interior and on the boundar}' then for every n
>m
same way
exactly the Cor every
.
number A as asserted
the existence of the z for every g> thus established. If z is restricted to lie in a circle |
,
2,
that a constant A' exists such
every
|
Thus
*[<(>
~ In-
fi,
I
for all these
1
/N.^ < A,
W
I
.
|
z's
and
441
A. Dirichlet's series.
58. Special classes ot series of analytic functions.
w's,
#(-) - _ + -^L + *'#nM -V*2
,
.
I
where C
is a suitable constant By 197, it follows that the series for K(z) indeed the series of absolute converges uniformly in the circle |*|<e, does so, values 2 gn (z) and, by 249, K(z) is analytic in the gn -l (*) whole plane |
1
Special classes of series of analytic functions.
58.
A. Dirichlet's
A
Dirichlet series
series. 52
a scries of the form
is
as exponential functions are analytic in the Here the terms chief The whole plane. question will therefore be to determine whether and where the series converges and, in particular, whether and where We have it converges uniformly.
Theorem number
1.
To every Dirichlet series there corresponds a real 255, as the abscissa of conrerf/ence of the series
known
X
such that the series converges when
number
The
I
may
the series converges
also
$R (z)
oo
be
everywhere,
>A
and
<
A diverges when >H z) in the former case (
.
+00;
or
in the latter
nowhere.
Further, if convergent in every circle of the half -plane $ft (z) ^> /' and accordingly the series, by Weierstrass theorem 249, represents a function analytic and regular in every sucn circle and hence in the half-plane 63 W (z) A.
jl
oo and
=|= -|-
X>h,
the
series
is
uniformly
1
>
The proof
follows a line of argument similar to that used in the
case of power series verges at a point Z Q> 9t (z)
it
>
SR (* ).
suffices,
We
93)
(cf.
first
at
converges
it
show
that
if
the series con-
every other point z for winch
As however
by 184, 3 a,
to
show
that the series
-
A
1
1
n=i 5<J
iMore generally, a sories ~- m~
form
or of the form
is
^an
called a Dirichlet series \\hen
e~^ nZ
,
where the p n 's are
it
positive
is
of the
numbers
any real numbers increasing monotonely to -f oo. The existence of the half-plane of convergence was proved by /. L. W. V. Jensen (Tidskrift for Mathematik (5), Vol. 2, p. 63. 1884); the uniformity of the convergence and thereby the analytic character of the function represented were pointed out by E. Cohen (Annales fie. Norm. sup. (3), Vol. 11, p. 75. 1894;
and
the A n 's 63
412 is
Series of complex terms,
Chapter XII.
Writing
convergent.
a fixed exponent
(for
ZD
the numbers
0-n
tainly bounded,
and the
\
-> (* 8-B
#<))>
^
< A, say.
|
'
The
series is accordingly
corollary, divergent at a point z real part is less than
= z^
9
i
the
it
is
M
tne Y are therefore cer-
J
term of the above
convergent
we have
As a
4
th
]
,0,
once seen
s at
Z
(z
"n
when
series is therefore
z
?R(z
)>0.
a Dirichlei series is at divergent every other point whose
that of z 1
statement:
.
If
Supposing that a given Dirichlet
series does not converge everywhere or nowhere, the existence of the limiting abscissa Jl is inferred (as in 93) as follows: Let z' be a
and z" a point of convergence of the series, and For z both real and xo thc y >5R(O> <SR(^) it will converge. Now the method series will diverge, for z apply y of successive bisection, word for word as in 93, to the interval x y on the real axis. The value i so obtained will be the / point of divergence
choose
=
a?
=
-
=
. . .
required abscissa.
Now
=
X>
oo, A' may therefore be any real a domain G in which JR^^Jl' so that in general G will take the shape of a segour series is uniformly convergent in that domain. us choose a point Z Q for which 3l(zQ )
i (for i z is restricted to lie in
suppose if
number); and |2|f^-R, ment of a circle,
A<
To show this, let we write
before,
More generally, we may
54
at
once observe that
if
|*|
<
Ct
and
if
we
the factor &, which depends on z and w, for all the values allowed for z and w.
W (l-f-z)
For every
z |
\
-^ ^^ = ^^^ 10
<
-^-
,
we
a)
therefore have 1
equality
1*1 is
at
<
/?,
once obvious
if
we
replace
remains
17
than a fixed constant
*
iy=
<
|
1
;
u
+/---.fl+.... o 4
hence
in
was denoted by
<* 2jR all
less
Proof:
With
,
the expression in square brackets, which
This
and \w\
write, taking the principal value,
#,
satisfies the in-
-
the quantities in thc brackets
by
theii
A. Dirichlet's series.
58. Special classes of series of analytic functions.
V
a convergent series of constant terms; by 198, 3 a
is
nz
show
fore suffices to
it
443 there-
that 1
1
converges uniformly in the domain in question and that the factors are uniformly bounded in G. Now, writing X d (> 0), 91 (z^)
=
1
1
i*~*
(n
+!)*-*
Using the evaluation given in the preceding footnote
expanding (1 see that a the
-|
=0
J
A
constant
in
certainly exists
modulus signs on the
hand
right
(or else directly,
powers of
(z
-
ZQ))
by
we now
such that the difference within above inequality is
side of the
in absolute value
<4 every z in our domain and expression on the right is thus
for
On
every
other hand, since
the
n
1, 2, 3,
T-,
the factors
The whole
are
uniformly bounded in G. By 198, 3 a, this proves that the Dirichlet series is uniformly convergent in the domain stated, and hence, in particular, that every Dirichlet series represents a function which is analytic in the interior of the region of convergence of the senes (the half- plane
9tf
(z)
> X)
.
From 1
yr <*LJ
follows at once that
a Dirichlet series converges absolutely at a any point z for which 9^(2) >SK(2 ), and if it does not converge absolutely at ZQ) then it cannot do so at any point z for which 9fi (z) SR (Z Q ) Just as before we obtain it
point Z Q ,
it
does so
<
Theorem also
be
+00
2.
or
if
at
.
There exists a definite real number I (which may oo) such that the Dirichlet series converges ab-
solutely for $l(z)>l, but not for ft(z)
=
following
=
Chapter XII. Series of complex terms.
444
Theorem
3.
Proof.
If
We
have in every case is
J
is
$1 (z
>1
ZQ)
.
>
9t (z)
9ft
(z )
+
1
,
then with
for
convergent,
absolutely
and
convergent
an
A<^1.
I
This proves the statement
at once.
Remarks and Examples. If a Dirichlet series
1.
.
not merely everywhere or nowhere convergent the the half-plane *)t (z) < A of divergence of
is
situation will in general be as follows,
<
<
I of conditional convergence of the series; the series is followed by a strip A $ft (z) the breadth of this strip is in any case at most 1, and in the remaining half-plane $R (#) > /, the series converges absolutely.
2. It may be shown by easy examples that the difference / A may assume and 1 (both inclusive), and that the behaviour on the bounding any value between lines 8t (z) = A and 9} (z) = / may vary in different cases.
The two
3.
series
2n
1
series
J?on~~ z anc*
z
provide simple examples of Dirichlet
which converge everywheie and nowhere. 4.
27
has the abscissa of convergence A
-
nz
=
1
thus
;
it
represents an analytic
function, regular in the half-plane fll (#) > 1. It is known as Riemann\ ^-function (v. 197, 2, 3) and is used in the analytical theory of numbers, on account of its connection with the distribution of prune numbers (see below, Rem. 9) 55 5. Just as the radius of a power series can be deduced directly from its co.
efficients
series
(theorem 94), so we
may
infer
what positions the two limiting
Theorem. The given by the
coefficients of a given Dirichlet
We
have the following
abscissa of convergence A of the Dirichlet series
formula A
from the
straight lines occupy.
=
_
lim
j
l
a u+l x log
-f
a^ 2
-f-
.
.
.
+
2
n z
is
invariably
av
where x increases continuously and
[eW] a n for a n in
Substituting
convergence 0.
series
A
this
=w,
0*]
=;.
formula, we obtain
I,
the limiting abscissa of absolute
66 .
concise account of the most important results in the theory of Dinchlet's in G. H. Hardy and M. Riesz, Theory of Dirichlet's series,
may be found
Cambridge 1915. 65
A
detailed investigation of this remarkable function (as well as of arbitrary is given by E. Landau y Handbuch der Lehre von der Verteilung der
Dirichlet series)
Primzahlen, Leipzig 1909, 2 Vols., in E. Landau, Vorlesungen uber Zahlentheorie, Leipzig 1927, 3 Vols., and in E. C. Titchmarsh, The Zeta-Function of Riemann, Cambridge 1930. 56 As regards the proof, we must refer to a note by the author: "Uber die Abszisse der Grenzgeraden emer Dinchletschen Reihe" in the Sitzungsberichte der
Berliner Mathematischen Gesellschaft (Vol. X, p. 2, 1910).
58. Special classes of series of analytic functions.
we
A. Dirichlet's
7. By repeated term-by-term differentiation of a Dinchlet series obtain the Dirichlet series
series.
F (*) ~
-
445
2
***>
n
(fixed,).
As an immediate consequence of
Weierstrass' theorem on double scries, these necescannot have a larger abscissa of convergence than the original series, and, owing to the additiopal factors log" n, they can obviously not have a smaller one sarily
They represent, in the interior of the half-plane of convergence, the derived functions F
either.
8. By 255, the function represented by a Dinchlet series can be expanded a power series about any point interior to the half-plane of convergence as centre. The expansion itself is provided by Weierstrass' theorem on double in
co
series.
If,
about ZQ
=
and
for instance, -j~
2 as centre,
required to expand the function
is
it
we have
continues to hold for k
this
the value
1.
Hence
for
n>0 -
for k
=
1
= 2,
(*)
1
= z *=1 k
3, ...
provided
we
as having
interpret (log lj
which gives the desucd expansion
9.
1'or
(*)>!, CO
the series
VJ
J
n=in 2
and
J
the product '
77 JI
l-/>'
2
values all the prime numbers 2, 3, 5, 7, (where p in succession) have everywhere the same value, and accordingly both represent the Riemann function (Euler, 1737; v. Introd. in analysin, p. 225) (z). takes for
its
.
.
>
Proof. Let z be a definite point such that 5H (z) = 1 -f-<5 1 By our remark 4 and 127, 7, the series and product certainly converge absolutely at this point. We have only to prove that they have the same value. Now
multiplying these expansions together, for denotes an integer kept fixed for the obtained is AT
where the accent on the
all
prime numbers
moment,
.
p< AT, _
whore
the (finite) product so
indicates that only some, and not all, of the terms taken. Here we have made use of the elemen2 can be expressed in one and tary proposition that every natural number only one way as a product of powers of distinct primes (provided only positive 15 * of the series written
2"*
down are
>
(G5l)
Chapter XII. Series ot complex terms.
446
exponents are allowed and the order of succession of the factors Accordingly
integral left
is
out of account).
*
7T
V>
V
On the right hand side \ve have the remainder of a convergent series, which when tends to co. This proves the equality of the values of the infinite product and of the infinite series, as was required.
N
-
+
By 257, we have
10.
where
for
ffi
-
(z)
>
1
-
--
= 1, M (2) = 1, p (3) = 1, p (4) = 0, p (5) = 1, . . . (1) 1, p (6) 1 according as n is divisible by the and generally /x (w) 0, + 1 or square of a prime number, or is a product of an even number of primes, all different, or of an odd number of primes, all different. The product-expansion of the ^-function also shows that for JR (#) > 1, we always have The curious coefficients (z) =t= 0. There is no superficial regularity in the fi (n) are known as Mobius* coefficients. H
+
,
mode
of succession of the values
Since f
11.
-
(z)
^n
z
0,
-f-
converges
among
1
1,
the
numbers
absolutely for 81
/LI
(n).
(?)>!, we may form
the square (f(*)) 2 by multiplying the series by itself term
arranging
in
order of increasing denominators (as
is
by term and reallowed by 91). We thus
obtain
in denotes the number of divisors of n. function explain the importance of the
where to
in
These examples may suffice problems in the theory of
numbers. B. Faculty series.
A /
(
is
faculty series (of the first kind)
n
a scries of the form
V
'
which of course has a meaning only questions of
convergence,
if
elucidated in
2
+ 0,
the
first
1,
2, ....
instance
The
by Jensen,
are completely solved by the following
258.
Theorem
of
/>/.rfu 67
.
The faculty
the exclusion of the points 0,
1,
ciated" Dirichlet series
^
series (F) converges
2, ...
with
wherever the "asso-
and conversely the latter converges wherever the series (F) conThe convergence is uniform in a circle for either series when it so for the other, 'provided the circle contains none of the points
converges, verges. is
0,
,
2, ... either in
1,
J Uber
6
Vol. 36, pp.
die
its
interior or
on
its
boundary.
Grundlagen der Theorie der Fakultatenreihen.
151218.
1906.
MUnch. Ber
58. Special classes of series of analytic functions.
Proof. series
at
We
1.
first
any
particular the faculty series at the
show
that
point
=j=
same
the
0,
gn (z)
by 184,
show
3 a, to
n=l is
t> all
convergent
2,... involves
that
of
As
point.
has the same significance as
447
convergence of the Dirichlet 1,
n*
>...(*+ n) if
B. Faculty series.
in
gn (z)'
1354, example 4,
it
is sufficient,
that the series
&(*)
8*
Now
+ i(*)
n =l
tends to a Sn
I
&W'&
finite limit
+i
as
(*)
w
I
increases,
namely
(2)
the value F(z); hence, in particular, this factor remains bounded for values of w (z being fixed). Hence it suffices to establish the con*
verge nee of the series
But
has been done already in
this
The
2.
fact
that
254, example 4. convergence of the faculty
the
series
at
any
point invoK es that of the Dirichlet series follows in precisely the same manner, as again, by 184, 3 a, everything turns on the convergence of
-!&,(*) -ft,+ila circle in which the Dirichlet series converges be 3. which none of the points 0, and contains 1, 2, ., absolutely either as interior or boundary points. We have to show that the faculty scries also converges uniformly in that circle. By 198, 3 a, this again reduces to proving that
Now
let
.
a fg\.a z Sn Bn \ ) *
+1
.
(z
\*.
uniformly convergent in $ and that the functions 1 The uniform convergence of uniformly bounded in
is
/
gn
(z)
remain
.
n=l
was already established
254,
in
4.
Also
it
was shown on
p.
440,
footnote 51, that there exists a constant A' such that 1
for every z in
theorem
Jf
and every
w.
This
is
all
that is required.
(Cf.
-H>,
3.)
The
converse, that the Dirichlet series converges uniformly in which the faculty scries does so, follows at once by uniform convergence of the series the a from 3 198, g n (2) g n + ,() and the uniform boundedness of the functions gn (z) in the circle, both 4.
every circle in
of which
were established
2
in
254,
4.
\
|
448
Chapter XII. Series of complex terms.
Examples. 1.
The
faculty series
S converges
1
'
at every point of the plane =(=0,
1, ...
For the
.
Dirichlet series
I
00
1
z
n=l is
evidently convergent everywhere.
As
.111 x+l
_
1
A/I* T =
/..
x(x+l)'
2! " 7~iT~/
_
?i\
*(*+l)(ar + 2)
x
__
1
x
x
.
> f
the given faculty series results simply, the series
_x
A* 1-
""' *>
/3
'
x
kl
(x -f 1)
Euler's
by
-
-
(x
+ *)
'
transformation 144, from
~~
_ __ """""
2. It is also easily
01
1
To show cessively
this,
left
(cf.
pp.
2656)
11
.
we have
from the
seen
(Stirling
only to subtract the terms of the right hand side sue side. After the w th subtraction we have
we
n\n z
1
..(*+)
z-n* *
when w-*oo, provided by 254, example 4, tends to Methodus differentialis, London 1730, p. 6 seqq.) C.
A
>
hand
). this,
$R (z)
(*-!)!
.
,
n\
and
that for
Lambert
Lambert's
series is a series of the
series.
form
58
again inquire what
is the precise region of convergence of the 1 can that for every z for which z n noted be series, it must of number the terms of an infinite the series bebe equal to zero, of the circumference unit circle For this the come meaningless. reason,
If
first
will
be entirely excluded
from consideration
59
while
we
discuss
the
A
68 more extensive treatment of this type of series is to be found in a paper author: Cber Lambertsche Reihen. Journ. f. d. reine u. angew. Mathem., the by Vol. 142, pp. 283315. 1913. 59 This does not imply that this series may not converge at some points ar t
of this circumference, for which zf =t= -f 1 for every n *Z will not consider the case here.
happen; but we
1.
This may
actually
58
C. Lambert's series.
Special classes of series of analytic functions.
question of convergence of these series, and the outside the circle will be examined separately.
We
which
theorem,
the
solves
completely
points inside and have the following
of
question
449
convergence
in
this respect:
Theorem.
//
whose modulus an z
4
s
converges, the Lambert series converges for every z ! a n * s n t convergent, the Lambert series
W^
precisely the same points as
at
converges
2an
is
n
+
"associated"
the
power
as before. provided Further, the convergence is uniform in every circle & which lies completely (circumference included} within one of the regions of convergence of the series and contains no point of modulus 1.
series
z
\
\
Proof. in that case
2a n
Suppose
1.
necessarily <^ 1
1
The
divergent.
and we have
to
radius
show
of
r
2 a n zn
that the
first
is
Lambert
and the assoaated power series converge and diverge together 1. 1, and that the Lambert series diverges for every \z\ \z\
series
<
for
>
Now
w
y
and
**
vi
it
Accordingly,
suffices,
z"
= y
n\
.(\
by 184, 3 a,
1
n
v-.
establish the convergence of
to
the two series
and -
for
The
|s|
first
M _
of these facts
7* I*
V
L
I
I I
obvious, however,
is
second follows from the remark that for
z |
\
< 1,
we have
1 1
while the z
n \
>
-x-
for all sufficiently large w's.
On where |
the other hand, if the Lambert series converged at a point zv 1 , the power series
ZQ
|
>
would converge to
=2 =+
for z
converge for z
1.
,
and by 93, theorem Hence the series
1,
would have also
would also have Finally, \z\ <*
Q <. r
to converge, which is contrary to hypothesis. the fact that the Lambert series converges uniformly in
inequality
once be inferred from the corresponding fact in n in virtue of the 46,2, power series \a n z \, by
may
the case of the
at
259.
450
Chapter XII.
The
2a n
is
diverges
thus completely dealt with.
2
n a n convergent, so that a n z has a radius suppose The Lambert series is certainly convergent for every z -< 1 and
Now
2.
?^>
case where
Series of complex terms.
1.
\
|
indeed uniformly so for
For
I
z
>
f
^> Q
I
1
all
values of z such that
z |
\
<^
>
we have
,
.-6T-, "!_(!)'
<
'
and as
I
z
this
1,
reduces the
assertions
later
the pre-
to
ceding ones, and the theorem is therefore established in all its parts. By the above, a very simple connection exists, in the case where _T a n is com ergent, between the sum of the series at a point z outside the unit
and the same sum
circle will
suffice if Accordingly it convergence of the series which
either the circle
<
z |
\
we
it.
thdt region of only the unit circle. This is
inside z |
2 a n zn
inside
point
consider
lies
? or the unit circle
the radius r of the series
the
at
<
is
1
\
<
or ^>
1
1.
according as Let r 1 denote the
itself,
radius of this perfectly definite region of convergence. The terms of a Lambert series are analytic functions
l^l^fj, and
for
the series
<: r lf
is
regular in uniformly con-
e\ery positive Q hence we may apply Weierstrass* theorem on double series to obtain the expansion in power series of the function rehave z r presented by a Lambert series vergent in
z
|
\
m
|
<
\
+a
.
We
+ a,2
a **
+
and we may add all these series together term by term. In the n row, a given power z will occur if, and only if, n is a multiple of or k a divisor of n. ing series, suffix
r
bolically
is
will
An
Therefore
be equal
to
the
,
the coefficient of z
sum
of those
a divisor of n (including 1 or w).
n
in the result-
coefficients
This
we
a v whose write sym-
60
An and we then have,
In words: the
for
z |
sum
\
=
ad
,
d/n l
,
of all a d 's for
which d
is
th
k,
a divisor of n.
58. Special classes of series of analytic functions
C. Lambert's series. 451
26O.
Examples. an
1.
example
=
1.
we
11)
An
Here
equal to the number of divisors of n t which (as in 257, denote by rn then is
\
= z
+
s 4 + 2 # 5 + 4 z* -f 2 3 7 -f 4 z* -f . In this curious power series, the terms #n whose exponents are prime numbers are distinguished by the coefficient 2. It was due to the misleadmgly close connection between this special Lambert series and the problem of primes that this series (as a rule called simply the Lambert series) 61 played a considerable part in the earlier attempts to deal with this problem. But nothing of importance was obtained in this manner for some time. Only quite recently AT. Wiener 62 succeeded by this means in proving the famous prime number theorem.
an =
2.
will
Here
n.
E
n
1
is
for |
2
as
3
-f 3
.
equal to the sum of
z
<
\
all
An
Zad
which we
the divisors of n,
8
relation
.
.
1
*---.= J7T n 's = sr+3s + 4* 8 + 7* + 6* + T L ~~ z n=\
The
3.
An
Thus
denote by rn '.
n
2 z*
-}-
1S
uniquely reversible,
e.
i.
12*'
+
----
An
for given
's,
the
din
a n can be determined in one and only one then have in fact
coefficients
We
way
so as to satisfy the relation.
denotes the Mobtui coefficients defined in 257, example 10, whose In consequence of this fact, not only can a Lambert 1. -|- 1 and series always be expanded in a power series, but conversely every power series - 0. may be expressed as a Lambert series, provided it vanishes for z = 0, i. e. A But it should be observed that a relation of the form
where
/i
(k)
values are 0,
need not remain true for 3 4. For instance, if A |
and we have the curious
> \
=
1
1,
even when both series converge there.
and every other an
-
J7
/iW
71=1
Similarly,
we
number
Z n
by diagonals allowed),
we
an
1
\n ~
.-
X
^~~ n = f(z)
pp.
61
Lambert,
Wiener, N., a
7,
pp.
|
1,
,
a
number
Z n
a n zn
g
(z),
and grouping the terms
1
in the double expansion of the Lambert series on p. 450 (which obtain
62
1100,
|
<
on
and
/(*)*(*) + *<*) + ...-
Vol.
2
of integers less than n and prime to
^n
oo
Writing
6.
S
r
* ~~
find the representation, valid for
(n) denotes the introduced by Euler.
where 9
0,
identity =.
6.
An =
(),
/*
27
is
*(").
JH. Anlage zur Architektonik, Vol. 2, p. 507. Riga 1771. new method in Taubenan theorems, J. Math. Massachusetts, H>1184, 1928, and Taubenan theorems, Ann. of Math. (2), Vol. 33, 1932.
452
XIL
Chapter bor a n
7.
=
1
"1
I)* /
( \
=
,
complex terms.
Series of
=
n,,
I)"" /
( v
(_i)n-i ^
1
1
=
v
=--,== w
, '
,
n
an
obtain in this way, successively, the following remarkable identities, 1 to oo : # 1, in which the summations are taken from n | zn zn n~ L --( l) a) 7 n .-__ 1
.
.
,
.
valid
,
we for
<
|
2
b>
2
^
^
^"i^h
^
l)
-^(1
'
W
---
Z*log(l
1 ~
Qt
-^i*
"
-
27
e)
>
===
(I OC
n
/^/ i
1
^ n),
4-
I
etc. 8. In the two identities d) and e) we have on the right hand side a series thus simple conof logarithms (for which of course we take the principal values) nections can be established between certain Lambert series and infinite products. ;
from the two
E. g.
identities in question:
zn
^.1
II (1 MO
--
9.
As an
0,
HI
sequence
Z
=
y
- Z -' 1
with
",
-
/
(cf. 6, 7) 1,
2,
1,
3,
21,
13,
8,
5,
34,
....
55,
then have l
,-*
* i.+
1
3
.!*... + + 8
1
+
1
- V'5\ A= f r /3 -V + - - V5 [L (J
where a and
ft
55
'
series
3
JC ,
are the roots of the quadratic equation x 2
174. Suppose z n ->
and b n -> b
_ es
- __ 3 A /7._ ^
.
1
The
proof
=
(Cf. Ex. 114.)
0.
is
based
64 .
conditions
may W3
infer
-* tfl
175. Suppose zn -> X)
we then
xn
x
Under what
4= 0.
,
n
_
Exercises on Chapter XII bj*
r
.
.
21
where L (je) denotes the sum of the Lambert on the fact, which is easily established, that
that
.
, I
71
interesting numerical example we may mention the following: Taking w n _i + w n-2 w ^ obtain Fibonacci's for every n > 1, */ n I, and
0,
We
h *")
(i.
e.
|
zn
-> |
-f-
oo).
Under what conditions may
infer that a)
(l \
b)
zn
+ -
z-)* n/
(z l l sn
-
-> e*, 1)
-> log *?
Landau, E.: Bull, de la Soc. math, de France, Vol. 27, p. 298. 1899. In these exercises, wherever the contrary does not follow clearly from the context, all numbers are to be regarded as complex. 64
Exercises on Chapter XII. 176. value than
The
principal value of z* remains, for fixed bound.
some
=
zn
177. If
all
453
values of z
t
less in absolute
j?(- If (*), \v/
v
either
of
Csy
z n ->
or
->
sn
n ) when 9R (z) 178. Let 0, A,
-=
arbitrary.
0,
according as
9R (z)
>
<
or
What
0.
the behaviour
is
0?
c,
b c ^ be four constants for which a d and let # be numbers (z z lt z 2t ) given by the re-
Investigate the sequence of
,
currence formula
zn
What
are the necessary
and
~ ~-
a n
LI hi
_ _ C5r n
i
v
(n V
a
-t-
sufficient conditions that (z n ) or ( ) \xr n /
1 J
2 ^
>
> /
should converge?
And if neither of the two converges, under what conditions can z p become again for some index />? When are all the n 's identically equal? 179. Let a be given 4= and Z Q chosen arbitrarily, and write for each n
z
-3r
^
1
(zn ) converges
and only
if,
Z Q does not
if,
two values of
on the perpendicular
lie
Va
through its middle point. a nearest to sr What fulfilled, (zn ) converges to the value of (xn ) when s lies on the perpendicular in question? line joining the
V
180.
The scries 27- {+~*~
.
does not converge for anv
on the other hand, does converge for every
real
y
real
y
;
to the straight
If this condition is is
the behaviour of
the series 27
n
~
TV---"
--
,
log n
,
0.
=4=
The
refinement of Weierstrass's theorem 228 that was mentioned in footnote 13, p. 399, may be proved as follows in connection with the foregoing example: From the assumptions, it follows, firstly, that we may write
180
a.
(n
where the
Bn
's
are
bounded;
^^r
--
+
l
&
hence, secondly, that
(y
we may
1 -H -~x
also
test
184,
3.
2 an
If
were to converge, then
(A> 2)
>
X) '
write
(C sumptions of the
- Min
~ N"" )
1
satisfy the as-
2 an bn =
E
- would
have to converge, contrary to the preceding example and theorem 255. 181. For a fixed value of z and a suitable determination of the logarithm,
does
tend to a limit as n -> -|-QO? 182. For every fixed x with lim
1
+
tt
,
< -h
-,
SR (z)
<
1,
+ ...+
exists (cf. Ex. 135).
The
may be expanded
in a
power
a n z n for z < 1 if we take the principal value for the logarithm. z 1. Jhat this senes stjjl converges absolutely for
Show
183. series 27
function (1 |
|
z)
-
sin
Hog
=
1
,
J
J
454
Chapter XII. Series
184. angle",
x tends to
If
we have
+1
l_, + .,_,,<> + ,t_4.
b)
(l-
.
e)
complex terms.
from within the unit
a)
d)
ot
circle,
and "within the
_!.. .
..-
_
(1-,
2 a *n z n--*lim ,.
n y ^&*
an 5 bn
provided the right hand limit
bn
exists,
is
positive for each
w,
and
Sb n
is
divergent.
185. Investigate the behaviour of the following power series on the circumference of the unit circle:
V
1
(
^)
e)
2
2i
186*
f" If
S
**
v
n '
n
tt
*">
wnere
n
has the same meaning as
a n z n converges for
for all such values of *, then
187. The power
+
^|an
f
in
Ex. 47.
-< 1 and its sum is numerically 1 converges and its sum is
z |
a |
\
<
.
series
h)
J-__
have the unit circle as circle of convergence. On the circumference, they also converge in general, i. e. with the possible exception of isoLited points. Try to express their sums by means of closed expressions involving elementary functions; separate the real and imaginary parts by writing 2 = 7 (cos x -\-i sin x), and write down the trigonometrical expansions so obtained for r 1 and for r = 1 separately. For which values of x do they converge? What are their sums? Are they the Fourier series of their sums? 188. What are the sums of the following series: coswccsinwy Y7 cos nx cos n y H ft -_. sin n x sin ny X' ^, all
<
.
f\ /
-*-/
M rl
and of the three further series obtained by giving the terms of the above series the sign (- 1) B ?
455
tixercisee on Chapter XII.
z* as in Ex. 187, but leaving
189. Proceeding with 1
we
,
the geometric series obtain the expressions
a)
V
b)
>7 ?" sin
rn
cos
nx =
- --r
1
-
-,
1
cos x
2 r cos x-}-r* ;
rsma;
n a;
=
1
Deduce from them
the further expansions
costta;
f
cos x) n w^i (2
and indicate the exact intervals of validity. 187 a the following- expansion will have been obtained,
19O. In Exercise among others.
rn
y. ^Tj n
Deduce from
it
sin
nx
tan
(
f
rsin x
\ I
\ 1
r
cos x J
-f-
cot x)
.
the expansions
00
y,
~ .x
1)"
"*
rn
sm n
x-
sinn
a;
=
tan
-x
(r
( f
&
\
-5-
x\
,
n=L
and determine the exact intervals of
validity.
191. Determine the exact regions of convergence of the following series
[log log n]
where
(#>) is real
and increases monotonely to
1!>2. Establish the relations
*"
-'
l
+*
where the summation begins with n
= l.
+OO.
45G
Chapter XII. Series of complex terms. 193. Corresponding to Landau's theorem (258)
Dirichlet series 27 (~ I)""
are convergent
"
1
we have
the following:
and the so-called binomial coefficient series 2a n
=
and divergent together, the points z
1, 2, 3,
.
. .
The
(
being disregarded.
194. For which values of z does the equation
hold good?
195. Determine the exact regions finite
of
convergence
of the following in
products:
77(1
196. Determine, by means a)
The second
a;.
2~
Does
2
g
of these has the value
cos
[cosh (jtx^ 2)
this continue to hold for
complex values
197. The values form
of the products 195, i) of a closed expression by means of the
198. For
1 1
1< 1
199. By means
0)^(1 + 5).
*>//(!+)
//(l+J*),
for real values of
of the sine product, the values of the products
(jr
x
\/
2)
]
.
of xl
and k), can be determined F- function.
in ihe
,
and the expansion of the cotangent and product may be evaluated in the + form of closed expressions; x and y are real, and the symbol f( n ) indicates of the sine product
in partial fractions, the following series
<
2 n=
sum
of
the
two
series
2
n=0
f( n )
2
and
k=l
f(-~ k )> & nd similarly
product:
+ 00 a)
<0
2
n=s-
J
*'
oo
+00
-f CO
the
+00 b)
-
for
the
g 59.
General remarks on divergent sequences.
Chapter
Divergent
457
XIII.
series.
General remarks on divergent sequences and the
59.
processes of limitation. of the nature of infinite sequences which we have 8 11, is of preceding pages, and especially in a strict for and recent construction date; irreproachable compai ativdy of the theory could not be attempted until the concept of the real number had been made clear. But even if this concept and any one general convergence test for sequences of numbers, say our second main
The conception
set forth in all the
criterion, were recognized without proof as practically axiomatic, it nevertheless remains true that the theory of the convergence of infinite sequences, and of infinite series in particular, is far more recent than
the extensive use of these sequences and series, and the discovery of the most elegant results of the subject, e. g. by Euler and his con-
temporaries, or even earlier, by Leibniz, Newton and their contem To these mathematician?, infinite series appeared in a very poraries.
way as the result of calculation, and forced themselves into e. g. the geometric series 1 -\- x -\- x* -\ocso to speak curred as the non-terminating result of the division l/(l x); Taylor's series, and with it almost all the series of Chapter VI, resulted from the principle of equating coefficients or from geometrical considerations. natural notice,
It
was
and the
:
in a similar
all
manner
that
infinite
other approximation processes
symbol
for infinite sequences
products, continued fractions occurred. In our exposition,
was created and then worked with;
originally, these sequences were there, and the question be done with them. what could was, On this account, problems of convergence in the modern sense Thus were at first remote from the minds of these mathematicians 1
it
was not so
it
is
.
not to be wondered at that Euler, for instance, uses the geometric
series
i-y-^x-f-*/
even lor
x =
1 or
x =
-,
2, so that
\
X
he unhesitatingly writes 2
41. Cf. the remarks at the beginning of This relation is used by James Bernoulli (Posit, arithm., Part 8, Basle 1696) and is referred to by him as a "paradoxon non inelegans". For details of the violent dispute which arose in this connection, see the work of R. Reiff mentioned in 69,8. 1
a
458
Chapter XIII. Divergent
or
1
from
similarly
-2+
22
-
2J
+
series.
...
= ^
= 1 + 2x+ 3x + 2
f^_
-)
.
.
.
;
he deduces the relation
i-a + 3-*+-... = i; and a great deal more. It is true that most mathematicians of those times held themselves aloof from such results in instinctive mistrust, and recog3 But they had nized only those which are true in the present-day sense no clear insight into the reasons why one type of result should be admitted, and not the other. Here we have no space to enter into the very instructive discussions on this point among the mathematicians of the 17 th and 18 th centuries 4 We must be content with stating, e. g. as regards infinite series, that Euler .
.
always
let
when they occurred
these stand
naturally
by expanding an 5
analytical expression which itself possessed a definite value was then in every case regarded as the sum of the series. It is clear that this
convention has no precise basis. Even though, 1 H 1 ... results in a very simple
1 for instance, the series 1 manner from the division 1/(1
fore should be equated to
This value
.
^
+
x) for
there
is
x
and there-
1 (see above),
no reason why the same
should
series
not result from quite different analytical expressions and why, in view of these other methods of deducing it, it should not be given a different The above series may actually be obtained, for x value. 0, from the
=
function f(x) represented for every f(*\ J{)
or from
t
J? ir-J)-" "* -l
1
>
by the Dirichlet series
= i_L4_!_!-L 4* -*-' 2*t-3-
+
= putting x to take 1
=
x
1.
1
In view of this
+
.
.
.
=
2 g,
latter
method of deduction, we should have
and in the case of the former there
mediate evidence what value /(O)
may have;
it
no im-
is
need not at any rate be
+ -2
.
3
Thus d'Alembert says (Opusc. Mathem., Vol. 5, 1768, 35; M^moire, p. 183): "Pour moi, j'avoue que tous les raisonnements et les calculs fond^s sur des series qui ne sont pas convergentes ou qu'on peut supposer ne pas 1'fitrc, me paraitront toujours tr&s suspects". 4 For details, see R. Reiff, loc. cit. 5 In a letter to Goldbach (7. VIII. 1745) he definitely says: ". . so habe ich diese neue Definition der Summe einer jeglichen seriei gegeben: Summa cujusque seriei est valor expressions illius finitae, ex cujus evolutione ilia series oritur". .
59.
General remarks on divergent sequences.
459
Eulers principle is therefore insecure in any case, and it was only Enter* unusual instinct for what is mathematically correct which in general saved him from false conclusions in spite of the copious use \vhich he made of divergent series of ihis type 6 Cauchy and Abel were the first to make the concept of convergence clear, and to renounce the use of any non-convergent series; Cauchy in his Analyse algebrique (1821), and Abel in his paper on the binomial series (1826), which is expressly based on Cauchy s treatise. At first both hesitated to take this decisive step 7 , but finally resolved to do so, as it seemed unavoidable if their reasoning were to be made strict and free from gaps. .
We
are now in a position to survey the problem from above, as it were; and the matter at once becomes clear when we remember that in whatever form it the symbol for an infinite sequence of numbers has, and can have, given, sequence, series, product or otherwise no meaning whatever in itself) but that a meaning was only assigned This convention consisted to it by us, by an arbitrary convention. is
in allowing only convergent sequences, i. e. sequences whose terms approached a definite and unique number in an absolutely de-
firstly
consisted in associating this number with the or in regarding the sequence as no its value, more than another symbol (cf 41, 1) for the number. However obvious and natural this definition may be, and however closely it may finite
sense; secondly, infinite sequence, as
it
be connected with the way in which sequences cessive approximations to a result which cannot a definition of this kind must ne\ ertheless in all sidered as an arbitrary one, and it might even
occur (e. g. as sucbe obtained directly), circumstances be conbe replaced by quite different definitions. Suitability and success are the only factors which can determine whether one or the other definition is to be preferred; the nature of the thing itself, that 8 sequence , there is nothing
is to say, in the symbol of (s w ) which necessitates any preference. We are therefore quite justified in asking whether the complication which our theory exhibits (in parts at least) may not be due
in
an
infinite
6
Cf.
on the other hand
7
So
far as
p. 133, footnote 6.
concerned, cf. the preface to his Analyse algibrique, Cauchy other things, he says: "Je me suis vu force d'admettre plusieurs propositions qui paraitront peut-ctre un peu dures, par exemple qu'une serie divergente n'a pas de somme". As regards Abel, cf. his letter to Holmboe (16. I. 1826) f in which he says: "Les series divergentes sont, en general, quelque chose de bien As already fatal, et c'est une honte qu'on ose y fonder aucune demonstration". mentioned (p. 458, footnote 3), J. d'Alembert had expressed himself in a similar sense as early as 1768. in
which,
is
among
8 (s n ) may be assumed to be any given sequence of numbers, in particular, therefore, the partial sums of an infinite series 27 a n or the partial products of an
infinite
product.
infinite series are
We by
s, with its reminder of the word "sum", because most important means of defining sequences.
use the letter far the
460
Chapter XIIT.
Divergent
series.
to our interpretation of the symbol (s n ), as the limit of the sequence, however obvious assumed convergent, being an unfavourable one, and ready- to-hand it may appear. Other conventions might be drawn up in all sorts of ways, among which more suitable ones might perhaps be found. From this point of view, the general problem which
presents
itself is
some way,
either
A
as follows:
particular sequence (s w ) is defined in the terms, or by a series or
by direct indication of Is
product, or otherwise. in a reasonable way?
it
possible to associate a "value" s with
it,
"In a reasonable way" might perhaps be taken to mean that the s is obtained by a process closely connected with the previous s. concept of convergence, that is to say, with the formation of lims w This has been found so extraordinarily efficacious in all the preceding that we will not depart from it to any considerable extent without
number
reasons.
good
"In a reasonable way" might also, on the other hand, be interpreted as meaning that the sequence (s n ) is to have such a value s associated with
that
it
wherever this sequence
occur as the final
may
result of a calculation, this final result shall always, or at least usually, be put equal to s.
Let us
The
first
2:(-l)"
5862. i.
e.
these general statements by an
illustrate
example.
series
the geometric series
= l-l+l-l+-..., or the sequence 2xn for x = 1,
0, 1, 0, 1, 0, ...,
(sjel,
has so far been rejected as divergent, because its terms s n do not approach a single definite number. On the contrary, they oscillate
unceasingly between 1 and 0.
This very
fact,
however, suggests the
means
idea of forming the arithmetic 9
Sc n
Since
S||
=~ + [1
n (
l) ],
we i
~"~
2 (n
so that s n
f
(in
find that
+ C- p.]
+ 1)
2
'
4 (n
+ 1)
the former sense) approaches the value
'
:
-^
By this very obvious process of taking the arithmetic mean, we have accordingly managed, in a perfectly accurate way, to give a meaning
to Euler's
paradoxical equation
11 +
1
--[-... =
--.,
to
General remarks on divergent sequences.
59.
associate with the series on the
hand
left
side
461
number
the
^
as
its
LI
number from
or to obtain this
"value",
always equate the
of a calculation
result
final
whenever
to
l)
(-
,
expansion
so; in the case of
_ j\n /
-~
--^i ~~
x
1
for
= 2xn x = 0,
=
x
for
is
it
described above
We
c s n
__
'
tend to a limit will
>
The
+ --
s
be said
and a great association of
.
+
... ----*~
*i -f
make
this
strated in connection with
new the
Two
reasonable.
2(
new
this
.
9
'
"
series
,
sum
which seems
-=-, j
remarks
further
de-
has already been demonn which now becomes l)
definition
series
convergent "in the new sense", with the thoroughly vantages of
,
__ ~~
,
the following
in the previous sense, the sequence (S M ), or to "converge" to the "limit", or "sum", s.
suitability of
may
Lt
"reasonable"
SQ
as
9
might therefore, as an experiment, If, and only if, the numbers
finition.
2 an
is
certainly
obtained in the manner
with the value
1, ...
is
equally true,
fairly bimple means (cf. Exercise 200); more evidence can be adduced to show that the
1,0,
it
1,
be shown by
the sequence 1, 0,
ap-
cannot of course be determined off-hand.
In the case of the
deal
it
Ct
n
foim
pears in the
Whether we can
the series.
will
the
illustrate
ad-
definition:
in the former sense and of (s n ), convergent so constituted, in virtue of Cauchy's theorem 43, 2, that it would also have to be called convergent "in the new sense", with the same limit s. The new definition would therefore enable us to accomplish at least all that we could do with the former, while the
Every sequence
1.
limit s,
is
example of the
series
n l)
-Z*(
shows
new
that the
definition is
far-reaching than the old one. 2.
two
If
series,
convergent in the
2bn = B, are multiplied together by 2cn E 2(a Q bn + !&-.! H-----t-0n
old
Cauchys
sense,
=A
and
rule, giving the series
we k now
fy))>
-2*0 n
more
tli at
mis
series
is
not necessarily convergent (in the old sense). And the question when c n does converge presents very considerable difficulties and has not been satisfactorily cleared up so far. The second proof of theorem 189, 9
From
--
the series (see above) for -
duce the value
~2
somewhat more therefore l
+O
for
o
x
=
1
carefully, l
+ l+O
We
.
is
have only
1 4- 0-a;
l+H-----
also
^
we can
accordingly de-
to observe that the series, written
x-
-\-
for
a;
X9 +
=
J
f
0#
l
a;
5
-f--l
--
,
and
is
Chapter XIII.
4:62
Divergent
series,
however, shows that in every case
C
2c
partial sum of n . The meaning of this is n denotes the n that <Scn always converges in the new sense, with the sum AB. Here situation which, the advantage of the new convention is obvious:
if
th
A
the insuperable difficulties involved, it was impossible to owing clear up as long as we kept to the old concept of convergence, may be dealt with exhaustively in a very simple way, by introducing a to
slightly
We
more general concept of convergence. shall very soon become acquainted with other
investigations
however, we shall (see make some definitions relating to several fundamental matters: Besides the formation of the arithmetic mean, we shall become acquainted with quite a number of other processes, which may with success be substituted for the former concept of convergence, for the purpose of associating a number s with a sequence of numbers (s n ). These processes have to be distinguished from one another by suitable of this kind
61 in particular);
first
of
all,
In so doing it is advisable to proceed as follows: The former concept of convergence was so natural, and has stood the test so well, that it ought to have a special name reserved for it. Accordingly, the expression: "convergence of an infinite sequence (scries, If product, .)" shall continue to mean exactly what it did before.
designations.
.
.
by means of new rules, as, for instance, by the formation of the arithmetic mean described above, a number s is associated with a sequence (s n ), we shall say that the sequence (s n ) is limitable* by that process, and that the corresponding series 2a^ is summable by the process, and
we shall call s the value sum also). When, however, as
of either
will
case of the series,
(or in the
we
occur directly,
are
making use
its
of
several processes of this kind, we distinguish these by attached initials 10 . shall say A, B, . ., F, ., and speak for instance of a F- process .
.
that the
We
.
sequence
(s n )
is
limitable
F, and that the series 2a n is summ be referred to as the Flimit of the
F; and the number 5 will sequence or Fsum of the series; symbolically able
F-lim s n ==
When
there
is
no
s,
V-2a n
fear of misunderstanding,
=s
we may
* German: hmitierbav. In the case of the concept of integrability similar and it was prob.ibly in this connection that was first introduced. 'Ihus we say a function is according as we are referring* to integrability in 10
sense.
.
also express the
the situation is somewhat the above type of notation integrable /? or integrable L Riemann's or in Lebesgtte's
General remarks on divergent sequences,
50.
463
former of the two statements by the symbolism
which more precisely implies that the new sequence deduced from the F-process converges to
(sn )
by
s.
When, as will usually be the case in what follows, the process admits of a &-fold iteration, or can be graded into different orders, we attach a suffix and speak of a V^-limitation process, a V^-summation process, etc. In the construction and choice of such processes we shall of course not proceed quite arbitrarily, but we shall rather let ourselves be guided by questions of suitability. We must give the first place to the fundamental stipulation to be
made
in this connection,
We
must not contradict the old one. proccss which
may be
namely that the new
definition
accordingly stipulate that any
F-
introduced must satisfy the following permanence
condition : I. Every sequence (s n ) convergent in the former sense, with the must be limitable V with the value s. Or in other words, lim sn
limit
=s
in every case imply
u F-lim s n
s,
must
s.
In order that the introduction of a process of this kind may not be superfluous, we further stipulate that the following extension condition
At
II.
must
to hold:
is
least
one sequence new
be limitable by the
Let us
call
the
particular process the that
implies
wider range
totality
(sj, which diverges in the former sense, process.
of sequences which are of this process.
range of action
limitable
The
by a
condition
II
only those processes will be allowed which possess a of action than the ordinary process of convergence. It
precisely the limitation of formerly divergent sequences and the of formerly divergent series which will naturally claim the greater part of our attention now.
is
summation
employed together, say a V- process and a HP- process simultaneously, we should be in danger of hopeless Finally,
confusion
if
if
several processes are
we
did not also stipulate that the following compatibility
condition should be
fulfilled:
one and the same sequence (s n) is limitable by two different simultaneously processes, applied, then it must have the same value by both processes. In other words, we must in every case have III.
//
= W'\ims n
11
We
,
if
both these values exist.
if some convergent sequences at least are by the process considered. This is the case e. g with the E - process discussed further on, provided the sutfix p is complex f
might also be
satisfied
limitable with unaltered value
263.
464
Chapter X11I. Divergent series.
We
shall only consider processes which satisfy these three con Besides these, however, we require some indication whether the association of a value 5 with the sequence (sj effected by a parti-
ditions.
is a reasonable one in the sense explained above Here widely -varying conditions may be laid down, and the
cular V- process
460). processes which are in current use are of very varied degrees of efficiency in this respect. In the first instance we should no doubt require
(p.
that the
elementary rules of the algebra of convergent sequences (v. 8) should as far as possible be maintained, i. e. the rules for term-by-term
and subtraction of two sequences, term-by-term addition of a and term-by-term multiplication by a constant, and the effect of a finite number of alterations (27, 4), etc. Next we might perhaps require that if, say, a divergent series 2a n has associated with it the value s, and if this series is deduced, e. g. from a power series 2cn xn by substituting a special value X L for x, then the number 5 f(x) should bear an appropriate relation to f(x^) or to Mmf(x) for x^x^\ and similarly for other types of series (Dirichlet series, Fourier series etc.). In short, we should require that wherever this series appears as the final addition
constant,
=
result
a
of
calculation,
the
The
should be s.
result
the
greater
number similar to the above which are satisfied by a us call them the conditions F, without taking let 264. particular process to formulate them with absolute precision and at the same pains the the of the of action time, greater range process, the greater will be its usefu