This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
=/(&)
Mi its anticipation of much in functional operators of the closed cycle class was not generally recognized until its republication in Acta Mathcmatica in 1926.* These studies culminated in a treatise on distributive oper
This memoir was published in 1886, but
of the
work
ations published in 1901 in collaboration with U. Amaldi. Perhaps the first to give a general impetus to the investigation of differential equations of infinite order was C. Bourlet (18661913), who in 1897 contributed fundamental results in a paper entitled: Sur les operations en general et Us equations differentielles lineadres
d'ordre infini.
This paper was amplified in certain important details
in 1899.f
In 1917 (1
J. F. Ritt
_ z/ot)
(1
considered the infinite product,
z/a2 ) (1
under the assumption that
z/a,)
2


(1
z/an )
,
l/\a n converges, and derived important \
n=i
properties of its inversion. Valiron and G. Polya.J
This approach has been extended by G.
The problem presented by the differential equation with constant coefficients was studied in 1918 by F. Schiirer under general conditions. I. M. Sheffer in 1929 devoted two memoirs to the subject,
imposing the condition that
*See Bibliography: Pincherle (1). tSee Bibliography: Bourlet (1) and (2). JSee Bibliography: Ritt, Valiron and Polya.
THE THEORY OF LINEAR OPERATORS
24
lim n
\f
(n)
(x) Y
/n t==
constant. *
j
This limitation was removed by the author, f who extended the domain of admissible solutions to functions summable in the sense of Borel. The relationship between operators regarded as Laurent expansions, valid in different annul i, of the same analytic function was also interpreted. An extended domain of validity was given to these operators by N. Wiener through the use of the Fourier transform. The work of Wiener is especially notable because of the highly rigorous treatment of a subject, which for all of its formal power has been regarded as
having suspicious origins.t
Next to the case of constant coefficients, the differential equation of infinite order of Laplace type has received special attention. In this equation the coefficients are polynomials of bounded degree, " a x p where p is a positive inM .T tl2 x " a,,,, fOn(x) ,
+
+
(.t
*n
tt
,
teger and not all the quantities a,,,, are zero. It is obvious that the 0. case of constant coefficients is included by setting p The study of the Laplace equation, apart from the calculus of differences in which it may be regarded as having had its origin, is
due to T. Lalesco in 1908. Incidental to a consideration of the problem of the inversion of Volterra integrals, he applied to the homogeneous case [in which f(x) 0] of the Laplace equation the transformation
^
u(x)
Lf
e
rt
v(t)dt
in which the path is a conveniently chosen one depending upon the coefficients of the equation. He made special application to the exx u (x) t= 0, panded form of the difference equation, u(x\\)
which defines the
gamma
function.
E. Hilb in 1921 was the first to solve the nonhomogeneous case of the Laplace equation by means of an unlimited differentiation of (6.1). The system thus obtained was found to come under the general Hilbert theory of linear equations in an infinite number of unknowns as applied to Laurent forms (see section 2, chapter 3) and a solution of the system was thus explicitly obtained. Hilb also developed in more detail the method which Lalesco had used in the ho
mogeneous
case.
*See Bibliography: Schiirer and Sheifer (2) and (4). fSee Bibliography: Davis (4). JSee Bibliography: Wiener (2). See Bibliography: Lalesco (1). \\SeeBibliography: Hilb (1).
LINEAR OPERATORS
25
At the same time
0. Perron" considered the general Laplace a limitation upon the number of solutions by equation and derived means of an application of known results concerning the solution of
the linear system f
2
+ b mn
(a n
)
xm.n
t

cm
m
,
t_
0, 1, 2,


.
two papers published in 1929f considered the 1. In the first of Q and p p %A (z)}  u(x) these, writing the equation in the form {^4 (<:') z =3 /(#) z a, and (b) d/dx, he discussed the cases (a) A^ (c) ^(2)^ (2 n) (z b). He further showed that if A, (;:) has r zeros I.
M. Sheffer
in
details of solution for the cases
+
l
,
of multiplicities p T p a p,, then the equation of infinite order can p2 be replaced by an equation of finite degree m, where m =7), 

,
,
+
+
"'
+ Pr.
In the second paper Sheffer employed methods similar to those used by S. Pincherle, reducing equation (G.I) by means of a l^aplace transformation to a contour integral equation and expressing the resolvent kernel by means of a second contour integral. He further proved that if /(#) is expansible in a scries of Appell polynomials, then a solution u(x) can be expressed simply in terms of the coefficients of the
expansion.
Since Appell polynomials have been important in the theory of differential account of them. By an Apj)ell equations of infinite order we shall give a brief that such a mean we A polynomial (x), polynomial, n
dA n (x)/dx = nA a _
l
(x)
.
The most general set of such polynomials has the form
where n C r
is
the rth binomial coefficient and the coefficients p r are arbitrary.
Moreover,
if
we have a(h)
~Vp
r
/?Vr!
,
r~o
then
it is
easily proved that
a(h)
r^=nV A
?J
(x) h"/n\
.
NO
The function a(h)
is
called the generatrix of the polynomials. first defined by P. Appell in 1880 and
have been the subject of a number of investigations. The following bibliography has been furnished the author by R. D. Carmichael: Such polynomials were
A. Angelesco: Sur une classe de polyn&mes et une extension des series de Taylor et de Laurent. Comptes Rendus, vol. 176 (1923), pp. 275278. Sur des ceux de M. Appell. Ibid., vol. 180 (1925), p. 489. polynomes qui se rattachent
*See Bibliography: Perron.
^^Bibliography.
Sheffer (1) and (3).
THE THEORY OF LINEAR OPERATORS
26 P. Appell:
Sur une classe de polynomes. Ann.
Sci.
Norm.
Sup., vol. 9 (2nd
ser.), (1880), pp. 119144.
On
Generalizations of the Bernoullian Functions and Numbers. vol. 47 (1925), pp. 277288. Invariant Sequences. Proc. Nat. Acad. Sci., vol. 14 (1928), pp. 901904. Certain Invariant Sequences of Polynomials. Trans. American Matk. Soc., vol. 31 (1929), pp. 405421. Exponential Polynomials. Annals of Math., vol. 35 (1934), pp. 258277. S. Bochner: See Bibliography: Bpchner (1). G. H. Halphen: Sur certaines series pour le development des fonctions d'une variable. Comptes Rendus, vol. 93 (1881), pp. 781783. Sur quelques series pour le developpement des fonctions a une seule variable. Bull, des Sc. Math., vol. 5 (2nd ser.) (1881), pp. 462488. Sur une serie d'Abel. Bull, de la Soc. Math. vol. 10 (18811882), pp. 6787. P. Humbert: Sur une classe de polynomes. Comptes Rendas, vol. 178 (1924), pp. 366367. R. Lagrange: Sur un algorithme des suites. Comptes Rendus, vol. 184 (1927), pp. 14051407. Sur certaines suites de polynomes. Ibid., vol. 185 (1927), pp. 175178; 444446. Memoire sur les suites de polynomes. Acta Mathematica, vol. 51 (1928), pp. 201309. H. Leaute: Developpement d'une fonction a un seule variable. Journal de Math., vol. 7 (3rd ser.) (1881), pp. 185200. W. T. Martin: On Expansions in Terms of a Certain General Class of Functions. American Journal of Math., vol. 58 (1936), pp. 407420. L. M. MilneThomson: Two Classes of Generalized Polynomials. Proc. London Math. Soc., vol. 35 (2nd ser.) (1933), pp. 514522. N. Nielsen. Traite elementaire des nombres de Bernoulli. Paris, (1923), 398 p. N. E. Norlund: Memoire sur les polynomes de Bernoulli. Acta Mathematica, vol. 43 (1922), pp. 121196. S. Pincherle: Alcune osservazioni sui polinomi del prof. Appell. Atti dei Sulle serie procedenti sccondo le Lincei, vol. 2 (4th ser.) (1886), pp. 214217. derivate successive di una funzione. Rendiconti di Palermo, vol. 11 (1897), pp. 165175. See also Bibliography: Pincherle (7), (10), (15). I. M. Sheffcr: See Bibliography: ShefFer (3), (5). J. Touchard: Sur le calcul symbolique et sur ^operation d'Appell. Rendiconti di Palermo, vol. 51 (1927), pp. 321368. M. Ward: A Certain Class of Polynomials. Annals of Math., vol. 31 (2nd ser.) (1930), pp. 4351. E. T. Bell:
American Journal of Math.,
The general equation of Laplace type was considered by the author* in 1931 from the operational point of view and the resolvent generatrix determined for it. The domain of solutions was extended to include formal expansions summable by the method of Borel. Comparably little attention has been given to integral equations of infinite order, due most probably to the fact that the methods of Volterra have proved both powerful and satisfactory. T. Lalesco in 19 10f considered the questions invoked when a differential equation JP
MaOte^^O
pO is
replaced by an integral equation,
*See Bibliography: Davis (5). fSee Bibliography: Lalesco (2).
,
LINEAR OPERATORS
=
27
u (n) b ,(x) = & n v(x} and y His main result showed that if are bounded with b n within a common domain the coefficients p (x)
where
}
.
oo
72,
and
a set of constants
if
Cm
exist such that
2 C B m converges 1
tll
m^i
where the B m dominate the coefficients in R, then a unique solution of the integral equation exists for which the integrals of different orders take the preassigned values C m The author in 1930* reduced the Volterra integral equation .
f Ja
an integral equation of infinite order and obtained its inversion by the method of generatrix functions and the operational product of C. Bourlet. Special application was made to the case of the closed
to
cycle,
i.
e.,
for kernels of the form
K(x
f).
Closely related to the problem of differential and integral equations of infinite order is the problem of infinite systems of differential equations, that is to say, systems of the form
du
; 1 llX
2X;(aO =
j
,(*)
=/,(),
(it=l,2,,
oo )
.
i
where the functions a i} (x) and /, (x) are given. Such systems were first studied by H. von Koch in 1899, who obtained a general existence theorem by means of majorant functions and applied his theory to the solution of certain types of partial differential equations. Although T. Lalesco discussed the problem of such systems in connection with his solution of Volterra integral equations in 1908, the next memoir on the subject was published by F. R.
Moulton in 1915. Incidental to his development of his theory of general analysis, E. H. Moore in 1906 had included the abstract theory of infinite differential systems, but he made no attempt to study such systems independently. Under the stimulus of the work of Moulton and Moore, W. L. Hart from 1917 to 1922, T. H. Hildebrandt in 1917, and I. A. Barnett in 1922 published a series of papers extending the general theory in several ways. W. T. Reid in 1930 and D. C. Lewis in 1933 again attacked the problem and applications were made to the theory of partial differential equations. Other contributions were made by A. Wintner in a series of papers published between 1925 and 1931, by L. Lichtenstein in 1927, by W. Stcrnberg in 1920, and by M. R. Siddiqi in 1932.f The demands of the Heaviside calculus naturally focused attention upon systems of equations with constant coefficients and the lit*The Theory of the Volterra Integral Equation, of Second Kind. University Studies (1930), 76 p.; in particular, pp. 2735. fFor the specific titles of these contributions see the Bibliography.
Indiana
THE THEORY OF LINEAR OPERATORS
28
erature of this subject contains numerous discussions of this subject. I. M. Sheffer in two papers published in 1929 stated existence theorems for infinite systems of differential equations with constant coefficients.
7. The Generatrix Calculus. In his article on "Probability" in the 13th edition of the Encyclopedia Britannica, F. Y. Edgeworth (18451926) makes the following remark: "It has been said that there is no book equal to Laplace's 'Theorie des Probabililes' for a comprehensive and masterly treatment of f
probability, but this ne plus ultra of mathematical skill and power* as it is called is not easy reading. Much of its difficulty is connected with the use of a mathematical method which is now almost super" seded, namely 'Generatrix Functions/ With the first sentence no one who has looked into this great treatise would quarrel; the second sentence, however, is open to
doubt since the generatrix calculus is essentially the calculus of the Laplace transformation. Notations for this transformation may change, but the method remains today one of the most effective tools
numerous theoretical and applied problems. We for may aver, example, that the method of saddle points in the theof ory asymptotic series on the one hand and the Heaviside calculus on the other are at heart applications of the generatrix calculus. This calculus as developed by Laplace depends essentially upon two operators defined as follows:* If f(x) is a function represented by the series
in the solution of
f(x)
on
=
a(n)x rt 
where

#(?*) is a function of n, then
the operators
G and D
Obviously write
n ,
OO
we
shall
have as a definition of
the relations
Ga(n)f(x)
and
we can
summation by integration and thus
replace
f(x)
= j
Df(x)=a(n)
a(n)x n dn
,
.
(7.1)
from which we seek the inversion Df(x)
= a(n)
.
*An account of this subject with examples showing its application to the problem of interpolation, the expansion of functions in series, the asymptotic development of the probability integral and the solution of difference equations will be found in a paper by Dr. Irene Price: Laplace's Calculus of Generatrix Functions, American Mathematical Monthly, vol. 35 (1928), pp. 228235.
LINEAR OPERATORS
29
Both the discrete and continuous problem can obviously be united by means of the StieltjesLebesgue integral
into a single one
=
f(x)
f x n da(n)
.
<*>=;.
Equation (7.1) was probably the first example of an integral equation and has been made the basis of many investigations. Replacing x by
f .'
3
We shall make a brief resume of some of the most important contributions to this subject. N. H. Abel (18021829) employed the equation in the more general
form* f(x,y,z,
)
=
C'V+W+SP
+
.
.

y (u,v,p,


)
tin
dv dp >
.
He
developed a number of fundamental properties of the transformation, but discovered no general method for finding its inversion. M. Lerch (18601922) in 1892 discussed the homogeneous case, to oo.f g(t) c=r 0, where the path of integration is the real axis from He showed that if g(t) is zero for an infinity of values such that x b km (m ), then in general g(t) will be iden0, 1, 2, tically zero, except over a set of measure zero. The possibility of
+
=
expressing sin kt, cos ki, l/P(kt), etc., as an integral of Laplace type is thus excluded. A. L. Cauchy (17891857) was familiar with the inversion l
fj(t)dt
^0 a, is chosen sufficiently large, and this formula was effecemployed by G. F. B. Riemann (18261866) in his celebrated memoir: fiber die Anzahl der Prltn::ahlen wntcr cuter gegebeneii
provided
tively
Grossc, published in 1859.$
*Sur
les fonctions generatrices et leur
dotevminantes.
Oeuvres,
vol.
2,
pp.
6781.
fSur un point de la Tbeorie des Fonctions Generatrices d'Abel. Ada Mathcmatica, vol. 27 (1903), pp. 339351. First published, Rozpravi/ ecske Akadamie, 2nd class, vol. 1, no. 33 (1892), and vol. 2, no. 9 (1893). tMonatsberichte der Berliner Akademie (1859). Werke, 2nd cd., Leipzig (1892), pp. 145153.
THE THEORY OF LINEAR OPERATORS
30
This problem was further studied by H. Poincare (18541912) in a paper published in 1912 employed it in discussing the, at that time, novel theory of quanta.* H. Hamburger in 1920f discussed the inversion for certain types of discontinuities in connection with the Riemann problem of prime numbers and this was again extended by found the inversion of J. D. Tamarkin in 1926.$ V. Romanowsky use in connection with sampling problems in statistics and contributed the inversion
who
f(z)
=A'^We^7r[y2(s
i)]
,
under the restriction
An extensive discussion of the inversion of the Laplace integral has been given by D. V. Widderil in a long memoir published in 1934. In this he considered the StieltjesLebesgue integral /(,*)=
f Jo
Obtaining the inversion of the integral by various limiting processes, Widder considered the type of function thus represented. The zeros of both f(x) and its inverse were investigated and the results extended to the complex plane. The paper concluded with the establishment of general conditions for the solution of the moment problem discussed below.
The problem which
was
satisfies
first
of the selfreciprocal function, the equation
i.
e.,
the function
discussed by H. Weyl** and later became the subject of in
*Sur la theorie des quanta. Journal de Physique, ser. 5, vol. 2 (1912), pp. 534; in particular, pp. 2324. ttiber eine Riemannsche Formel aus der Theorie der Dirichletschen Reihen. Math. Zeitschrijl. vol. 6 (1920), pp. 110. JOn Laplace's Integral Equations. Trans. Amer. Math. Soc., vol. 28 (1926), pp. 417425. 0n the moments of standard deviation and of correlation coefficients in samples from normal. Metron, vol. 5 (1925), No. 4, pp. 354; in particular, p. 9. The Inversion of the Laplace Integral and the Related Moment Problem. Trans. American Math. Soc., vol. 36 (1934), pp. 107200; see also: Generalization of Dirichlet's series and of Laplace's integrals by means of a Stielties integral. Trans. African Math. Soc., vol 31 (1929), pp. 694743. 
A
**Singulare Integralgleichungen. Dissertation, Gottingen, (1908), 86
p.
LINEAR OPERATORS
31
vestigations by T. Carleman,* J. Hyslop,t G. H. Hardy, and E. C. Titchmarsh.J The solution appears in the form
= Aar + fl
/(.r)
where A,
/>, A,
and
a,
Ba;"
1 ,
are subject to the restrictions
A r(l
a)^B
2
2
r(a)
,
sin
a
.T
=
A
2
n
.
The use of the Laplace transformation as a means of solving certain types of difference and differential equations is probably too well known to require more than a passing reference here. This subject will be extensively developed in chapter 10, where a brief resume will be found. P. Humbert in 1914 considered the problem of inverting the Laplace transformation by means of known solutions of the associated differential equation. His suggestive paper generalized the problem by considering the inversion of inhistorical
tegral equations of the
form
f(x)
=
C Jo
Closely associated with the differential equation problem, we find the theory of asymptotic expansions from the standpoint of E. Borel. This theory culminates in what is referred to as the method of saddle points, or the method of steepest descent. Some demands upon knowledge of this powerful tool of analysis will be made in the en
suing pages and the reader is referred for a description and bibliography to the author's Tables of the Higher Mathematical Functions, vol. 1 (1933), part 2, pp. 4164. Fundamental references include the work of Borel, P. Debye,** E. W. Barnesft and G. 
*Sur les equations integrates singulieres a noyau reel et symetrique. Uppsala Univcrsitcts Ar*skrift (1923), 228 p. fThe integral expansion of arbitrary functions connected with integral equations. Proc. Cambridge Phil. Soc., vol. 22 (1925) pp. 169185. t Solutions of Some Integral Equations Considered by Bateman, Kapteyn, Littlewood ,and Milne, f'roc. London Math. Soc., 2nd ser., vol. 23, pp. 126; also: Solution of an Integral Equation. Journal of the London Math. Soc., vol. 4 (1929), pp. 300804. 0n Some Results Concerning Integral Equations. Proc. of the Edinburgh Math. Soc., vol. 32 (1914), pp. 1929. \\Legons sur les Series Diver gentes. Paris (1901) ; in particular, chap. 4. **Naherungsformeln fur die Zylinderfunktionen fur grosse Werte des Arguments und unbeschriinkt veranderliche Werte des Index. Mathematische Annalen, vol. 67 (1909), pp. 535558. ft A Memoir on Integral Functions. Phil Trans, of the Royal Soc., vol. 199 (A), (1902), pp. 411500. The Asymptotic Expansion of Functions Denned by Taylor's Series. Phil. Trans, of the Royal Soc., vol. 206 (A), (1906), pp. 249297.
THE THEORY OF LINEAR OPERATORS
32
N. Watson.* chapter
A
resume of the method
will be
found in section
4,
5.
The famous moment problem of analysis
is essentially an outthe of calculus. In the generatrix growth StieltjesLebesgue form we seek the inversion of the infinite system of equations
f tda(t) ,71=0,1,2,3,..,
(7.1)
Jo
=
6 where the quantities fi n are given. The transformation t
orem for the homogeneous
case,
O^i '
a
,
V,
9 o O,
i i., Ltj
,
(0 is analytic in the interval, then
if
;
results of great elegance.
t
n
II
Considering the
ip(t)dl=Q
,
tt~^0,
finite
1, 2,
,
problem,
m
1
,
he obtained the solution
where ^(x) is a function which does not vanish at either a or 6. Laurent obtained the differential equation satisfied by tp(x) and developed the theory of the Legendre and Laguerre polynomials. * Theory of Bessel Functions. Cambridge (1922). Chapter 7. Also: An Expansion Related to Stirling's Formula, derived by the Method of Steepest Descents. Quarterly Journal of Math., vol. 48 (1920), pp. 118. fSolution d'un probleme d'analyse. Journal de Math., vol. 2 (1857), pp. 12.
JSulle equazioni integral!i
f JJ a I
0(.c)dx~0
,
Atti del Lined, vol. 30 (1), (1921), pp. 1719. Sulla theoria di chiusura dei sistemi di funzioni ortogonali. Rendiconti di Palermo, vol. 36 (1913), pp. 177202. Sur le calcul inverse des integrates defmies. Journal de Math., vol. 4, series 3 (1878), pp. 225246.
33
LINEAR OPERATORS
The most extensive researches on to T. J. Stieltjes (18561894),
who
the
moment problem
are due
proved the following fundamental
result:*
Consider the
moment problem n
da(t)
If a(t) is a solution, then
in the
w
,
=
form
0,
1,2, 3,
we have formally
/ 00
da
(t)
.
/ (x+t) c=tio/x
PI/X*
}
fa/a*


/
Hence there
exists a corresponding continued fraction
i I ,
_
:
,
\x
proved the theorem: The continued fraction has integrals or only one, according as the converges or diverges. If 2 b n diverges, then the fracvalues of x which do not belong to the negative real axis,
Stieltjes then infinitely
series
many corresponding
2 bn
tion for
all
including zero, converges and equals the corresponding integral. F. Hausdorff in 1923 proved the theorem f A necessary and sufficient condition that equation (7.1) have a bounded nondecreasing solution d(t) is that the sequence of moments shall be completely :
monotonic.
Researches on this subject have also been made by H. M. Riesz, R. Nevalinna,** T. Carle
Hamburger,* E. Stridsberg,
*Recherches sur les fractions continues. Annales de Toulouse, vol. 8 (1894), pp. J, 1122; in particular, p. 71 et seq.; vol. 9 (1895), pp. A, 147. fMomentprobleme fiir ein endliches intervall. Mathematische Zeitschrift, vol. 16 (1923), pp. 220248. tuber eine Erweiterung des Stieltjeschen Momentenproblems. I, II, and III. Mathematische Annalen, vol. 81 (1920), pp. 235319; vol. 82 (1921), pp. 120164, 168187. Nagra aritmetiska undersokningar rorande fakulteter och vissa allmannare koefficientsviter. Notes 2 and 3. Arkiv for Mat., vol. 13, No. 25 (1918), pp. 170; vol. 15,
No. 22 (1921), pp. 126. le probleme des moments.
Notes 1, 2 and 3. Arkiv for Mat., vol. 16, 1pp. 123 and 19 (1922), pp. 121; vol. 17, No. 16 (1923), pp. 52. Also Sur le probleme des moments et le theoreme de Parseval correspondant. Acta Literarum Ac. Scienliarum Regiae Universitatis Hungaricae Francisco Josephinae, vol. 1 (19221923), pp. 209225. **Asymptotische Entwicklungen beschrankter Funktionen und das Stieltjesche Momentenproblem. Ann. Ac. Sclent. Fenn., vol. 18, No. 5 (1922), pp. 153. II
Sur
Nos _j. 12 (1921), :
THE THEORY OF LINEAR OPERATORS
34
man,* D. V. Widder,f I. J. Schoenberg,J T. B. Hildebrandt, E. Hille and J. D. Tamarkinii and others. An extensive account of the problem will be found in Oskar Perron's Die Lehrc von den Kettenbriichen, Leipzig and Berlin (191*3), chapter 9.
We
The Heaviside Operational Calculvs.
8.
have already spoken
in preceding sections of the controversy which has centered around the question of the validity of the methods initiated by Oliver Heavi
side in the theory of electrical communication. Since the problem of the Heaviside calculus, as it pertains to the general theory of opertors, will be systematically treated in chapter 7 we shall content our
selves at this place with a its
meager description and a brief
outline of
history.
The
first
problem attacked by the Heaviside calculus
is
the de
termination of functions which represent the charges, (?/, in an electrical network. Since the charges are connected with the currents by the formula, /, dQ,/dt, the problem is equivalent to finding the currents in the various circuits which comprise the network. If we repd/dt, as is cusresent a derivative with respect to time by p, p 2 L i} p fl/C ; tomary in this theory, and if we write Z n (p) <= R ,p where R, L and C represent resistence, inductance and capacity resolution spectively, then the Heaviside calculus is concerned with the
=
=
t
+
,
of the system,
Z^(p) >Qi
+ ^,
Z
(P)
*Qi
+#
Z nl (p)
*Qi
+Z
21
2
(p)
(P) *
nZ ('p)
Q2
*Q,
ln
(p)
Q = A(<)
2n
(p)
^>Q n = f
nn
(p)
*Q n
+ "
+Z +Z
+
+Z
*Q* H
z
,
(t)
,
fn(t) (8.1)
are known electromotive forces (e. m. f. s.) which are regarded as continuous (except at the origin) and specified functions of time. These equations are to be solved subject to initial con
The functions
ditions,
f
t
(t)
which may be imposed without
loss of generality at
t
t=
0.
*Sur le probleme des Moments. Comptes Rcndus, vol. 174 (1922), pp. 16801082; Sur lex equations integrales siugnlieres a noyau reel et symetrique. Uppsala (1923), 2128 p., in particular pp. 204220. fLoc. cit., p. 191 et seq. JOn Finite and Infinite Completely Monotonic Sequences. Bulletin Amer. Matli. Soc., vol. 38 (1932), pp. 727G. On the Moment Problem for a Finite Interval. Bulletin Amer. Math. Soc., vol. 38 (1932), pp. 2G9270. Proc. Questions of Relative Inclusion in the Domain of Hausdorff Means. Moment Funcof National Academy of Sciences, vol. 19 (1933), pp. 573577; On tions. Ibid., pp. 902908; On the Theory of Laplace Integrals. Ibid., pp. 908912. 
LINEAK OPERATORS
35
The second problem of the Heaviside calculus is concerned with the solution of a class of linear partial differential equations of which typical members are the equation of wave motion, the equation of heat conduction, and the equation of telegraphy. (See section 6, chapter 7).
To those familiar with the history of differential equations it seems curious to regard system (8.1) as presenting a problem that is sufficiently novel and difficult to warrant the invention of a special calculus for its solution. Classical methods of great power were in it long before the time of Heaviside. The system appears in many types of dynamical problems and is found as early as 1788 in the Mecanique Analytique of Lagrange (vol. 1, p. 390). The novelty consisted in two facts. In the first place the functions
existence to attack
known electromotive forces, are discontinuous at I 0. One of the simplest of these, for example, is the unit e. m. f., f(t) 1, t 0, f(t) 0, t < 0. In the second place a problem of this kind is actually solved by a single operational process which yields not only particular integrals of the system, but those particular integrals which satisfy as nearly as possible the discontinuous boundary conditions imposed upon the system. fi(t) in (8.1), representing
=
^
^
;
It should be particularly noted at this time that integrals with the specified discontinuity cannot be constructed from the analytic functions which satisfy the differential system. All that can be ex
=
to as high an order pected is to find integrals which vanish at t as possible. This simple fact is essential in interpreting the Heaviside calculus and differentiates the problem of electrical networks
from other dynamical systems which employ
identical differential
equations but other boundary conditions. The source of information regarding the network problem is to be found first of all in the original papers of Heaviside and particularly in his classical
work on Electromagnetic Theory
(see Bibliog
This work should be consulted not only for its deep insight into the problems of electrical theory, but also for its numerous excursions into philosophy. One may not be expected to agree in all details with the thrusts made by Heaviside at mathematical rigor, but the evidences of a sparkling intellect are to be found on every
raphy)
.
page.
Probably the principal result of Heaviside as it pertains to the circuits defined by (8.1) is what is called the Expansion Theorem. In the second volume (p. 127) of his Electromagnetic Theory, Heaviside describes the efficacy of this theorem in the following words: "It does not require special investigations of the properties of normal functions. It is very direct and uniform of application. It
THE THEORY OF LINEAR OPERATORS
36
avoids, in general, a large amount of unnecessary work. The investigation of the conjugate property, and of the terminal apparatus in detail in order to apply it to the determination of the coefficients, is wholly avoided. It applies to all kinds of series of normal functions,
as well as Fourier series.
And
it
applies generally in electromagnetic
problems, with a finite or infinite number of variables or more generally, to the system of dynamical equations used by Lord Rayleigh in the first volume of his treatise on Sound, which covers the rest of the work, and upon which he bases his discussion of general prop;
erties."
The methods of Heaviside were unappreciated for some years, but by 1916 it began to appear that they were destined to play a leading role in modern electrical research. Among those who have been most influential in investigating and interpreting the Heaviside calculus should be mentioned K. W. Wagner, T. J. Bromwich, J. R. Carson, V. Bush, N. Wiener, H. Jeffreys, G. Giorgi, H. W. March, L. Cohen, E. Berg, T. C. Fry, W. H. Eccles, W. 0. Pennell, J. J. Smith, H. Salinger, H. Pleijel and R. Liljeblad, F. Sbrana, J. B. Pomey, B. Van der Pol and P. Levy.* It would seem fair to say that four methods have been used prominently in discussing the Heaviside calculus. The first of these is a direct use of formal operators, the actual expansion being that of the outer Laurent annulus of the inverse operator. The significance of this statement will be discussed in chapter 6. The second method was initiated by Bromwich and forms the basis of the exposition published by Jeffreys. It is founded upon the use of complex circuits of the form, Q(t)
=
(1/2 n
i)
j
e<*
V(s)ds
.
This method goes back essentially to A. Cauchy and has the advantage not possessed by other methods of furnishing a solution for general boundary conditions as well as the special conditions of the Heaviside problem. The work of March and Fry in particular bear upon this type of approach.
The third method is due essentially to Carson and consists in reducing the Heaviside problem to the inversion of the Laplace integral, f*
GO
A(t)e~ pl dt
.
y o
*The reader is referred made by these men.
tions
to the bibliography for
an account of the contribu
LINEAR OPERATORS
37
This method has the advantage of stating the problem in a form which can make use of all the accumulated knowledge classified under the heading of the generatrix calculus. An extensive evaluation of integrals of the Laplace type has been made by Carson and others and this furnishes a simple tabular solution of problems which would otherwise present great formal difficulties. The fourth method is associated with the Fourier integral and possesses the usual power of this great analytical tool. Among its chief exponents have been Wiener, Giorgi, Sbrana and Bush, although, as one might expect, there is a very close connection between it and the methods which depend upon the Cauchy integral and the Laplace equation. Probably the highest rigor has been attained by the use of this method. It also possesses the advantage of simple application, particularly since the publication in 1931 by G. A. Campbell and R. M. Foster of a table of Fourier integrals.* We turn next to a consideration of the second Heaviside problem which is devoted to the solution of problems pertaining to wave
propagation in cables. As we have previously stated these problems are formulated in terms of partial differential equations and are characterized, as in the first problem, by the discontinuities at the time origin imposed by the instantaneous introduction of electromotive forces.
A
characteristic problem is that of a noninductive cable with R and capacity C per unit length subject to an
distributed resistance
impressed voltage V Q (t) at the point x the differential equations
It will
=
0.
This problem leads to
be seen in chapter 7 that this problem presents two very
interesting questions. The first and simpler of these is the interprehave already discussed tation of the symbol #*, where p d/dt.
^
We
this fractional operator and will give a much more extended account of it in chapter 2. It may be added that the special interpretation
arrived at by Heaviside is entirely justified by the general theory of such operators. The second question is much more profound and concerns the interpretation of certain divergent series which result from the formal application of operational symbols. Although such series have appeared from time to time since the days of Euler and have been the subject of extensive investigations in the present century, it is doubtful whether they are yet admitted without some suspicion * Fourier Integrals for Practical Applications. The Bell Telephone System, Technical Publications (1931), Monograph B584, 177 p.
THE THEORY OF LINEAR OPERATORS
38
comments:* "It is a strange which early in the nineteenth century have been banished once for all from rigorous
into mathematical literature.
F. Cajori
vicissitude that divergent series,
were supposed
to
mathematics, should at
comment further
its close
be invited to return."
We
at this point on the subject of divergent
shall not
and asymp
totic series.
was natural that the novel and empirical methods of Heaviwhich were designed primarily to reach practical results with a
It
side,
minimum
of computational labor, should arouse objections among the mathematicians who were then deeply engrossed in the problems of rigor which were being vigorously pursued by Weierstrass and the German school. The controversy was unfortunate, but probably salutary in the end. At any rate, one reads with great interest the
following appraisal made by E. T. Whittaker in an article which anyone should consult who wishes to interpret the significance of Oliver Heaviside in modern electrical theory :f "Looking back on the controversy after thirty years, we should now place the Operational Calculus with Poincare's discovery of automorphic functions and Ricci's discovery of the Tensor Calculus as the three most important mathematical advances of the last quarter of the nineteenth century. Applications, extensions and justifications of it constitute a considerable part of the mathematical activity of today". 9. The Theory of Fimctionals. Proceeding to the fourth phase of the historical development we can do no better than to quote from Volterra, who above all others has vigorously pursued this important
subjectrj
"Now, if we consider the isoperimetric problem and if we regard a plane area as dependent upon the curve which encloses it, we have a quantity which depends upon the shape of a curve, or as we say today, a function of a line. Since a line can be represented by an ordinary function, the area may be regarded as a quantity which depends upon all values of a function. It is evidently a function of an infinite number of variables. In fact, we may regard it as a limiting case of a function of several variables by supposing that their number increases without limit in the same manner as a curve may be regarded as the limiting case of a polygon the number of whose sides increases to infinity. "But the area is only a special case.
On
all
sides
we
are able to
^History of Mathematics. New York (1919), p. 375. fOliver Heaviside. Bulletin of the Calcutta Mathematical Society, (19281929), pp. 199220. %Legons sur les fonctions de lignes, p. 14.
(See Bibliography).
vol.
20
LINEAR OPERATORS
39
examples of functions of lines. Thus the action exerted a flexible filiform electric current upon a magnetized needle, deby the pends upon shape which we give to the circuit and consequently is a function of a line. find other
"In order to unite in a general concept all the different particular cases, it is sufficient to imagine a quantity which depends in a given arbitrary manner upon the shape of a curve. A general function of a line will bo one which corresponds to a quantity depending
upon all the values of one or more functions and would always be regarded as a function of an infinite number of variables." The general concept thus presented by Vol terra is sufficiently broad, it will be observed, to bring most of the problems of analysis within its domain. A principal consideration is the determination of the order of generality which will lead to the most fruitful spe
As
apprehended, the notion of definite integraencountered in mathematics of a function of example a line since an integral, as contrasted with a derivative, depends upon all the values of a function within an interval. It is not, however, until we encounter the integral, cialization.
is easily
tion is the first
f*
I)
r
Ja
f(x,y,y')dx
is the concern of the calculus of variations, that we are able to grasp the rich possibilities inherent in the idea. It is natural, therefore, in the generalization of the concepts of continuity, differ
which
and the analytic expansion of functions, that the notion of variation in the sense in which it is found in the calculus of variations should be extensively employed in the theory of functions of lines. entiation, integration,
The calculus of functionals in the modern sense began in 1887 with a series of papers by Volterra published in the Rcndiconti de la R. Accademia dei Lincei. The term functions of lines was used to designate these researches, but it was J. Hadamard who first employed the term functional (fonct'ionnelle) .* Since there is some confusion as to the precise use of these two designations it may be illuminating to quote the following from Volterra [See Bibliography: Volterra (1), p. 74] "The name function of a line was initially taken to mean what in general is now called a functional. In this sense the term 'function of a line' has been used by many writers, and in particular by Volterra, who was the first to introduce this concept, in his Paris :
*Sur 351354.
les operations
fonctionnelles.
Comptes Rendus,
vol.
136
(1903), pp.
THE THEORY OF LINEAR OPERATORS
40
(1913) and in many earlier works. At present, however, having adopted the term 'functional' for this general concept as being more convenient because less specific, we shall reserve the name
lectures
'functions of lines' for those particular functions, of a geometrical nature."
more
strictly
The first treatise on the subject of functional was the Paris lectures of Volterra published in 1913 under the title: Lemons sur les f auctions de lignes, although a work by J. Hadamard Lccom sur :
Je
calcul des variations,
which appeared
in Paris in 1910,
had adopted
the point of view of the theory of functional as a foundation for the calculus of variations.* The first comprehensive treatment of the subject, embracing on the one hand concepts of general analysis and on the other the theory of permutable functions, was due to G. C. Evans who published Functional* a,nd their Applications: Selected Topics, including Integral Equations, in 1918. The well known work of P. Levy: Legons d'analyse fonctionnelle, was published in Paris in 1922. The volume of research stimulated by the fruitful concept of fonctionals is now very great and for a more comprehensive survey of the subject than is possible in this brief historical sketch the reader is referred to Volterra's Theory of Functional $ and of Integral and Integrodifferential Equations published in 1930, a translation of lectures delivered at the University of Madrid in 1925 and published in Spanish in 1927. In order to orient ourselves in the material of this field, we shall begin with a definition of a functional. Let us designate by the symbol,
a functional in the interval (a ^ t ^ 6). P. G. L. Dirichlet's well known definition of function in the ordinary sense is then extended as follows: // a la?w is given by means of fined within an interval (a ^ i ^ b)
which ,
to every function x(t) demade to correspond
there can be
one and only one quantity z, then z is said to be a functional of x(t). This general concept has been further generalized by E. H. Moore and M. Frechet who applied it to abstract aggregates, the eleetc. Defiments of which may be any quantities whatever, A, J5, nitions of continuity both for functional and abstract aggregates
depended upon the primary concept of length, which is made to depend upon the following postulates: The length between a pair of (B,A) > elements, A, B, is a number (A,B) such that (1) (A, B) (B, C) 0; (3) (A, B) (5, B) (2) (A, A)
0,A^B;
=
=
*See in particular chapter
7,
book
2, pp.
=
+
281312, of that work.
^
LINEAR OPERATORS
41
(A, C). If a sequence of elements, A,, A.., element A is the limit of the sequence provided
A
lim (A.
^0
n )
,
An
is
given, then an
.
n=oo
at
In terms of these concepts the continuity of a functional, A then follows provided,
an element
lim
FU,,]^ F[A\
,
WOO
whenever lim
An = A
.
Uniform continuity then implies the existence of an arbitrarily small positive value, 6, to match an arbitrary
F\_A>]
<*

,
whenever
U,
A')
<
d
.
a large measure of freedom is left in these defithe by concept of length, which has been specialized in a number of ways. One of the most fruitful definitions has been the integral, It is clear that
nitions
(X,Y)
= P
[X(0
Jo
where X(t) and Y (t) are functions of integrable square in the sense of Lebesgue. Then (X, Y) implies that X(t) 0, exY(t) cept over a set of points of measure zero. This concept reveals the fundamental significance of the definition of Lebesgue measure and necessitates the use of Lebesgue integration throughout arguments
=
~
which depend upon
this definition of length.* Functional employing this concept are called continuous in the mean and limiting processes are referred to as limits in Ike mean. One also speaks of such functionals as being continuous almost everywhere, that is to say, except over sets of points of at most measure zero.
Another
definition extensively
(X, Y)
=^MaxX(0
employed
is
the following:
Y(t)
When a functional, F[x(t)~\ has continuity with this definition it is said to have continuity of order at the element x(t). If, moreover, sur
*This integral was discovered by H. Lebesgue in 1900. See Lebesgue: Leqons Integration. Paris (1904).
I'
THE THEORY OF LINEAR OPERATORS
42
(X,Y)=Max\X(t)
Y(t)\
,
I
X'(t)Y'(t)\
,
and a functional F[x(t)] is continuous with this definition, it is said to have continuity of order 1. In a similar way we may define continuity of any order.
The explicit representation of f unctionals was naturally one of the first problems to be studied in the new calculus and special attention was given to those obeying the postulates of linearity, that is to say, to f unctionals of first degree. They are of the type, K(t) Ja
where K(t) is a given function. The general representation of linear functionals has occupied the attention of numerous writers, one of the first being given by Hadamard in 1903.* Another was given by F. Riesz in 1909 1 as follows:
Let F[x(t)] be a linear functional, with continuity of order zero and limited in the field of functions that possess at most a finite number of
finite discontinuities.
where X(t;s)
is
Then
define the function,
specified as follows:
X(t;s)
= 1 forays = for < t^ b s
,
s
Then the most general functional of the by the following Stieltjes integral:
.
specified kind is defined
In a similar way, more general linear functionals with continp have representations of the form
uity of order
F[x(t)]=
Functionals of
:
f K(t) x(t)
first,
second, third, etc. degree appear in the
form,J *Sur
354.
les operations fonctionelles. Cotnptcs Rendus, vol. 136 (1903) pp. 3&1See also: Lemons siw le calcul dcs variations. Paris (1910). fSur les operations fonctionnelles lincaires. Comptes Rendus, vol. 149 (1909),
pp. 974977. JSee P. Levy: Leqons d' analyse fonct/lonnelle.
Paris (1922), vi
+
442
p.
LINEAR OPERATORS
(*K(8)x(s)ds
43
,
Ja
K
I
I
Ja
** a,
Z
(SI, s 2 )x(Si)x($ 2 )dSi
r c JarK
3 (
J a Ja
ds 2
,
SM
b
("K nVSj,S (
f
J
J
2,
X ^ (5 The term regular functionals
Fm >(0]
G n [x(t)']<=
n
of degree
,
Fo
ds n
W)
d^!
is
applied to the series
(fe 2
.
= a constant,
W=Q
and functional power
series
provided the series
convergent for
The processes
is
by the sum,
of differentiation

<
x(t)
Q
.
and integration have been gen
eralized for functionals, following closely the analogous operations in ordinary calculus. Thus for a continuous functional the variation
will be arbitrarily small
with \d x(t)\
<
e.
.
Now suppose that t varies between the limits s and area under the variation function 6 x(t) will then be
^ The
f J&
dx(t) dt
s
+
^.
The
.
derivative of the functional at the point
t
=s
is
then de
fined to be
lim
= F'[>(0;s]
In terms of this derivative the can be written in the form,
C
first
variation of the functional
b
F'\x(t)\s\ dx(s)ds Ja
.
THE THEORY OF LINEAR OPERATORS
44
If 6 x(s) is replaced
we
by e
limit of 6
F/e computed,
obtain,
f F'\x(t)
;
s\
Ja
and
in general,
lim [d F<*=0
This idea, which is due primarily to Volterra,* has been genby Frechetf who considered functionals of the form F[x(t,a)'], where x(t,a) is differentiate with respect to the second eralized
parameter.
For the regular functional of degree n the derivative may be computed to be,
where we abbreviate, b
,=
(
K
2 (
Ja 2
^
("
Jb
f
etc.
Jb
The problem of constructing a logical theory of the integration was attacked in several ways. R. Gateau was the first to suggest such a theory which he did by generalizing the concept of mean.%. For an ordinary function in n variables the integral of the of functionals
function coincides with its mean provided the integration is taken over the unit cube The reader is referred to the treatise of Levy for a discussion of the difficulties involved and the success achieved
by
this generalization.
P. J. Daniell
extended the general concept of the Stieltjes inte
le funzioni che dipendona da altre funzioni. Rendiaonti del Lincei, (4th series), (1887), pp. 97105, 141146, 153158. fSur la notion de diflerentielle dans le calcul fonctionnel. Comptes Rendus dn Congress des Soc. Sav., (1912). $Sur la notion d'int^grale dans le domaine fonctionnel et sur la th^orie du potential; avec note de Paul Levy, Bulletin de la Soc. de France, vol 47 (1919), pp. 4770. Also: Sur diverses questions de calcul fonctionnel. Ibid., vol. 50 (1922), pp. 137. General Form of Integral. Annals of Mathematics, vol. 19 (2nd series) , (19171918), pp. 279294; Integrals in an Infinite Number of Dimensions. Ibid., vol. 20 (2nd series), (19181919), pp. 281288.
*Sopra
vol. 3
A
LINEAR OPERATORS
45
grals to functional integration and important contributions were made to the theory by N. Wiener.* The concepts of integration in
the generalized sense are also found in the work of fi Borelf and M. FrechetJ as early as 1914. An extensive account of the origin of the generalizations and their subsequent development will be found in the third part (pp. 261439) of Levy's Leqons cV analyse fonctionnelle. Specialization of these general ideas has led in a number of interesting directions. Foremost among these was naturally the theory of integral equations, which we shall treat in more detail in a later
Closely associated with the development of the Volterra integral equation was the theory of functions of composition and the technical treatment important subclass of pcrmutable functions. of these subjects will be found in chapter 4.
chapter.
A
In order to discuss problems in magnetic hysteresis, elasticity, and other forms of hereditary physics, Volterra introduced the principle of the closed cycle, which may be described as follows F > u(x). Now let u(x f T) Consider the operation, g(x) and Then if G(x) F > U(x), the operU(x) g(x T) G(x). ator F is an operator of the closed cycle. The principle is simply il
+
=
=
:
=
=
lustrated in the case of elastic torsion.
If a> represents the angle of torsion and P the torsion couple, the relationship between them is to a first approximation
where
A:
is
from physical considerations. But more complicated than this since eo debut also upon the history of the elastic body
a constant determined
actually the relationship
is
pends not only upon P the torsion of which is being studied. This second approximation expressed in the form of an integral equation
is
(9.1)
where K(t
s) is the coefficient of heredity.
*The Mean of a Functional of Arbitrary Elements. Annals of Mathematics, 21 (2nd series), (1920), pp. 6672; Differential Space. Journal of Math, and Physics, vol. 2 (1923), pp. 131174; The Average of an Analytical Functional and the Brownian Movement. Proceedings of tlw National Academy, vol. 7 (1921), pp. 294298; The Average Value of a Functional. Proceedings of the London Math. Soc., vol. 22 (2nd series), (1922), 454467. vol.
^Introduction geometrique a quelques theories physiques. Paris (1914), vii 137 p. JLes singularite's des espaces a un tres grand nombre de dimensions. Congres de V Association, francaise pour V Advancement des Sciences (Le Havre). (1914), pp. 146 147; Sur 1'integrale d'une fonctionelle etendue a un ensemble abstrait. Bulletin de la Soc. Math de France, vol. 43 (1915), pp. 248265.
+
THE THEORY OF LINEAR OPERATORS
46 If
P(t)
is
periodic of period T,
+
=
P(t T) P(t), then a)(t) and the operator which coman operator of the closed cycle
satisfies the criterion of the closed cycle
prises the right member of (9.1) is for all periodic functions of period T.
As one may easily apprehend, operators of the closed cycle form a restrictive subclass of operators which embody the general concept of hereditary mechanics. This concept has been ably set forth by E.
whom we quote:* "In all this study (of classical mechanics) the laws which express our ideas on motion have been condensed into differential equations, that is to say, relations between variables and their derivatives. We must not forget that we have, in fact, formulated a principle of nonheredity, when we suppose that the future of a system depends at a given moment only on its actual state, or in a more general manner, if we regard the forces as depending also on velocities, that the future depends on the actual state and the infinitely neighboring state which precedes. This is a restrictive hypothesis and one which, in Picard,
least, is contradicted by facts. Examples are numerous where the future of a system seems to depend on former states here we have \hcwedity. In some complex cases one sees that it is necessary, perhaps, to abandon differential equations and consider functional equations in which there appear integrals taken from a distant time to the present, integrals which will be, in fact, this hereditary part. The proponents of classical mechanics, however, are able to pretend that heredity is only apparent and that it amounts merely to this, that we have fixed our attention upon too small a number of variables. But the situation in this case is just as it was in the simpler one, only under conditions that are more complex." The actual representation of hereditary mechanics in analytical
appearance at
:
form led to integrodiffcrcntial equations. The case of torsion discussed above furnishes an elementary example if we consider the dynamical case and study the oscillations of the elastic body. Then mrf 2 P/dt and we obtain co(t) must be replaced by o(t)
10.
The Calculus of Forms
hi Infinitely
Many
Variables.
The
stage of the theory of operators was ushered in by Fredholm's classical papersf on the solution of the integral equation fifth
*La mecanique classique
et ses approximations successive. Scientia (Rimsta (1907), pp. 415; in particular, p. 15. fSur un nouvelle methode pour la resolution du pr obi erne de Dirichlet. o/v. of Kong. Sv. Vetens kaps Akad. Fdhr., vol. 57 (1900), pp. 3946. Sur une classe d'equations fonctionelles. Ada Mathematica, vol. 27 (1903), pp. 365390. di Scienza), vol. 1
L1NEAK OPERATORS f K(x,t)u(t)dt
47
=
Ja
a solution attained heuristically as the limiting form of a set of algebraic equations
where the t are equally spaced points A ^=L t l+ l t, The solution appeared in the form

%
in the interval
r?,
t
(a,b)
and
.
where D(x,t\X), called Frcdholm's first minor, and Z>U), called Fredholm's determinant, are functions of A .
However, the use of algebraic guides to obtain transcendental results was not original with Fredholm, this method having been employed effectively as early as 1836 by J. C. F. Sturm (18031855),*
who
investigated the properties of a differential equation of second order by means of the limiting form taken by the solution of the difference equation
L u l+l t
The
+ M, u
t
f N, u,
!
=0
.
solution of the integral equation
was originally attained by Volterra in 1896 in this manner, f More recently R. D. Carmichael has indicated the scope and power of this heuristic guide by deriving oscillation, comparison, and expansion theorems for various types of functional equations.! The values of
A
determined from the equation
are called principal values (Eiyenwerte) spectrum of the integral equation.
and as a
set
form the
*Sur les equations differentielles lineaires du second ordre. Journal de Mathematiqnes, vol. 1 (1836), pp. 106186; see in particular, p. 186. See also: M. Bocher: Lemons sur les methodcs de Sturm. Paris (1917), 118 p. fSulla inversione degli integral! defmiti. Atti de Torino, vol. 31 (189596), pp. 311323, 400408, 557567, 693708; in particular, p. JJ11 et seq. J Algebraic Guides to Transcendental Problems. Bulletin of the Soc., vol. 28 (1922), pp. 179210.
Amer. Math.
THE THEORY OF LINEAR OPERATORS
48
Under usual
conditions the principal values
form a
discrete
and
this set is called the point spectrum of the equation. Under singular conditions, solutions may exist for the homogeneous 'equaset
and this set is called the continuous spectrum of the equation. A simple example of such a spectrum has been given by E. Picard,* who considered the integral tion for a continuous set of values of i
,
equation ^ A
u(x)
f JS)
e\*"u(t) dt
,
which has the spectrum ;i
The general
= y (l + a 2
2 .
)
solution of this equation
u(x)
=A e + B atf
is
e'
lir .
Associated with the discrete spectrum we have a set of principal functions (Eigenfunktionen) ii^x), u z (x), u^(x),  which satisfy the homogeneous equation ,
,
f K(x,t) u(t) dt
=
Ja
when
A
assumes the corresponding principal values. and symmetric, K(x,t) K(t,x), and the kernel may be expanded in
=
If the kernel K(x,t) is real then the principal values are real
terms of the principal functions, as follows
:
on
XV
\X jT/ )
'
^
7.
Ui\'X) Ui \v) / A
provided the series in the right hand
member
is
i
f
uniformly conver
gent.
Many years before the introduction of the concepts of integral equations into analysis, special systems of functions called orthogonal functions had been employed. These functions have the property b
G(t) u>(t) Uj(t) dt=^0
,
i^j
,
where G(t)
is a weighting function. It is obviously possible to set without affecting the generality of the situation since the G(t) system of orthogonal functions can be written
=1
*Sur une example simple (Tune equation singuliere de Fredholm. Annales de normale supericure, vol. 28, 3rd ser. (1911), pp. 313324.
I'ecole
49
LINEAR OPERATORS
Wellknown systems of such functions include sin mx, cos mx, (the Legendre polynomials), H m (x) (the Hermite polyL rn (x) nomials), (the Laguerre polynomials), etc. (See section 5,
Pm (x)
chapter 12). If an arbitrary function f(x) malized orthogonal functions
expanded
is
in a series of nor
oo f ( /y \ J \A> J
f
7
, '
.
ni
,
(
/y
\
Ji W/i\*tjJ
j
,
where by normalized we mean that b
~
f Ml Ja then the coefficients
/,=
Cf(t)
u,(t) dt
Ja
satisfy
what
A
2
is
known
as BesseVs inequality
+ A + /r + A 2
2
hf
H
*
:
C U(t)Y dt
,
/a
where n
is
any
integer.
If the equality sign instead of the inequality holds for
every function f(x) of integrable square, then the set of orthogonal functions is called closed; otherwise it is said to be incomplete or open. The set of functions 1, sin mx, cos mx, has been showli by A. Liapounoff* and A. Hurwitzf to be closed. In 1907 E. PischerJ arid F. Riesz both demonstrated the follownow known as the FischerRiesz theorem
ing result
:
CO
If a series
Sf?
converges, then there exists a function /(#)
of integrable square for which the
f,
are the coefficients of the ex
pansion CO ~f
* First
( IT
~~ i
'<
^ f '
11
(IT}
published in 1896 in the Proc. of the
Math
Soc. of the University of
Kharkov and discussed by W. Stekloff: Sur un probleme de ia theorie analytique de la chaleur. Comptes Rendus, vol. 126 (1898), pp. 10221025. See also E. T. Whittaker and G. N. Watson: A Course of Modern Analysis. 3rd! ed. Cambridge (1920), pp. 180182. tiiber die Fourierschen Konstanten integrierbarer Funktionen. Math. Annalen, vol. 57 (1903), pp. 425446; in particular, p. 429. JSur la convergence en inoyenne. Comptes Rendus, vol. 144 (1907), pp. 10221024. vol.
Sur les systemes orthogonaux et 1'equation de Fredholm. 144 (1907), pp. 615619, 734736.
Comptes Rendus
,
THE THEORY OF LINEAR OPERATORS
50
where the u,(x) form an
infinite
system of normalized orthogonal
functions.
The
situation which was so spectacularly revealed by the theof integral equations immediately challenged generalization. ory This was forthcoming in 1908 in a series of monographs by D. Hil
bert which were later collected into the treatise: Grundzuge einer allgemeinen Theorie der Lincaren Integralgleichungcn, Leipzig (1912). The pathway was laid open through the theory of systems in infinitely many variables. E. H. Moore (18621932) in his fruitful concept of general analysis* saw a powerful unifying principle which he expressed as follows :
"The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features".
M. Frechet in his thesis: Sur quelques points du calcul fonctionParis (1906), Red. Circ. Mat. Palermo, vol. 22 (1906), pp. 174, and in his more recent volume: Lcs E spaces Abstraits, Paris (1928) has explored the consequences of very general definitions. S. Pincherle in his numerous contributions and in particular in his book written in collaboration with U. Almaldi in 1901 (see Bibliography) gave an unrestricted view of the depth and power of the concepts of linear operations in the abstract. nel,
classical memoirs have born much fruit, which we more recent contributions of N. Wiener, A. Wintner, M. H. Stone, J. von Neumann, H. Weyl, S. Banach, and many others.
These now
find in the
specific reference, see Bibliography). In view of the highly technical nature of these generalizations, it seems best to postpone further discussion to chapter 12. In the last two sections of that chapter a brief account of the history and present status of the theory of forms in infinitely many variables and of the abstract theory of linear operators is given.
(For
^Introduction to a form of General Analysis. Lectures delivered in 1906 in published in 1910; On the Foundations of the Theory of Linear Integral Equations. Bulletin Amer. Math. Soc., vol. 18 (19111912), pp. 334362; Proc. of the Cambridge Inter national Congress (1912), vol. 1, pp. 230255.
New Haven and
CHAPTER
II
PARTICULAR OPERATORS 1. Introduction. In the first chapter we have attempted to set forth a general view of the concept of operator. We have traced the
development from its early origins in algebraic analogy, through the formalism of the last century, to the deep and broad current of modern speculation. We have attempted to show the cen
historical
which the concept occupies in numerous applications. In the present chapter we shall set forth in some detail a description and classification of several types of operators which will concern us in later pages. We shall mention somewhat lightly the definition attained by Fourier transforms and the inclusive generalization of the StieltjesLebesgue integral. Not, let it be said, to minimize the importance of these matters, but because they have been treated elsewhere far more extensively than could be done in the compass of this volume/ Our own approach to the subject will be made by mecins of expansions in terms of the elementary operator z d/dx. In spite of what might be regarded as an unhappy formalism thus introduced, we shall find that a majority of the specific applications are easily attained in this manner. Linear differential and difference equations are naturally included and the theory of both Volterra and Fredholm integral equations, at least so far as their formal aspects are concerned, may be discussed without difficulty in terms of the elementral position
tary operator. Since we propose in later pages to throw an unusual burden upon the operator z, it is not out of place to say a word about its generality. The derivative of a function is generally defined as lim [/(z+fc) ?l
f(x)]/h
.
=
By pushing this limit to its logical conclusion, K. Weierstrass (18151897) was able to construct a function which is continuous over an interval and yet does not possess a derivative at any point of the interval.f The derivative is thus envisaged as a property im kleinen of a function. At a point it may inherit nothing from other portions of the function. It depends entirely upon the oscillatory character of the function within infinitesimal intervals. Let it be *See, for example, references in the Bibliography under N. Wiener, M. H. Stone, and J. von Neumann. fFirst published by P. du BoisReymond Journal fur Mathematik, vol. 79 (1874), p. 29. :
51
THE ^HEORY OF LINEAR OPERATORS
52
added, however, that supplementary hypotheses may entirely alter the situation. If, for example, we consider the mode of approach of h to zero and assume that the limit is the same for every path, then the function is defined throughout the region of its analytic continuation.
An
integral,
restrictive
on the other hand, as the limit of a sum avoids this
im kleinen aspect of the
derivative.
from
It inherits
all
parts of the interval. Its generality in the StieltjesLebesgue form is a strange contrast to the narrow limitations of its inverse. It is a property im grossen of the function which defines it. Thus to base a theory of operators upon z rather than upon 1/z would seem to impose an unfortunate blemish upon it. On the one
hand, the use of z in general permits an easier formal manipulation as one may surmize from the fact that differential equations preceded integral equations by two centuries. On the other hand, its definition seems to limit application to a class of functions far more restricted than the class to which its inverse applies. One may reflect, however, upon the fact that 1/z may be expressed in terms of z by means of the formal equivalence
l/z=
e**)/z
(l
* .
first operator, which may be interpreted as an integral of Lebesgue type, has a very general application. The second operator, since it is nonsingular and expansible only as a power series in z,
The
is
limited to the class of unlimitedly differentiate functions.
How
ever, in the domain of functions common to both operators the results obtained are identical. Is there any way in which the generality
of the first can be wholly or partially restored to the second ? It is seen that this may be accomplished in large measure if the function to which the transformations are to be applied is defined by
a Fourier integral, f% 00 I
JOG 00
e^'f/O*) dfidi
.
(1.1)
/CO
If then an operator of the form F(x, z) be applied to f(x) we can give meaning to the transformation by means of the following formal equivalence:
F(x,z) >/Gr) (l/27i)
(1.2).
f
"VfoU)
Juo
e^'dJi
f
e^'/OO dp
.
Joo
It is clear that in this manner a certain measure of generality has been restored to the differential operator and that the limitation
*See section
6.
53
PARTICULAR OPERATORS
upon the range of functions to which to the convergence of the integrals.
it
applies has been transferred
2. Polynomial Operators. In this and ensuing sections we shall the foundations of our study by considering a few common types lay of operators.
If we designate by the symbol z the differential operator d/dx, we may then define the elementary polynomial operators to be z, z 2
,
z3,
,
zn,
from which the general polynomial operator
through their linear combination
Fp (x,z)^=A
(x)
is
constructed
:
+Ai(x)z + A*(x)z*\
\A p (x)z*
,
where the AI(X) are functions of x which share a common region of definition. It is tial
obvious that this
is
the operator of ordinary linear differen
equations.
3. The Fourier Definition of an Operator. As we have stated in the preceding chapter, the definition formulated in (1.2) goes back to J. Fourier, by whom it was stated in a slightly different form to apply to the case of fractional derivatives. An instructive application of it was made in 1895 by T. LeviCivita in solving the Volterra
integral equation,
f(x)
= f K(x
r
t)
u(t) dt
with special reference to the inversion of the Abel integral in which the kernel
is
a
t) .*
(x
N. Wiener has suggested the following extended definition :f Let us consider the function
F(x,z) *f*(x)
where S
in
which
(A) is defined to
9?
(A)
is
(3.1)
be
a function possessing the following properties:
*Sull'inversione degli integral! definiti nel campo reale. Atti di Torino, vol. 31 (1895), pp. 2551. See also the author's study: A Survey of Methods for the Inversion of Integrate of Volterra Type. Indiana University Studies, Nos. 76, 77 (1927), pp. 5457. fSee Bibliography: Wiener: (2).
THE THEORY OF LINEAR OPERATORS
54 (a)
of
(A) is defined
orders
all
over the interval (0,1) and has derivatives
;
(b) (w)
(c)
An
q?
(0)
=
(l)
=0
for
example of such a function
=
all
is
positive integral values of n.
given by
&'* dt
f J\\
.
Jo
The kernel, S&(i), is thus seen to be a function of the type exhibited in the figure where the transition curve
Al
O
TV
FIGURE
1
Then, if f(x) is summable and of summable square over every range and if there are numbers A and k such that f(x) ^ A x* for every x of sufficiently great magnitude, it can be proved that
finite
1. i.
m. F(x,
z)
>/s(#)
6* uo
where
exists,
limit is to
l.i.m. is
the abbreviation for limit in the mean.
be regarded as the operational equivalent of
A
is
sequence of functions of summable square, f l (oc) said to converge in the mean toward a function f(x) if lim w
A every
( ~^ Ja
for
f 2 (x),
'
This
* f(x). '
'
,
/ n (^)>
[/(*)
necessary and sufficient condition for convergence in the number e there exists a number n such that
positive
mean
is
that for
,
L vol.
r
F (x,z)
2
If ,(*)
dx\
<e
,
values of n' > n. This theorem is due to E. Fischer (Comptes Rendus 144 (1907), pp. 10221024) and F. Riesz (Ibid., pp. 615619; 734736).
all
In order to prove the theorem
where
A
By
let
us
first define
is the measure of the point set over which the integral is taken. hypothesis there exists for every e an integer m(e), such that
t
PARTICULAR OPERATORS e
Now
a
=
We e/j/2
A
>
~b
a
55
> m()
w, n
>
.
assume that
see that the point set over which this inequality holds cannot exceed 3 or if e it cannot exceed a ?/ Now let 77 ti ij ?7 n 2 3
=
,
>
^>
.
,
>
>
be a set of arbitrarily small quantities for which the sum ra2 (e), be a set of integers such that converges, and let 1 (e),
TJ
m
l/w
/nl
<
V
m,n >
mp (e)
p =:
,
fc
k
,
+ 1,
I
+
> +
^2
'
*
.
It will be evident that these inequalities hold simultaneously over a set of + A a k where i) 7/ points of measure A k Evidently the set k fc+1 A k is a subset of A k ^ From this we conclude that the sequence {f m } k (e) , converges uniformly in A k and converges at every point in the limit set A M This limit set is obviously equal to A, with the possible exception of a set a w of measure zero. Hence we obtain a limit function
=
,
fc
=
+
.
.
m> m
,
.
f(x)=limf k (x)
,
k=^
when x
is in
A^, which we
define to be zero over a^
In order to show that f(x) *A t (f m ,fn)
is
.
a function of integrable square, consider
^ ^(C/n) < An
Since the convergence of f n upon and hence derive
e
is
,
,
uniform,
n
> m(0
we can
.
allow n to become
infinite
/
Since, moreover,
Ak
u (/,n ,/)
is
<*
m>m(s)
,
included in the set
A k+lJ we
.
can
let
A:
become
infinite
and thus obtain
'A(/m./>
From
this
we
1>t()
'
derive the inequality
f f"dx^2IAao (f m ,f)
J A<x>
<
+
2
f J Ac
which proves that f(x) is a function of integrable square. From the above argument we see that f (x) is uniquely determined to within a function defined over a set of measure zero, that is to say, a function g(x), not identically zero, which satisfies the equation
Such a function we shall call a null function. Two functions which most by a null function will be said to be equivalent almost everywhere.
A
differ at
definition similar in type to that proposed by Wiener has been the basis of the study of G. Giorgi and F. Sbrana. These writers, however, interchange the role of the operator and the function
made
THE THEORY OF LINEAR OPERATORS
56
and thus
define /*
oo
F(z)f(x)=
/
oo
t)dt=
G(x
f(t) J00
f(xt)G(t)dt, JU3
(3.2)
where we abbreviate 00
I
(
F(w)e wt dw
(l/2ni) Ji
(3.3)
.
oo
Their mode of derivation is as follows It is obvious that we can write the function f(x) in the form :
r*
Ux
f(t)dt
oo
i
f(t)dt
(3.4)
,
Jx
oo
\
provided proper restrictions are imposed upon f(t) /<*) ,= (**), fc> 1.
as,
for example,
Recalling the identity I( t )
= (l/n)
f Joo
we can
(siiUi/A)cn=(

]' 1 
'
I
l>n' <x U t
.
write (3.4) in the form /
/(a;)
00
=1^2*
f(t)I(xt)dt
.
J00
If
we now
operate with F(z), this becomes
f(t)G(xt)dt Xoo 00 where we use the abbreviation G(x
we can
Since
t)
,
~V%zF(z)
> I(x
.
also write
(1/2*0
e)/w}dw
f{V*
,
Jion ioo it is
clear that
G(x)
=y
2
G(x) can be expressed
zF(z) >I(x)
in the
form
= (l/47i i 00 l
ii)
f Ji
provided the proper convergence properties are possessed by F(w).
PARTICULAR OPERATORS
57
A more extended account of the Fourier operator will be
given in
section 5 of chapter 6 in connection with its application to the inversion problem of differential equations of infinite order with constant coefficients.
PROBLEM Given the Fredholm transform /&
K(xt)
T(u)
n(t) dt
,
and a function f(x) such that is
(a) f(x) (6)
(c)
K(x) 'Ti(z)
integrable in (,<*>), /(.v)
is finite, /
=
=
0,
continuous, and intcgrable
dt
k(z)
,
T(M)
if
/(#),
=
u(t)
we have
(0
;
K(t) sinirzt dt
I
,
=
z in
(0,oo).
the inversion '
Real part of
d^
M
;
Jo
do not vanish simultaneously for any value of that
(0,c)
a
un
/
K(t) cosirzt
^
a,
in
OO
Jo
Show
<
x
0,
t
o
f(x)dx,t>a
:
,
*^ o
.
(LeviCivita:
loc.
cit.).
Hint: Write u(t)
dz ^
I
Jon >
then show that
/'
coB'7rz(y
I
t)
u(y) dy
;
Jo
when F(je,2/,z)
Real part of
6 iir ^''V[/i()
rcc K( rooK(x I,
F(x,y,z) dx
'
f)
+
?
.
4. The Operational Symbol of von Neumann and Stone. A still more comprehensive definition has been given by J. von Neumann and M. H. Stone. In the work of the latter the operator is represented by a LebesgueStieltjes integral of the form
J /'
dQ(EJ,g)
,
<.
where / and g are elements
in abstract Hilbert space, E\ is a family of special operators defined for all real values of /, Q(f,g) a numerically valued function, and F(l) a function which represents the operational interpretation of the symbol. If we set
=F(x,ti)
THE THEORY OF LINEAR OPERATORS
58
and r\
ru
e^dr J on
*J

the operator of Stone will reduce to (1.2). The operator of von Neumann and Stone was developed in connection with their generalization of the spectral theory of integral equations and infinite quadratic forms initiated by the researches of D. Hilbert. An account of the spectral theory is given in chapter 12. From the formal results which are developed there the reader can
determine the specializations of the general operator which apply in particular cases. The full generality of the operator in the form given above can only be appreciated by a consultation of the original sources. In view of the importance which the Stieltjes integral has assumed in modern generalizations of the linear operator a brief resume of its special features will be given here. The integral was first defined by T. J. Stieltjes in connection with his classical researches on continued fractions (see section 7, chapter 1). Extensive accounts of the integral will be found in E. W. Hobson's Theory of Functions of a Real Variable, vol. I (3rd ed.), Cambridge (1927), pp. 538546, in H. Lebesgue's Legons sur rintegration, (2nd ed.), Paris (1928), pp. 252313, and in C. J. de la Vallee Poussin's Integrates de Lebesgue, Paris (1916) pp. 127. Formulations of the integral most useful in the theory of linear operators have been
given by M. H. Stone [See Bibliography: Stone (i2), pp. 158165; 198221] and A. Wintner [See Bibliography: Wintner (5), pp. 74105]. The original definition of Stieltjes, valid in the domain of Riemann integration, was extended by J. Radon: Theorie und Anwendungen der absolut additiven Mengenfunktionen, Sitzungsberichte der Akademie der Wissenschaften, Wien, (1913), vol. 122, pp. 12951438, in particular, pp. 13421351. This extension enlarged the domain of functions integrable in the RiemannStieltjes sense to include those integrable in the Lebesgue Stieltjes sense. The Lebesgue Stieltjes integral thus formulated is often referred to as the Radon Stieltjes integral. The literature of the subject is now
very extensive.* By a Stieltjes integral of a continuous function, f(x), with respect to a function of limited variation, v(x), we mean an integral of the form
1= which
is defined
as follows
Consider the
f(x) dv
:
sum
*In addition to the sources already cited the reader will find the following instructive references: E. B. Van Vleck: Haskin's Momental Problem and its Stieltjes' Problem of Moments, Trans. Amer. Math. Sov., vol. 18 (1917), pp. 3126330; G. A. Bliss: Integrals of Lebesgue. Bull, of the Amer. Math. Soc., vol. 24 (191718), pp. 147; T. H. Hildebrandt: On Integrals Related to and Extensions of the Lebesgue Integrals. Bull, of the Amer. Math. Soc., vol. 24 (191718), pp. 113144; 177202; H. E. Bray: Elementary Properties of the Stieltjes Integral. Annals of Math., vol. 20 (2nd ser.) (191819), pp. 177186.
Connection with
PARTICULAR OPERATORS where x lt x 2


ar
,
,
w _1
are a set of points in the interval
< and such that (x *i
!

.r^)
l
59
<
<
^2
'
n 
as
<
'
_!
<
&
,
The points
oo.
such that
(a, b)
t l
belong to the intervals
*ii
We
then define as the Stieltjes integral the following limit: J
= lim/
n
.
WW3
The proof that this limit and v (x) can be established ,
If v(x)
is
exists,
under the assumptions made concerning f(x]
easily.*
constant except for a denumcrable set of discontinuities of positive t t tm then / reduces to the series 2
saltus {5 t } at the points
.
t
,
,
,
.
.
,
,
More generally, the function v(.r) may be regarded as a density function defined over measurable sets of points of the continuum between a and b. Designating by A v the value of this density function over an interval A^, and f t a t
point interior to
A^, we
define the integral as the limit

provided the limit exists. 5. The Operator as a Laplace Transformation. It will be convenient later to express an operator as a Laplace transformation. For this purpose let us define f(x) as follows:
=
f
c**v(t) dt
,
JL
where v(t)
is
a function which generates f(x)
when L
is
a properly
defined path in the complex plane. The operational equivalent of F(z) > f(x) is cleai*ly contained in a formal manner in the integral representation
F(z) ~>f(x) If the
path
L
in particular is the real interval
lim
e
ft
{F(t) v(t)}
(>1)
(
(5.1)
.
oo,
a)
and
if
=Q
J00
for
all
values of n, then (5.1) can be expanded in the following series of continued integration by parts
by means
F(z) >f(x)
:
=e
ra
fV*{F(t) v(t)}'dt/x
F(a) v(a)/x
*See, for example, Lebesgue: loc.
cit., p.
253.
THE THEORY OF LINEAR OPERATORS
60
(Fv)Y# a + (FvY'
v(a)/x
f
1)"
*/0
a) V (a)/x
A vided
(Fv)'/x
+
2
(FvY'/x* 
}
(5.2)
.
similar series in terms of inverse factorials is obtained prc that, for all positive integral values of n,
we assume
lim e
(
*+ n)t
w n (t)
=0
,
*=00
where we employ the abbreviation
w(t)=F(t)v(t) Under
this
we
assumption
w(0={^wi(0}e' at integrate by parts and thus obtain a
ef*
f
Wo (a)/x
e<
M w >
(t)
dt/x
*^uo
e
a
/x(x}l)
+iV
2
(a)
\
}
(5.3)
.
6. Polar Operators. By the term polar operator we an operator expansible as a power series in 1/z
shall
mea
,
+
.
(6.1)
,
where the b (x) are functions sharing a common region of and where, by definition, t
\/z*u(x)
('u(t)
dt
definitio
(6.2)
.
Jc
We should first notice the significant fact that the polar operato 1/2 can also be represented as a nonsingular operator of infinite or der. This is accomplished in the following manner: Let us refer to the transformation u(t)=u(x)
+
(tx) u'(x)+ (tx)*u"(x)/2l
+ . (6.3)
as the Taylor transform of u(x). If
we apply
this transformation to u(t) in (6.2),
l/z > u(x)
=
C*e*>* dt * u(x) I r
we
get
PARTICULAR OPERATORS
^{1
l/z*u(x)
e (c
~x
61
^}/z*u(x)
.
Hence we see the essential equivalence for all finite values of c of the two operators 1/z and {1 e (c ~* )z }/z The equivalence is seen to have limitations, however, if we observe on the one hand the great generality that can be assumed for u (x) when (6.2) is a Lebesgue integral and on the other hand the ~ restriction that the function operated upon by (1 e (c x}z }/z must have derivatives of all orders. The equivalence can be partly restored, however, if we make use of (1.2) and replace F(x, i A) by .
We
shall
now prove n
l/z *u(x)
that the function r*X
=
ds
~8
S
r
ds
I
u(s) ds
I
I
can be reduced to the single integral n l/z * u(x)
*
=
(xs)*
f
u(s)
ds/(nl)
!
(6.4)
.
Jc
Let us assume that the formula
is
true for
n=
k.
We
shall
then
have
l/z>{l/z**u(x)}
= C dx Jc
fc+1
)
(x
s)^u(s) ds/(kl)l
to this integral the Dirichlet s ^= c, s x, x x, we find
Applying
=
=
over the triangle (l/2
J(* c
*u(x)
= C u(s) =
f*
formula for integration
C* (x
ds
(xs)*u(s)
.
s)*
ds/kl
l
dx/(k
1)
we now
establish
!
.
Jc
Since the formula its
general validity by
is
obviously true for n
=
1,
induction.
Because of the frequent application of Dirichlet' s formula in the theory of operators we adjoin a brief discussion of it here. Consider a convex closed curve which lies within a rectangle formed by the lines x d, and for which the points PR and QS are the c, y 6, y a, x extreme values of x and y as shown in the figure. (Figure A). Let y^ y^(x) and y2 y2 (x) be continuous functions representing the two arcs PQR and PSR rex 2 (y) be, similarly, the equations of the spectively and let x x^(y) and x 2 two arcs SPQ and SRQ. Now consider the double integral
=
=
=
=
=
=
=
r= where K(x,y)
is
cb
7
A.
"a J
r^2 K(x,y) dy i
J *Jui
a function integrable within the contour.
definition of a double integral that 7 with respect to x and this leads to the
may
It is clear
from the
be evaluated also by integrating
new form
first
THE THEORY OF LINEAR OPERATORS
62
rd
1=
Jc
p* \
dij
\
2
K(x,y) dx
.
Jxi
Equating these values we obtain the general formula rd rb (* px dx K(x,y) dx dy K(x,y)di/= * Ja Jc J*! 2
.
ill
FIGURE B.
FIGURE A.
the general formula =by applying x x = for which we
Dirichlet's formula follows as a special case to the triangular region bounded by the lines x
t,
a,
y,
have (Figure B) I
dx
Ja
I
K(x,y)du
=
"a
I
dt/
Jo.
I
K(x,y)dx.
Jy
This formula was
first given by P. G. L. Dirichlet (18051859) in 1837: Sur ^dont le term general depend de deux angles, et qui servent a exprimer des fonctions arbitraires entre des limites donnees, Journal fur Math., vol. 17
les series
(1837), pp. 3536, in particular, p. 45. useful generalization of the formula was made by W. A. Hurwitz: Note on Certain Iterated and Multiple Integrals, Annals of Mathematics, vol. 9 (1908), pp. 183192, who established the validity of the interchange of the variables of integration for a kernel of the form
A
K(x y)
(t
9
where f(x,y)
is
o?)*i
(ya)/ii (xy)*if(x,y)
,
X,
ft,
v,
>
,
a continuous function.*
Let us now without loss of generality set c 0.* Applying the to the formal transform we find Taylor u(s) equivalence of the following operations
l/z*>f(x)= n
> f(x)
:
{!(! + xz)
= (1
e*'
Mz
= [x /n n
xn
n
n
n+2
z
2
/2
 f(x)
!
*For a more extended account of the conditions under which this interchange may be effected see J. Pierpont: Tlw Theory of Functions of Real Variables, vol. 1 (Boston) 1905, pp. 395398, and E. W. Hobson: The Theory of Functions of a Real Variable, vol. 1 (2nd edition), Cambridge (1921), pp. 479 et seq.
of limits
63
PARTICULAR OPERATORS
The functions
= {1
Q n (x, z)
[1
xz
f
+ x z /2\ + 2
2
t
j
e }/2"
=
n
,
1, 2, 8,
,
be conveniently referred to as generatrix functions. satisfy the differential equation*
may
They
L(Q n ) ^z9 2 Q,,/9z
2
+
(1f xzAn)
3
+n
Q./3
a?
=
Q
,
(6.5)
as
may be
proved by direct substitution of the integral
Q
n
*
=
(x, z)
f (xt)*
e<**>*
Jo
_ and
its first
*
f Jo
5 ni e
two ^derivatives

M
dt/(nl)
^/ (n_i
)
!
!
in the left
member
+ n)
+ n.r s"
of (6.5)
.
We thus get L(Q n )
=
f {z s n + l
(
Jo
1+
xz
s
n
1
}
e 82f
d^/(^
1)
!
Integrating the second term by parts once and the third term we obtain
twice,
L (Q n )
*
=
f
{z S
M
(1
Jo
+ xz f n) z
+ x z*s +
{
(1
+ xz + n)/(n\~l)
f
1
n+l
1)
/(n\l)
+ xz/(n+l)
}
} e
x
8Z
ds/(n
1)
er** /(<&!)
which
!
I
,
is seen to reduce to zero. Since there thus exists a forma) equivalence between the polar 2 operators \/z, l/z l/z" and the generatrix functions which possess convergent Taylor's expansions throughout the entire zplane for all finite values of x, the analytic theory of operators will obviously exhibit fundamental points of difference with the analytic theory of functions. The only exception to this is the case where c oo for which the generatrix function, Q n (x,z), reduces to n l/z and the polar operator has a genuine singularity. It will be important also to notice that the polar operator (6.1) may be replaced by an integral of Volterra kind as follows Making use of (6.4) we write ,
,
=
:
*This equation
is
due to L. F. Robertson.
THE THEORY OF LINEAR OPERATORS
64
B(x, 1/z) >u(x)
(Ma)
=
C* K(x,
+b
2
(x)
u(l) dt
t)
(xt)
+b
3
(x)
(xt)*/2l
,
Jo
where we abbreviate
A
variant form of considerable use n placing l/z by the integral t
We
_l)
dt
(6.1)
by
re
.
thus obtain for (6.1) the expression
B(x,l/z)
= f
7.
Branch Point Operators. One
quire whether an interpretation can be points at the origin, as, for example, log
!}
found for
is
finds
dt it
(6.6)
.
interesting next to in
made of operators with branch z* z^ (// a positive fraction), 1
,
z, etc.
A long history
(see section 5, chapter 1) is attached to the probto the operators z** and zrt, but the inter
lem of assigning meaning
pretation of these symbols may now be said to have attained a logical and satisfactory form. Slightly generalizing a definition stated independently by both N. H. Abel and G. Riemann, we shall mean by the fractional integra^ tion of a function u(x) the operation
=
v~ l
J('{(* c
t)
/r(v)}u(t) dt
r>0,
,
(7.1)
and by fractional differentiation, z m+v
 u(x)
c
Dxm+v u(x) y)}ii(t)dt
m <= 0,
1, 2,
;
<
(7.2)
,
v
<
1
.
a vital part of the operational symbol and is in his original indispensable in certain problems. Abel let c definition and J. Liouville, as we shall see later, stated a definition
The constant
c is
=
PARTICULAR OPERATORS
~
65
oo which was equivalent to setting c The latter choice served to remove some ambiguities that can arise in application. .
The
real significance of the definitions just given attaches to the the symbols obey the index law that fact :
J}^ D x c
~v
=
u(x)
This can be proved as follows By definition we have
X
C
ZV
(
'" V >
u(x)
(7.3)
.
:
J\' c
Applying to this integral Dirichlet's formula for the interchange of variables,
we
get
f*M(t) dt
i"((x
t
S)M' (s
Making the transformation y
=
(s
t)/(x
t)
and recalling
that 1
J(V>(1 o
we
shall
dl/i=rOi)r(y) //'(/*
I/)'
+ r)
,
have *
*=
(x
Jfc
X
f
VMi
y)^dy/rdi) r(v)
^0
(x
^=
u(t) dt
t)v*
c
D
f

v
t)i
u(x)
*u(t)
dt/r(p+v)
.
Formula (7.1) is seen to be an immediate extension of the polar operator (6.4) where n is replaced by v and (n, 1) by F(v). This obvious generalization was the one used by Abel in solving the tautochrone problem.* Liouville without specific reference to Abel formulated a second definition. Since is is clear that !
*
1,
Solution de quelques problemes a Paide d'inte#r,ales defmies.
pp. 1127; in particular, p. 17.
Oeuvres,
vol.
THE THEORY OF LINEAR OPERATORS
66
e
dx n
=a
* n
e ax
and
f
e at t n ^dt
=
a~
n
e at
,
R (a) >
,
J.oo
we may assume zv
as a definition that
^ e _a ax
v
e ax
^
R (a) >
,
Hence any function expressible
f(x)=
c n e"*,
or negative.
v positive
in the series
R(an )
>0
,
W=
possesses the formal derivative
71^0
But
be in agreement with the definition oo and proper uniformity conThus we have
this definition is seen to
of (7.1) provided c ditions are fulfilled.
Making
is set
equal to
the transformation s
=x
t,
Cn
we
get
dt/r(iv)
n_
An elegant method of attaining the same result is found in a generalization of Cauchy's wellknown integral formula, f (f(t)/(tx) Jc
M }dt
,
is taken as a closed curve in the com#.* plex plane about the point t Let us generalize this to fractional values by considering the function
where the path of integration
=
x and where the path of integration is now taken about the point t has one branch to infinity so chosen that the integral converges.
From
the relation
*This derivation
is
found
in Laurent.
(See Bibliography).
PARTICULAR OPERATORS
r(
we
=
v)
n/sin n v
67
T (v+l)
,
see that
=
1)
(
If in this
v+1
7i
formula
2jiyisin2nv
(cos
we
=
set v
n,
where
reduces to the proper limiting value, 2
FIGURE Let us assume that
ji
1) /[sin n v ?i
i/n
!
is
a positive integer,
is
identical with the Liouville definition.
v is a negative
number
of course
is
it
2.
this integral over the path indicated in the figure.
which
.
.
a negative number and
v is
r(v+l) ]
removed
us evaluate thus get
let
We
The assumption that manner
in the obvious
m
times. It will be convenient for us differentiating the integral later (see sections 4 and 7, chapter 8) to refer to the open circuit of
by
figure 2 as a Laurent circuit in contrast to the closed Cauchy circuit employed in Cauchy's integral formula. The integral  X X~ V x~ n is easily computed as follows
D
:
_jVx= r
(
J 00
Making the
substitution x
l)
(
If
1
v
we by
^,
xs/(s
t
v
1),
we reduce i
f Jo
this to
ds/r(v)
take the first derivative of this function and replace we obtain Liouville's result,*
_/ = ar
(
1)
v
F (?H/z) / [r (n) x"^]
*As a matter of fact, Liouville derived his formula manner by means of the special Laplace transformation r(n)/x n
=
/"*
I
=r
somewhat
different
[see (5.1)],
GO t w i e***
Jo Since the Liouville Le definition assumed that *00
in a
(7.4)
.
dt z/
1
.

e^
= aM
e aj? ,
we
get
THE THEORY OF LINEAR OPERATORS
68 It is
thus clear that any function of the form
/(*)=
f an x~ n~i
n
possesses a derivative (or integral) in the Liouville sense, GO
xn If nition,
~]
.
(7.5)
on the other hand we use the generalized AbelRiemann
we
defi
find
c
Making
D vx n =
r(xt)
x
the transformation
v l
t
n
x
t
(x
c)s, w>e
can write this
as
f
c)
s v ^[(x
c)
(1s)
Jo
+ c]
n
ds/F(v)
where we abbreviate
(x,c)
= 1+
(n+v)c/(xc)
(
(n+v)
Hf
From
the special value
= 1 we obtain the familiar for
mulas
.
Formulas
(7.6)
were derived by Peacock from an obvious generalization of
the case of integral order.
but
is
(7.6)
He
properly deduces the values
unsuccessful in his attempt to perform the differentiation Z\
From
these results

(1
X 2)J
.
we can immediately
construct the general
derivative (or integral) for the function
We
thus obtain
=xrfi
c^a;r(w+l)/r(^/i+l)
.
(7.7)
69
PARTICULAR OPERATORS It is
now apparent
that there exist essentially two definitions
of fractional operators which have different domains of usefulness. great deal of confusion has arisen over this fact as may be seen
A
from arguments advanced by Kelland stead of formula (7.7). For negative values of the new formula
When n when n n and //
n
in favor of
formula (7.5)
in Louville's definition
in
Kelland devised
an integer and /* a fraction, this yields the value 0; an integer and n a fraction, the value is GO when are both integers, sin n n/sin (n fj)n is replaced by is
//is
;
=
I/cos // TZ 1)^ ( Kelland's argument runs as follows: .
Since F(p)
P(
r(n
p) t= n/sin p n
F(l n) f /i)
n,
we can
write
= V[sinwjir(l + n)] = 7i/[sin( n + ^nT(l + n ,
Using these values been replaced by
,
formula (7.5) in which n has obtain (7.8). The obvious difficulties which
in the Liouville
we
=
oo due are introduced in the fundamental definition (7.1), c to the behavior of x n at infinity, substantially nullify this procedure. This leads to the general observation that the Liouville formula, v a>iV /(#), is applicable in the case where
^
f(x)
/V V
and the AbelRiemann formula,
= = a constant.* \f(x) \/
where the statement f(x)
Km ar*
,
f(x),
when
has the customary meaning that
O(
00
The
by Fourier on pages Chaleur was obtained from his integral
definition of fractional operators given
561562 of his Theorie de representation of f(x)
la
,
(l/27i)
f" J 00
f /
n
OO
=
s Since d n cos s(a; t)/dx gral values of n, the generalization
n
cos {s(a;
^)
+ n ji/2}
for inte
*0ne should observe that these are sufficient conditions for the existence of the integrals involved provided f(x) is otherwise integrable.
THE THEORY OF LINEAR OPERATORS
70
= (l/2ji
f
)
is easily
One
f (t) sv cos{s
f
Joo
(xt)
+ n n/2} dt ds
'oo
made. is
interested in the question whether this formulates the
Liouville or the
AbelRiemann
definition.
The answer
is
ambiguous
since proper specialization leads to either.* 8. Note on the Complementary Function. A great deal of confusion has been occasioned in the history of fractional operators by the question of the existence of a complementary function for fractional operators. Since the solution of the differential equation
d n y(x)/dx n t= is
n an
,
integer,
given by
y(x)
= C + CiX + Cx~ (>
h
\
Cni
x "~ l
>
a function which must therefore always be added to the operator z~ n  f(x), it was argued that a corresponding complementary function should be added to zrv > f(x) This function would be the solution of the equation .
ZP >
i/'
(#)
=
,
11
a fraction.
complementary function was
Liouville argued that the
Ida* his reasoning being as follows
,
C
arbitrary,
(8.1)
:
Since
let
us assume that y(.c)
is
expansible in a series of the form
V(a)t=
A n /x
.
ncc
We
then obtain 00
=
which must be identically zero. But this happens only when n and / is a fraction. Hence positive values of n are 1, 2, be excluded and we reach (8.1) as the arbitrary function. ,
0,
to
*In the theory of electrical circuits Oliver Hcaviside found frequent use for > 1 to mean I/ d'dt. He interpreted pi the operator p, where p (ir t)*. Since 1 is of the AbelRiemann type it is obvious that Heaviside's operator f(t) must be interpreted by this theory. Specialization of formulas (7.6) shows that this has been done.
=
=
PARTICULAR OPERATORS
an example Liouville
AJS
Juo V
71
cites the case
(cos ax/ (1

#2 )
}
d#
=%n
^
a >
for which he obtains ZM * f(a)
/on = {cos (ax = (_l)/i i/^e y(a) I)/*
(
ff//(l
*/> /<TI)
+
d#
2
a;
)}
A)
To compute and thus gets
If 1,
y;(a), Liouville
_l
then lim
1,
then lim
^
(8.2)
.
1
makes the transformation ax
^ f(a)
=
and y(a)
=
0.
If
in (8.2)
we
=
/i
t
=
a^uo
^

/(a)
=A
(a constant) and
i/'(
a ) ^^
^.
tt^UQ
Unfortunately for this analysis, {COS
1/2
fi
if
we
n X^/(l
J'oo o
= {_l/
2
^

COS
1/2 /t
Peacock has a similar tion.
71

!/> 71
CSC
difficulty
let
a
+X
!/> (//
2
) }
+ 1)
dx] 71}
(
1)
(!)/*
find
/*
=
.
with the AbelRiemann defini
It is clearer in this case that the
equation
=
has no solution except the trivial one v(t) 0, since we are here concerned with a Volterra integral equation the theory of which is quite complete.
The naive point of view of Peacock is dominated by an extenwhat he calls the principle of the permanence of equivalent
sion of
forms. Although it is stated specifically for algebra he assumes its validity in all symbolic operations. This principle he formulates as follows
:
"Direct proposition: Whatever form is algebraically equivalent to another ivhen expressed in genercd symbols, must continue to be equivalent whatever the symbols denote.
"Converse proposition: Whatever equivalent form is discoverable in arithmetical algebra considered a$ the science of suggestion, uthen the symbols are general in their form, though specific in their value, will continue to be an equivalent form when the symbols are general in their nature as well as in their form."* *Peacock's report: (See Bibliography) pp. 198199. Algebra, vol. 2, Cambridge (1845), pp. 5960.
Also his Treatise on
THE THEORY OF LINEAR OPERATORS
72
this principle
Applying
observes the function
first
= X M (Co,T(n) + d/x F(n~l) + C

zr
Peacock

4
h
C n i/
n
~
l
/'(I)
}
>
and hence he concludes that there
^a
2 2 /.r
r(t?
2)
positive or negative integer,
exists a
complementary function
in
the general case, r/*
=
>
+ C /aTOi
2)+..}
2
.
His arguments are ingenious but often misleading and occasionalFor example, he considers the
ly erroneous, as in the present instance.
identity
d'(ax
+ b)/dx' = r(l f n)
a r (a f b) 1 // 1 (l
where r may have positive or negative integral
+w
values.
Generalizing this formula for fractional values of 6 tains, for r 1, the result 1/2 a
=
= =
2
/37i
Replacing
(,T
+
I)
2
by
2
x'
4~
1/2
2.r
+ Aor" + A! 1
4
1>
we
r)
a;"
get by
3/2
he then ob
r,
4
.
means
of (7.6)
the unambiguous expansion d*
If
= 8x* /3ji^ + 4x A = x (l + 1/x) (x 4 1) /2
(x+l)*/dx*
we expand
3/2
J
B/2
172
'
4 ar*/*
/2
1/2
by the binomial the
orem and compare the terms
we we choose
ten down,
provided
of this series with the result just writsee that the two fractional derivatives are the same,
n
>
.
9. Riemann's Theory. In a paper developed during his student days but posthumously published, Riemann essayed a theory of fractional operators by seeking a generalization of the Taylor expansion CO
h n (d"u/dx n )/nl
tt(a?4ft)=
.
n=o
For the form
this purpose he
assumed the existence of an expansion of
PARTICULAR OPERATORS
73
oo
k v h v (d v u/dx v )
h) t=
(9.2)
,
VQO
where the r's form a set of numbers which differ from one another by integers, and the kv are constants to be determined. If
now u(y)
transform
in the Taylor
=u(t)
+
(yt)
+
u'(t)
(3/_/)= % "(f)/2!+...
,
(9.3)
where we
u(x
set
x
y
f to
+ h) = u(k) +
and k
(x
t
=
A:,
we can
+ h)u'(k) +
write (9.3) in the form
k
(#
+ Expanding
(#
Since, however, ,
fc
+
11
/i)
as a series in
we have l/T(fw + may be written
fe,
+ to)V(fc)/2!
we next
l)=0
(9.4)
.
obtain
for mc=:
1,
2,
the above expansion
r=cx>
Substituting this value in (9.4),
+
u'(k)(x
k)*'/r(2
r) f
we
obtain the expansion
u"(k)(x
k)
Since the derivative with respect to x of the coefficient of r we may assume, reis the coefficient of & /r(rfl) in is the desired derivathe braces to that expression ferring (9.2), tive. If we abbreviate this by u (r) (x) we then obtain, by means of a
h r~*/r(r)
,
differentiation with respect to
fc,
du(x)/dk*=u(k)
(x
k)"/F(r)
,
and hence we derive
dk/r(r)
THE THEORY OP LINEAR OPERATORS
74
X
ii^(x)= Riemann then saw
fit
to
(x
t)^u(t) dt/r(
add
to this definition a
r)
.
complementary
function of the form
where the the limit
The
K
n are arbitrary constants, because of the ambiguity in of the integral. essential nature of this complementary function can be
c
easily ascertained. Let us consider the difference
C (x)
When SEE 0,
C(x)
c'
but
=
c
,Dx
~v C
D X^ u(x)
u (x)
JO^v
.
c we have already proved in establishing (7.3) that when c' = c this will not in general be true. In other
words, C(x), the complementary function, measures the deviation of our integration symbol from the index law, this deviation depending clearly upon the choice of c'. In order to obtain a complementary function which depends up
on as many arbitrary parameters as we choose, to proceed as follows
np
,
it is
only necessary
:
Let the number n be resolved into p positive fractions nlf the sum of which equals n. Then we shall have
>
The
u
=
c
.J)i^
&+* u f C
2
n>2



,
(x)
successive substitution of these symbols in the one preceding
gives us finally C
/V u = ZV Cl
Ml
C./V"
1
cA" Wp ^+C (x)+ 1
Cl Z?x
ni
C
2
(x)
which, since each Ct(^) depends upon the choice of an arbitrary C M a function that depends upon p arbitrary parameters. A few examples of formulas obtained by an application of these fractional operations to several of the elementary functions are appended below.
is
PARTICULAR OPERATORS
75
TABLE OF FRACTIONAL OPERATIONS
v
x
x"
=r
)v x  n
/)/ log
(
n
+ l)xw/r( n + v +
xv+n
a;
=
j;v
(_!) v r (71
[log
1)
+ o /r (n
,
)
,
a?
where we abbreviate
/*a? r
t v i
z^z e
e ai
,
Jj
>
)
," sin
^
= (a;Vr
(2 ~f p)
}
,
(
(2
(2
D/ sin
.e
=
v
lz>'/r (2
cos x
= {xVr
(1
+
v) (3
}
{1
v)
") (4
+
)
(5
^ (g __ y) 0(4
0(3
2
a;
f)
+
+0(3 +
)}(x
(
Z)^
+
") (5
2

(1
(1
+
+
v) (2
i')
(2
p)(2
+ + r)
(3
r) (3
+
") (4
^(4
+
THE THEORY OF LINEAR OPERATORS
76
a2 x 2
ax
(1+?)
+
(1
a*x 2 (l
,) (2
(i
i>)
__^)(2
where B(m, n)
~ r(ra)r(n)/T(ra
+
(JO
v {
iO(3
/
0
n o
D
v)
n)
,
'/">
'V
fl
't'
n l n~: fx~ v /r(
^) 1( "S
1
B (n
PROBLEMS !2.
Verify the first five formulas in the table of fractional derivatives. Discuss the function
3.
Prove that
1.
u(x) =cDg v
x*
.
\ ^^(lx^^^fJL1
Vwar\
4.
Prove the formula
5.
Show c
that
D
(x
*u(x)
c)'
v
oo
2 =;r(r + l) ^
(
l)
m
y
_uW(a;)
y
T y+m
(a;
c)
m
.
m!
Functions Pcrmutable With Unity. The fractional operajust discussed can be regarded also from the point of view of the permutable functions of Volterra, the theory of which we shall consider more adequately in chapter 4. For our present purpose let us define a function of composition of first kind by means of the symbol JO.
tors that
we have
rv
f(x,t) g(t,y) dt
y>(x,y)
,
J$
which we can denote
in the convenient
form y(x,y)
=/ * g.
If it
PARTICULAR OPERATORS
happens that the two functions /
/
* g
and g are
=g * /
77
so related that
,
then / and g are called per mutable functions.
The group of the closed cycle consists of functions permutable with unity; i.e., the class of functions (f(x,y)} which satisfy the ,
equation
P/<*,*)ld*
V <*,!/)=
=
("lf(t,y)dt
Jx
.
Ja?
Taking partial derivatives with respect to y and x we see that y(x,y) satisfies the equation
from which we deduce that v' is a function of the form \p(y x). Moreover since f(x, y) is related to y by the equation dy>/dy /(#, y), it follow, at once that f(x, y) is also of the form f(y x) Hence if f(x, y) and g(x,y) are functions belonging to the group .
of the closed cycle
we
shall
have
g(ty)dt=
t)
its
y
+ x)
=
1 is a member of the group, successive powers of composition and thus obtain
Moreover, since f(x, y)
compute
f" */(*) ff(s JQ
1
*
1
=
*
c*
I2
I
y~
ds
=y
x
ds
.
we can
,
^
*sds=
Vt= r
*s n
~2
ds/r(n
1)
=
X ) 2 /2l
(y
(y
x)
n~ 1
/T (n)
,
.
Jo If
we employ /
* i
x
the further abbreviation y
f\f(s)
n
u) */P(n)
(s
}
=u
ds
we
see that
,
Jo
which is the AbelRiemann definition of fractional integration when n is allowed to assume all positive values.
We are thus able to bring the theory of permutable functions of the group of the closed cycle into relationship with the theory of fractional operators through the conclusion /# 1"=!" */
,
u
=y
x
,
v
>
.
THE THEORY OF LINEAR OPERATORS
78
Logarithmic Operators. Little exploration has yet been of the properties of the logarithmic operator
11.
made
=
logz
,
 is a function analytic about in
where
(z)
2
1
A
= P
(x
/o
we
obtain
=
(z"\ogz)
x that
t)
is to say,
(11.1)
For
v
1,
we
thus get
=
f
Jo
where C (Euler's constant)
equal to 0.57721 as follows
is
.
We may then define log z > f(x) log
2;
Cx
*f(x) =^lim
I
v
(vy(v)
logs}
:
sv
1
s) ds
f(x
,
Jo
v=o
(11.2)
where we use the customary abbreviation For f(x) 1, this becomes
=
log z * 1
= lim v=o
(vy (v)
I
~T'(v) /F(v)
v'(*0
v v log s] s
+ \/v\
x v log x}
vo
the equation defining the
and the fact that y (1)
=
ds
Jo
lim (x v [y(v)
From
~l
C
,
log z  1
we
=
y
function,
easily derive
C
log x
.
.
.
PARTICULAR OPERATORS Similarly
we
find
log z * x n log z > x
79
= lim
c
=
Cx
x log x
,
x
v log s} s 1 '^ (x
{>/'(>')
s)
n
ds
J
V=0
= lim x
n+v
+ 1/V]
l>(y)
x n log x
C xn Thus for the function f(x)
# w+v log
.
+ fix + / ^
/
a;
2
2

H
we
are able to
obtain the interpretation
logz/(s)
=
C/(.r)
/(a;) logo:
Proceeding to more general considerations
(11.3)
.
we
find as
an analogue
of (7.3) the identity zv> (z ']ogz) *f(x) =z
(11.4)
Proof
we
shall
^
:
Since
have
>
[^ log z}  /(a;)

(x
X X
f"{v(")
iogr(i
s)} (x
ty*
(t
=
(^
s)
Js
By means
of the transformation y
s)/(x
s), this
be written
where we have
s)
log [(.T
f'log
JQ
1
s)^ (1
T/] } (.r
[(a:
s)
2/1
(1
y)"
1
(
s)
may
THE THEORY OF LINEAR OPERATORS
80
log
log
v)
When
(
this is substituted in the integral
we
see that
we have
established (11.4).
Making use of this result the symbol
we
are immediately able to interpret thus get
We
.
logz {
log z
^
+ ViZ
1
log z
+
log z H
}
> f(x)
=
{Cf log#}/(x)
+ Jo(V(s
log (a
Q(3
)
t)}f(t)dt
,
where we abbreviate
It will also
be useful for us to have the identity
log z

(zv
 /(x)
t=
}
> (log
jr/*
(11.5)
which assures the commutative property Proof
Giving our attention to the
:
 f(x) log z > (ZP
= lim v=o
['(x
}
= log z >
tr^vvW
to the operator log z.
member, we have
left
1
t)v f(t)
J(*(x Q
rlog (x
dt/F(u)
t)}
Jo
X
= lim
("/()
r/(s) ds
J'(x f
O"
{>"/'(>')
1
J
s)'
(^
1
{v /'(") y log (
*)}
s)(l
y)]}
^log [(*
X (*
s)
"^^ ( 1
V)
" 1
y^
81
PARTICULAR OPERATORS
=
r/(s) "0
But the
M,)
log
s)]
<r
last expression is seen to
be equivalent to
which was the object of the proof. in the theory of fractional operators there exists a second oo for the definition for log z which corresponds to the choice of shall derive this formula following a lower limit of integration.
As
We
method used by Sbrana.
we
Referring to formula (3.2)
= (logz)/z and, noting
F(z)
let
the identity log
we
obtain for G(t),
,?
=
(3.3)
*
c^)/x} dx
r
{
Jo
(c
,
the expression
,
/*oo
{c"/w} dw
Jioo ioo {e*
(e~*
{
*^o
(1/2
7i
/)
(1/2
^e w(t ' dw/w] x)
,1
J(
,
of the identities
Making use
<
for
t
1
for
t>0
f or
t
1
for
t>x
f ; ^
we then
.
s_
i
<
x ,
obtain '
G(
*To derive
J^
c^y/7/
f Jo
(
we note
this,
v
"
I///
+
1 ) dx/.r
=
ey
tir
Jf t
and hence
Ju
log s
=
/
c/.v
(ei/'/aO i/o
j

/)
i
=
r ^ {
(
C")/.T} t^
.
/ o
See Riemann Weber Die Partiellen Differentialf/leichunyen der Mathematischen 23 (4). 25 (4) (1900), Physik, vol. 1 (1919), :
;
THE THEORY OF LINEAR OPERATORS
82
/oo
e*logxdx Jo
C
log
When
this value
(Euler's constant)
substituted in
is
(3.2)
we
.
get the desired
formula,
or by one differentiation,
logz^f(x)
=
+ 2nni) +z ^ r
(C
j
f(x) f(t)
co
The operator inverse to z~ v log z can be attained by solution of an integral equation due to Volterra. Since we have by definition z
v
log z * f (x)
means
of the
=
the operator inverse to this will naturally be the one which furnishes a solution of the integral equation zr v
Making use
log z * f (x)
of (11.4)
we
first
=g(x)
.
reduce the problem to the simpler
equation z l
Defining a
we
shall
~'
logz+f(x)=z v *g(x)=h(x) l
new
function
prove that the desired inverse z 2 *
/(<&)=
To show
r J
is
given by the equation
t)h(t)dt.
i(x
o
this let us note that
sr*logz^>f(x)
=
("[C
Jo c=
We
.
+ log
(x
)]
lim (d/dv) {!rv Vl
then form the following identities
:
/()
dt
(11.6)
83
PARTICULAR OPERATORS
S)
,D
s)
ds
f(r)dr \i(t Jr
(*'
Jo
J('f(r)dr J(* o
dy
('
s)(s
8)v(s
(t
*^ r
o
p /(r) dr
J"
JV
Taking the derivative with respect to bers of this sequence, we get
v
of the first
and
last
mem
v D, f(s) ds
s) (d/dv) o
= (d/dv)
rt
poo
f(r)dr
Jo
v
= Letting
v= 1,
and hence, since
we
z~ l
(' Jo
r)'f(r)dr/r(l
(t
+ v)
.
finally obtain
log z > f(s)
/()= or in terms of g(x)
r
{(t v
=h (s)
z 2 > I Jo
l(t
,
we
reach the desired result,
s)h(s) ds
,
,
f(t)=*+
(' i(t Jo
s)o(s) ds, (11.7)
= _^v i_^ +
(*
l( s )g(t
8)ds.
Jo
The
ideas
which we have
set forth above are capable of
tensive generalization, which we may describe as follows Let us denote by p the derivative operator p d/d v
=
(p) a
power
series in p.
*It will be noticed that
f(t)
= f(x)
this device
we
+
,
and by
We may then write*
we have
(tx)f'(x)
an ex
:
replaced /(t) in the integral by
+ (ta)*
/"(a?)/2!
+
=
e<**>*
are able to discuss the pure operational symbol
itself.
<**>*,

since
/(). By
THE THEORY OF LINEAR OPERATORS
84
J(
from which we derive
z>(
logz)
('*{
.(p) > (x
)>/T()}e
^^
(
.
(11.8)
For example, z~
v
2
log z
*=
(p) =
if q
p'
,
obtain from (11.8) the formula
*
=
?/>(>')
we
2
f
v
z^
s"
1
c
s~
+
log z
2
(log s
f V
1
2iy
c
a"
log s
(i>)
2 (log s
+y
2
i//(v) }
(v)
ds/r(v)
,
^o
(119)
We where
p
z
v
another place to consider the case
shall find it convenient in
log
n
zc=
n
from which we
,
n
l)
(
f*
J
obtain, v
t)
w ^{(a 9 v
V^(v)}
e (i x)s dt
.
(11.10)
The problem
of inversion
example, multiply rbtain Z
M
I
zr***
^M^
v
by
#(//)
^(/0
dfi
?
*^o
is
similarly generalized.
and integrate from
=z
X v+l
f Jo
e (t
^
s
l(x
Let
t)
dt
us, for
We
to oo.
thus
,
(11.11)
where we abbreviate 1(8) r=
(/<)
Sf'/r (/.+!)} dfi
.
(11.12)
^0
Let us assume that the desired inverse is the function F(z). The function $(/i) is then to be determined so that the left member of (11.11) shall equal F(z). That is to say, if we write z in the form e log ~, we are to determine $(//) as the solution of the integral equation,
f Jo This, we observe, is an integral equation of the Laplace type, the inversion of which has been extensively studied. An account of the methods available for its solution has been given in section 7 of
chapter
1.
As an example z~ v
(a
log z).
let
us determine the inverse of the operator
PARTICULAR OPERATORS
We
first
consider the equation,
f
gfiios*^) d[i=l/(a
which we note has the solution
The desired inverse
is
I
z~v
Jo
#(/<)
logz)
= e^
.
C* C/? l
$(ju)
(11.13)
,
then derived from (11.11),
f* (JO i+i
85
dju~z
c * los7} Pdu
v
(
=
z v /(a
log z)
Jo t)dt
\
a/ 1
JQO o
{
s/
i
/T(/i
(11.14)
,
+ 1)}
rf/
.
If we set ac^O, this result is essentially equivalent which we derived by a longer and more difficult argument.
12.
Special Operators.
special operators which (a) calculus
important
Of particular importance is
we
shall list a
(11.7),
number
of
to us in later chapters.
in the theory of the difference
the operator 6
see
In this section
will be
to
>/(*)= f(x
+ a)
(12.1)
.
This follows at once from the Taylor transform of f(x) as x by a and u by / in (6.3) replace t It is obvious that the difference
when we
we
.
can be written
J,/=(c* and
in general the nth difference, defined
n)
will
1)
by
nf(x
+n
appear symbolically .
(12.2)
It is also useful to consider another operator intimately (b) associated with the preceding one. If, in the series
(12.3)
THE THEORY OF LINEAR OPERATORS
86
we can
n~ replace u(x\n) by e > u(x),
we
y=
(i
+ e" + e + .c^H2
= !/(! Making use
write (12.3) in the form
e~)
*u(x)
)
*u(x)
.
of the wellknown expansion of 1/(1
we can
e~)*,
write this
y
=
(1/3
1/2
+ 5^/2!
+ 5 s /6! 

5
JS 2 sY4!
s
)
?i(o;)
,
(12.4)
B
where the ,
are the Bernoulli numbers
t
= 1/6, 5 = 1/30, B, = 1/42, B = 1/30, = 5/66, B = = 7/6, # ^ 3617/510, B = 43867/798, 691/2730, = 1222277/2310 J5 5
4
2
G
8
7
J? 10
.f
In order to explore the convergence of this series
we make use
of the f ormulaj
from which we have, lim
in the limit,
{JW (2p + 2)
X
!}
2 (2p) !/,} z
{
*Sce Whittaker and Watson: Modern Analysis, 3rd
ed.,
Cambridge (1920),
p. 127.
of the f Adams in the report of the British Association for 1877 gives tables 62 Bernoulli numbers. These have been extended to 90 by S. Z. Serebrennikoff. See H. T. Davis; Tables of the Higher Mathematical Functions, vol. 2 first
(1935).
JTo achieve
%
i
cot
V2 ix
formula we notice that
this
=V
2
i
cos
V2
ix/sin
But we also know that
IT
V2
cot
ix
TT
= z
Vz (r
=
1
+
'
1/z 4
e
V
2z/ (z~
n 2 ) and hence
GO
/ll
CO
2
{1/4T
1
'??
2
o:
2
/(4'7r
2
H
2
2 )
4
ar
4
/(4'7rH
2
i
)'
}
.
Comparing
this
with the
Wcrl
expansion (12.4)
we
at once obtain
from which we derive the desired formula 1/32P f
given by James Bernoulli in Ars Conjectandi (Basel, 1713, p. 97). He exhibited their usefulness in the summation of the powers of numbers and boasted: "Huius latcrculi beneficio intra semiquadran
The Bernoulli numbers were
first
tem horae reperi, quod potentates decimae
numerorum ab
imitate in
summun
sive quadralosursolida mille
collecta efficiunt
91,409,924,241,424,243,424,241,924,242,500
."
primorum
PARTICULAR OPERATORS
87
We
thus see that the series converges for values of z in the interval < z < 2n, a conclusion that we might have reached directly from the fact that the singularities of 1/(1 e s ) are 2jti, 0,
=
kni,
.
Equation (12.4) because of its central role in the integral calculus of finite differences has been the subject of much study, but most treatments of it introduce an arbitrary constant because of the presence of the polar operator l/z. It will be shown later that this inoo. determinacy can be removed by assuming the integration from We shall then replace (12.4) by the more comprehensive formula
(_i
+ i/2 s_ B z*/2 + B zV4! B
As an example 2 u(x) t= l/x
let
We
.
!
t
2
(12.5) /6!+) *u(t) dt us consider the summation for the case where 3
z
.
should then have
[_ l/x
2
2 !/2x 3
JS,3!/2 !x
B
=
1 /x
+ 1 l/2x~ + B
1/2 \x
3
7
...]
dx
3
Although divergent, this series is summable by the method of Borel (see chapter 5) applying to it the integral ;
00
e~ xt t n
we

get
y
=
00
f
e**
(1
Jo
+ t/2 + BJ*/2
B
!
4
2
/4 f !
B
3
t*/6
1

)
dt
I)} dt o
For x <=
More
.
o
1 this reduces to the
wellknown sum
generally let us write w(^)
= l/x m m > 1 ,
then have
__
l/ (
m
l)x
B m (m +
m~ l
2
+ l/2x
w
1)
f
(m +
2) /4
!o;
.
We
shall
THE THEORY OF LINEAR OPERATORS
88
=
f Jo
*t '
B c=
e**^
(
Jo
For
a;
Hence
2
{t /(*
!)
2t
tM
/4 1\
)
+t}dt/r(m)
= m = 2n, this formula yields the wellknown result 1,
u(x)
if
is
a function of the form
U(X) = M A + Um/X M + Urw/X M r
r
where r
is
a number greater than f
(&).=
{e^V(l
i,
we
shall
have
Wr.mfr
6')}{ m^o
*^0
(12.6) If
we adopt
the customary notation of the zeta function of Rie
mann, and note that y (x)
is
the series
y(x) t=u r
we obtain from
(r,x) fw
(12.6) the
m
wellknown formula
If series (12.3) be truncated at the nth term, so that
then the
finite
sum may be expressed
y=(l This function
is
symbolically in the form
c*)*u(x)
e*)/(I
(12.7)
.
readily reduced to derivatives
by means of the
identity oo ,
where 
is
(n)
=
*/&&*+ P C2 B n^ l
P
C
4
J5 2
(12.8)
n^ 4 + P C 6
the Bernoullian polynomial of order p.
*See Whittaker and Watson: Modern Analysis, 3rd chapter 13; in particular, p. 266.
ed.,
Cambridge (1920),
PARTICULAR OPERATORS
89
PROBLEMS The following problems contain many of the formal
results basic to the dis
The following
cipline generally referred to as the calculus of finite differences.
abbreviations are employed: "^
f
x
n+i
2 u(x) = Ai u(x)
=I
= Lim A

1^/''r J. ^ ^(/
'r^'r l/
x
^0
0"
= lim
1
~
*U{Xip)
"
r/'yil^ A i^ JL */ J/F
i?41^ ft T^ JL
*
^
J
^
/"
^
w 4 1 ^ 1(t\\.)
^ U/
= lim 
D> 0<>
A* x' 1
;
afel
9^.../'/y /j ^ ^ C
"
j
lim& l u(x)
u(x)
fcn+i
i
wr (n)
U (X j
x
J
n
a_o
These last two symbols are called respectively the differences and differential coefficients of zero. Numerical values for them will be found in Davis: Tables of the Higher Mathematical Functions, vol. 2, pp. 212 and 215.
Expand
1.
E/
>
=
u(x)
(ljA^p > u(x)
,
and obtain Newton's
inter
polation formula:
u(x+p)
+ &u(x) +
A2 lfc
(
x)
r n^ r(A O x
Ar{) r
As/2
(A
#n
4.
A r iO n
4
Ar O r
a
)
,
is
+ A3 /22
AV2
.
r!
,
A
On
=
1
.
the nth Bernoulli number.
Prove that
P(x)
if
H
(Herschel).
a polynomial with constant
is
+~P(E) This schel 's: 5.
D r QW
is
known
Examples
coefficients,
then
+
2
as Herschel's theorem, being found in Sir
J.
F.
W. Her
of the Calculus of Finite Differences, (1820).
Prove that
nD r O< w
>
f
rD riO< n
D r W )~r\ r
) ,
~
n
x(r}
A^O M /r!
Verify the following table of differences: (a)
A#<")
(6)
A
(c)
An
a&
<>
x n ^=w.!
,
1 n) t^
)
D 0< n = >
,
Establish the following expansions:
xn
7.
+
A3u(x)
Establish the following formula:
3.
6.
f
Prove that
2.
where
u(x)
,
(
l)
n
^(n
1)1
THE THEORY OF LINEAR OPERATORS
90
)
(/)
,
Aa mx+b =
(h)
A
(i)
/tv
.
.
(am
~ log
1)
=2
sin(axib)
A log x
(k)
=
&(ax + b)(~ n an(ax + &)<nD A ux vx ~ v x+ ^ A ux f ux &v x
(e)
aw
sir
(x+n)P (x+n
2)4 (tffn
r ,
D (x+n3)V (x+n5) R
where
are the binomial coefficients p, q, r, the binomial coefficients, n C t n C2 n C 3 , ,
n
C
n
,
C2
n
,
C4
,
,
and P, Q,
.
,
A
4)
p
(acf w
A" O n+2
0" +1
Verify the following table of inverse differences:
8.
(A)
AIZ< W =zx(
(C)
A
>
(D)
(E)
&*X'X\=ixl
(F)
Ai^^A^)^^^
(
G
Ai
)
k
afM+b
&ixa*
(H)
,
= k oj*+ V
Ai(^ (
a
= lJc~ a/ a
1)
,
((
1
+ 6) Ai C os(ax + 6) Ai sin(ax
(7)
(J)
(K)
zzr
.
sin(a6
&
+
C os(a
+
b
x J
a
x J
(^
1
V2 a)/(2sin
6
1
(a
I)
+
2 f 2 4 3 2 2 3 f (Find the value of the sum Solution: Making use of problem 6 and (A') of problem
8,
+
6)
I2
9.
22
x)2 J Hence, since
V x'
2
2X
= Ai
2
2 J (# 2
4^
n2 2 n
.
we have .
/ihi
a'
2
2^
JTl
we
get 2^+1 (n 2 10.
Show 1!
2n
+
3)
6
.
that 21 t
(x+n)
PARTICULAR OPERATORS 11.
we
If
abbreviate,
S n (p)
ln
+ 2"
compute S 2 (p) S3 (p) S, (p) S 5 (p) 12. Using the notation of problem ,
13.
+ Pn
,
11,
prove that
Prove that
m Making use sin 15.
+
f 3
.
,
,
n+l 14.
91
a;
of (/)
sm
j
cos
sc
f
cos
(
z
1)
+ 1)
(wf 2)
problem
,
2.c
f
cos
So;
+
m+n
show that
8,
of (J), problem
A third
(c)
m(?n
+ sin 3^c +
2.c
Making use
(n
^
sin
+
si n
8,
show that
n x sin ^2
l
/2
sin
+ cos
operator useful in
(
n \ 1 ) a;
%#
cos ^H.r sin 
?z.c
sin %a?
many
+ A:)^/(.c)=c V* l
places
is
t^/(^)]}
the following:* (12.9)
.
That this operator is valid for fractional as well as integral values of n can be proved as follows We first extend the wellknown formula of Leibnitz for the differentiation of a product to the case of fractional operators. (See sec:
tion 2, chapter 4). For this purpose
we
write
'}
U'(x)
Jf'{V(t)/r(v) C
V v \dt
(
at once obtain the desired formula,
v
cP r {U(x) V(x)} J
+v
r
Jc
7"(.r)/2!}
from which we
(It
(v
+ l)
= U(x)
V^(x)
vU'(x) V^*"(x)
U"(x) V^~^(x)/2l
(12.10)
.
Returning to the main problem, we then write e** z
*See
J.
Edwards:
m fi > {e** f(x)}
Differential Calculus, 3rd ed.
,
London (1904),
p. 70.
THE THEORY OF LINEAR OPERATORS
92
m
an integer and /u is a fraction between and 1. Then (12.10), setting U(x) equal to e** and V(x) equal to f(x), we get
where
is
a
=
gfc*
3
[^^ _ ^
_
k /Hin i
fc
__
j(l
(/l
 iU (//
+ i)
+ !)
fc
2
z
2
fc
/2
2
in
/H
!
This expression reduces to e~*
x
fcj z m > [c (1 f
( 2 _(_
>[(!
&)
which was
to
k/z)
' l
+ A;/^)/
1
 / ^ (
*
(a:) ]
= (s
/</*> (a) ]
+
fc)
be proved.
(Hadamard's Operator). In the third part of his classical on functions Hadamard considered an operator which analytic paper is closely related to the operator for fractional differentiation and ~v integration. Designating this operator by the symbol Q x f(%), we (d)
may
define
as follows:
it
Iog0 vl /(0 d(logt)/r(y)
.
(12.11)
By means of the transformation written in the form
t
= sx,
this operator
(log 1/8)*+ {f(sx)/s} ds/F(r)
The most
significant property of
that the index law holds for
it
;
that
Q~ x
v
f(x)
is
this
we
be
.
found in the fact
is,
(12.12)
.
To show
may
write
{
(log
x/ty^/t F(v) } dt
'
f
{
dog */) " 1 / () /s
r
(//) }
*^0
If we abbreviate the second integral by / (x, s) and apply to transformation log t == y, we then obtain
it
the
PARTICULAR OPERATORS
r v
I(x,s)=
IO
h
*(
y \log x)*
log s)
1
log
(y
s)^
S
this integral
r *=
transformation
Making the further ,
93
y
(
f
log x) / (log x
becomes
tf/s)^
fV
1
1
s)^
(l
Jo
which,
when
If f(x)
substituted above, is seen to establish the identity. is a function analytic about the origin and vanishing
there,
we
see
from the wellknown formula (log l/t)
1' 1
t"
l
~}
dt^=r(v)/m v
'
that the
Hadamard operator transforms f(x)
Qx
v
f(x)
=

fm x
m
as follows:
v
(12.13)
.
(e) (The generatrix operators of Laplace). The basis of the calculus of generatrix functions which P. S. Laplace adopted as the analytical method of his TMorie des Probabilites is created from two
operators G and D defined thus If f(x) is a function defined by the series :
00
f(x) e=
where a(n)
is
a function of
Ga(n)=f(x)
n,
we
and
n a(n) x
shall
,
then have, by definition,
Df(x)=a(n).
(12.14)
The function f(x)
is called the generatrix of a(n) and a(n) is determinate of f(x). The use of these symbols and their connection with other opera
called the
tors
may be illustrated in the following way Let us assume that f(x) is expansible in the series :
GO
f(x)
=
'O
a(n)x n UQ
(12.14),
.
Then
since
~m m f(x)/x =J^ a (n) x tl
MO
,
we
obtain from
THE THEORY OF LINEAR OPERATORS
94 Also, since
1) f(x)
= a(0)/x +
a(0)]
[a(l)
+
we have where
zl
=a(n+ 1)
a (ft) but in similar generally, fashion,
a(n)
More
.
we prove
In order to apply this formula to the calculus of
we write f(x)/x
m
that
finite differences
= (l + m(m
l
Operating upon both members with
= a(n)
f
D we
then get
m A a(n)
+
,
(12.15)
which we recognize as Newton's formula for interpolation. For a more complete account of the significance of this operator the reader
is
referred to section
7,
chapter
1.
(/) (The "C&lcwl de Generalisation" of Oltramare). An operator resembling in some ways that of Laplace was made the basis of a calculus by G. Oltramare which was developed and applied by C. Cailler and D. Mirimanoff in numerous ways. The calculus is based upon the operational symbol
or
more
generally,
Geau+bv+cw ^
From


= y(a? +
j/
+ 6, 2 +

c,
the expansion
au+ we
a.
(au)*/2l+
derive the fundamental operator
Gu*
=
n)
(x)
(12.16)
.
In illustration of the application of his calculus, Oltramare applied it to the solution of the functional equation
f(x,y)af(x
l,y
+ I)=0
.
(12.17)
PARTICULAR OPERATORS Since
we have by
95
definition
and
equation (12.17) can be written symbolically v*
ae)
(i
=0
from which we get aeu+v^Q
1
m
of this relationship between
Making use
n and v we write
Operating upon these functions and recalling definitions,
we then
obtain +*>
=f
( Xy
= tfy X + y) ^ a
y)
V
(
where y(z) and i)(z) are arbitrary functions. Since the last term may be written ax [a (f ^ #(# y)~\, it is clear that these solutions are
+
essentially equivalent. The calculus is also extended to functions represented
by
definite
integrals
)
Ja
dt
;
for example
G
I/ (u f a)
n
=
f* I
oo
c at t* 1
t)
dt/r (n)
.
"0
The Liouville definition of fractional differentiation is obtained as a special case of this formula. Making the specialization a 0, m and n and we then recalling (12.16) get /< f
=
,
m
=
\
^MI
t)
dt/r (m
^0
which we see
is
identical with the definition of Liouville discussed in
section 7. It will be seen from this description that the calcul de genercdisa^ tion is not the development of a principal of functional transformation but is essentially a table of symbol equivalents which is indispensable for its application. The table given below includes some of
the most
common
elements.
THE THEORY OF LINEAR OPERATORS
96
TABLE OF G TRANSFORMATIONS 0< n >(x)
(1)
Gu n
(2)
G(u
+
GUM
n
a)
n
t
=
e^t^^(xt)
__ ^ _*<,*>(*)_
pcot,*i0( 3
dt/r(n)
.
.
J (4)
(
6)
G^ Ge
= 0(a + a) = f "%' ^ [>
2
(
Ge
(5)
.
I/TT * )
2/
4
(
?y
J^ (7)
G
l/(e (lu 4
/
/
~%
e M )
(0(. f a?/i
I
/.:: GO
f
(9)
G
!/(. 4~ &
2 )
= (l/^)
0(** 4 2/0 4

00f
?/0
CO
X
r 7
au=
(10)
G
(11)
Gcos<Mt=^(0(*
(12)
G e M2 /4a
sin
2
(l/2i){0
(
+o
cos aii
^~ (/2<2
+
10 [
( J ~(jj
r>
(t
+
+a)i]
0[ar f (t
a)i]} dt
.
^ v)cosyvdv
0(ar j
.
Jo r **> 2
6 it f
4c)''}
I
e6f/2<{eW4rtc)"2/2c
*^
^r:
{2/(4ac
62)
1}
c&^/^sin j
{
(4ac
6*J )i
t/2c) 0(jc
t)
70 Jo
62
(15)
~
J
f
"Jo Jo X
I
(cosyz?/(a 4 4
dt
PARTICULAR OPERATORS
(16)
G
e/ M /u n
=
rs>
poo "TTi
97
I
e~ a '^/
I
r
T/"
1
n+1 (sin f/^
^o
^o
/oo (17)
aV4* a ]
c'0[jc
JLO
(18)
(19)
(20)
(21)
Ge & V(l
c M
~
)
V2 0(^
f
6)
/QO
%i
{(e^
+
6"/) 0(j
^co
(22)
Gl/0>
+
g0=
I
^t) dt
^"^(ar
.
/o
The efficacy of this table in the solution of certain types of integral equations will be evident from an example. Let us consider uO
qt) dt
1
e*'
J
f(x)
,
which from (5) and (22) can be written
Gl/(p+qu) From
we
this
=Ge
.
at once derive
G l^v(x)
^ G &*
(p
+ qu)
=pf(x)
+ q f'(x)
.
It will be clear, however, that this process is entirely formal and the operator gives no information regarding the validity of the solution thus attained.
The method within
its
formal scope, however,
is
general and
yields the solution of the integral equation (
^(x
Ja
where
T, 7\,
T
2
,
T
l9
y
T '")Tdt'=f(x 29
are functions of
t,
9
y '") 9
in the symbolic
,
form
THE THEORY OF LINEAR OPERATORS
98
(T* U
+ T 92 V
+...)
*
Tdt]
J]3 j
.
PROBLEMS 1.
2.
=
Given F(z) Prove that
ea
compute F(z) 
~,
e & V(lf x)
.
where we abbreviate 0(o)
4!
=1
0=+
2!
6! <* 4
+ "
a6
7TfT7 w /
75
'
v^
/
See C. W. Oseen: Sur une application dcs methodes sommatoires de M. M. Borel et MittagLeffler. Arkiv for Mat., Astro, och Fysik, vol. 12 (1917), No. 16, pp. 113. 3.
Denning
4.
If
xz,
xz,
prove that
show that a
Hence show that
=
__
3 02 f 03
0(4) 3^:
f
7 02 f
0(3)
0(5)
where the
~ + 15 02 +
coefficients are
denned
y
03
(J
_j_
04
?
4
25 03 f 10
f
5 ,
terms of differences of zero,
in
(For numerical values see Davis: Tables of the Higher Mathematical Functions, vol. 2, p. 5.
212).
If 6 =. xz, .<2^2
show that 0(2)__0
0(0__1)
^323= 0(0__1)(02) ^4^4
I=Z
^n ^n
=
0(0_1) (0_2) J nn
0(n)

(0
+
_
9
30< 2
0'>
where the values T r n are denned
f])
T.
in
Trn
+29
>
ZTZ0(4)__ 60(3)
(?<
3
>
+ T 2 W ^< 2 + >
,
f
H0(2)
TI"
__60
,
B
terms of the differential coefficients of zero, [A'()
!
.
(For numerical values see Davis: Tables of the Higher Mathematical Functions, vol. 2, p.
215).
0.
Prove that
7.
Prove that 0() ~> (x wl ?0
=x(0
f
m)^ n
>
and hence show that
m
(x u)
= x m [F(e + m)
PARTICULAR OPERATORS
99
The General Analytic Operator. Having thus surveyed what call the elementary operators, we turn to a more comprehensive theory which will include the preceding as special cases. This theory centers about the properties of what we shall refer to as the general analytic linear operator, which possesses the expansion. 13.
we might
+ z*{A*(x,z) + B The functions A
(x f y) the latter vanishing there l
2
(x,l/z)}
(13.1)
B>(x,y) are analytic in y about y
f
=
0,
;
=a B>(x,y) = A>(x,y)
lo
+a
(x)
tl
2 (x)y f a l2 (x)y' \
>
(13.2)
b> 1 (x)
The functions a tj (x), b lj (x) possess a common domain of existence (/?), such that if x is restricted to this domain, the functions AI(X, y) and Bi (x, y) exist for values of y within a second domain S.* Although it is our purpose to explore in the ensuing pages the many specializations of which the operator F(x, z) is capable and to claim for it an important role in the general theory of linear functional equations,
no attempt
is
made
or even contemplated to
endow
it
of the general linear operator. The intriguing generality of linear operation in the abstract has been made the basis of the school of mathematical philosophy inaugurated by the general analy
with the
title
E. H. Moore and is the object of the more recent investigations of N. Wiener, J. von Neumann, H. Weyl, and M. H. Stone through the medium of the Fourier integral and the StieltjcsLebcsgue integral. We should also observe that the operators B (x,l/z) are in sis of
l
n cases essentially included in the operators A^(x, z) since l/z can be replaced by the generatrix function Q n (x,z) [l (l } xz n 1 n 2 the con1) !}
many
+
=

!
,
stant limit of the integral is zero (or more generally any finite constant). In this manner the operators />,(#, 1/2 ) are transformed formally into operators without singularities. When, however, the constant limit of the integral is infinite, this the generatrix function then becomes
lim
Q n (x
c,z)
is
no longer the case since
=\/z
n ,
c=oo
and the polar character of the operator cannot be removed. One of the most interesting features of the operational calculus from the present point of view is the variety of approaches to basic formulas. Emil L. Post (see Bibliography) has employed the following definitions in an effective manner. * While this statement quent developments.
is
true in general, exceptions will appear in subse
THE THEORY OF LINEAR OPERATORS
100
'
\f(D) (%(,) Mo
A
= lim

Ar>0
/( VA.)1
+ !
=
#(,.)
l
im /(I

AJ'0
JA'u
I
4) P
(
/'/;)(
l/Ax)
0(X
AT
JGT)
4>(x)=A/lim>0
As an example we
set
f(D)
log
77,
=
<Mo)
1,
X
We
finite.
then have by
the first formula i*
log
l
(A/Ax) J
= As A x more, as p get the value
>
= log(l/Aa)l
1/2
1/p
To
(1
+
1/2
+
H
logp)
I/?)
log(pA.r)

log(D)
l*i=_C JO
X
log(X
ft
)
.
Furtherand we thus .
.
I
I
Limiting our
T/i^ Differential Operator of Infinite Order. ^^. attention for the present to the operator
A(x,z) =a<>(x)
we
X
0, p increases indefinitely and p A x approaches A' oo, the bracket has the limit C (Euler's constant)
+a
shall first consider the
1
(.T)
bounds
+a
2
2
(a?)s /2!
to be
+
,
imposed upon
(14.1) its
generality.
The content of this discussion is based upon the work of C Bourlet and S. Pincherle, the development of the former being closely followed.
S a linear operator and by S(u) the value obwhen S operates upon some member, n, of a class of functions which we may denote by ^4. The following definitions will be useful An operator will be called continuous when Let us denote by
tained
:5:
:
S\\imn(x,h)~\
,
/<>/<'
and regular when the result of operating upon every function analytic in a certain domain D is a function analytic in a second domain D'. An operator will be called uniform if it sets up a onetoone correspondence between analytic functions. Otherwise it will be multiform. For example, the definite integral is uniform but the indefinite integral
is
multiform.
The following theorem shows the formal scope (14.1)
of the operator
:
*Bourlet uses the word "transmutation" for & and "transmuee" for S(u). He says of the word operator "Operation etait un terine trop vague et qui se pretait mal a la formation de mots derives." :
PARTICULAR OPERATORS
101
Theorem 1. Every uniform, continuous, regular, linear operator can be expressed in the form S(u)
where
A (x,
ftm(T)
= S(x m
in
by (14.1) and the coefficients are
z) is defined
xS(x n
mCl
)
which m C l m C 2 ,
,
= A(x, z) +
> 1
)
,H
C2
x 2 S(x m ) 
x m S(l)
,
are binomial coefficients.
,
we let S(u) be a uniform, continuwe determine a function such that
Proof: For the proof of this ous, regular, linear operator and the difference
a u
S(u) will be
()
identically zero when u =
S (u) when u
is
a
We
a, (x),
thus find that a can be determined so that
u
u'
Similarly a second function, will satisfy the condition
1.
i
set equal to x. It will be readily seen that
Mff) =S(x) In similar fashion a 2
S (u)
when n = x
2
thus
;
a2
(x)
.
can be so constructed that
(a;)
a u
!
c)
we
xS(l)
a2 u"/2 c^
w'
!
=S(x m
)
,
get
^S(x
2
2xS(x)
)
+
Continuing this process we obtain without formula a m (x)
it
mCl
xS(x^
+
1
)
TO
C
2 a:
2
S(.r
difficulty the general
w2 )

x m S(l)
,
(14.2)
which can be established for the general index by induction.
Now
consider the series
T(u)
=A(x,z) >u(x)
which for every analytic function ti(x) that renders this series convergent will define a uniform, continuous, regular linear operator.
Then the
difference
a uniform, regular, continuous linear operator for which, moreover, 2 the transformation of 1, x, x namely, of all the integral is
,
powers of
x, is
equal to zero.
,
"
102
Now of #
THE THEORY OF LINEAR OPERATORS
suppose that u(x)
is
a function regular in the neighborhood
,
u(x)
c=A
~\A I (x
x
)
+A
2
2
(x
)
H\A m (x
w )
+
,
and consider the polynomial U m (X)
We
crrr
A
shall
()
f
AI (X
X
n)
all
A
2
(X  X
2
)
+
'
+ A m (X  X
m )
then have
S(u m ) for
f
T(u m )=V(u M )t=Q
values of m.
But since the operator was assumed
to be continuous, it follows
that
U(u) *=U(\imn m )
We
^lim U (u m
)
==
.
thus establish the desired identity
It is possible to extend the application of theorem 1 to special cases even when T(u) does not converge for analytic functions regular about x. Thus it is clear that T(u m ) will always exist and it may
r(lim?O existing also. In lim T(u m ) much in the the value S(u) same way as values are assigned to divergent series. If we indicate by means of a subscript the variable to which the operator applies, theorem 1 can be put in a form that is often useful happen that lim T(u m ) this case
we may
will exist
without
assign to
in application.
Thus replacing u(t) in the operator S (u) by its Taylor transform (6.3) and making use of the linear character of the operator, t
we
shall obtain
S
t
(u)
= +/
W
te<>(a0/n!
+
... ,
(14.3)
where we have used the abbreviation
PROBLEMS 1.
Derive Taylor's expansion from theorem
2.
Assume
1
by assuming S(u) ~u(a).
that
coefficients a m (x) by means of (14.2) and the table of fractional derivatives given in section 9. Now compare with the expansion given in problem 5, section 9, and hence establish the identity
and compute the
PARTICULAR OPERATORS
m
v v
rrr: j_
+m
y
m(m
1)
__
}
+1
O'+l)
103 ...
m
(*'j2)
X
2W
Define S(u)
3.
(
W )> anc^ compute the coefficients
,(***)
by means of
(14.2). Then compare the resulting expansion with formula (12.8) and derive a relationship between the Bernoulli polynomial of degree p and the sums ,r
15.
section
Differential Operators as a Cauchy Integral. In the last a differential operator of infinite order could be
we saw how
expressed in a form analogous to Taylor's series. We shall press S(u) as a contour integral in the complex plane. Thus let us consider the operator
S(u)
=A(x,z) >u(x)
now
ex
(15.1)
,
\^here A(x,z) is defined by (14.1) in which a,, (a*), ^(a:), are functions analytic in a domain R. Then employing Cauchy's formula,
zx) m+l
i
']
dz
c
we have upon
substitution in (15.1) the formula
f ln(z)y> m /
(x.z)/(zx)ldz,
c
where we abbreviate
y^(x
f
z)=a
(x)
+a
s
(x)/(zx)
m
we
increases indefinitely, If, as attain the formula
S(u)
= (l/2ni)
J
i/ wl
 a m (x)/(z H f
x)
M .
converges to a function y(x, z)
u(z)ti>(x,z)/(zx)] dz
.
(15.2)
which an operator of for transformation furnishes a every function analytic (15.1) in a given domain. This is supplied by the following theorem It is of special interest to note the case in
form
:
A
Theorem 2. necessary and sufficient condition tliat the operator defined by (15.1) furnishes a transformation for every function u(x) regular in a domain of radius Q around the point x ot is that the series y>(x,z)
a (x)
+
a,t
(x) /
(zx)
shall be convergent for every value of z
+a
2
(x)/(z
which
rr)'M
satisfies the condition
THE THEORY OF LINEAR OPERATORS
104
That the condition is necessaiy is proved as follows: If S(u) existed for every function analytic in a region R of radius Q around x , it would exist, in particular, for the function u(x) t=l/(z #). for this
Since 2
l/(z
we
,
tf)
>
u
(n)
function (x)
we have u(x)
nl/(z
.r)^
1 ,
,
= l/(z
x),u'(x)~
by substitution inS(u)
get
S{l/(zx)\= v (x,z)/(zx)
,
where y> (x, z) is defined as above. But this series must be convergent for every value of X Q \Z
=
z for
which
.
\
FIGURE 3
That the condition is sufficient follows at once from the fact that y(x,z) converges on the boundary of a circle of radius Q described around x as center, for if u(z) is a function analytic within and on the circle of circumference C, then T(u)
=
(l/2ni)
#)] y(x,z) dz
f Jc
and defines a transformation for ever function analytic in the region. Such an operator, S(u) will be called complete in the domain exists
9
R.
The Generatrix of Differential Operators. The function defined explicitly by (14.1) we shall refer to as the generatrix of the differential operator.* The central theorem relating to the generatrix of a uniform, 16.
A (x, z)
continuous, and regular operator
is
the following
:
3. The generatrix of a uniform, continuous, and, reguwhich is also complete in a certain domain around the for this value of x a transcendental entire function in
Theorem lar operator,
point X z of
let
,
is,
genus
1
,
or
0.
*This term is due to Lalesco. See Bibliography: Lalesco (1), p. 193. uses the term "fonction operative."
Bour
PARTICULAR OPERATORS
105
Proof From the definition of completeness the series :
must be convergent for \z\^= I/Q where Q main of regularity around ,r. Hence A(.\\ z) ,
value of z and, consequently,
A (x, z)
is
the radius of the doconvergent for every an entire function of this is
is
variable.
Moreover, small,
we
we
if
designate by
a m \/(o 4 e) m Also,
if
<
1
,
and hence
?w,
We now make Lemma. is
\a m \/ml
<
(Q
+ s) m /ml
.
is any positive number, we have for a sufficiently large i] the inequality (Q ~\~ s) m < (ml) *, so that
a m \/ml
mains
arbitrarily
1
value of
/I
number
a positive
e
shall have, for a sufficiently large w, the inequality
use of a
< l/(m)^
lemma due
to J.
an entire function in than l/(m!) 1/x the function is
less
//,
in
,
z,
(16.1)
.
Hadamard.* the coefficient of z m re
of genus less than A
where
not an integer.
Referring now to (16.1) we see that, since the coefficient is ~ smaller than l/(m!) 1 in absolute value, we shall have A 1/(1 >;), and since q is a number as small as we wish the function must be of class 1 or 0.
=
7
?
17.
ume
is
Five Operators of Analysis. The object of the present volto exhibit the formal unification attained in the solution of
linear functional equations by means of an exploration of the properties of the general analytic operator of section 13. This is most clearly seen if we translate five important operators of anaylsis into the form
given.
The
first
where the
A
l
operator
is
the polynomial operator of section
(x) are functions of x
2,
which share a common region of
definition.
The second operator
is
the operator of the Fredholm transforma
tion,
A
u(x) 4 (* K(x,
t)
u(t) dt
,
(17.2)
subject to certain limitations which will not be speto u(t) the Taylor transformation (6.3) we
where K(x,
t)
is
cified here.
If
we apply
can write (17.2) in the symbolic form les properietes des fonctions entiercs. Journal de mathenuitiques, 9 (1893), p. 172. See also section 2, chapter 5.
*Etude sur 4th
ser., vol.
THE THEORY OF LINEAR OPERATORS
106
(t t) e
K(x,
We
 x) ~
dt} *
u(x)
.
shall refer to the function
F
t
(x,z)=A}
I
Ja
.*>' dt
K(x,t)
(17.3)
=
Fredholm generatrix of first kind when A and the Fredholm generatrix of second kind when A 1. The third operator is derived from the Volterra transformation
as the
of a function, *
+
Au(x)
K(x,
f
dt
(0
t)
(17.4)
,
which can be written by means of the Taylor transformation of u(t) as
or by applying the Taylor transformation to K(x,
{A
f
K '(s, x)/z*
K(x, x)/z
t
~
as
t),
K "(x x) /z* K '"(x,x)/z* t
t
t
= {A+ If
A
=
0,
we
F or
its
f*00
e ts
Jo
K(x,xt) dt}u(x)
.
shall refer to the function
P
K(x,t)e< *^ dt 1
(x z) =^A\9
(17.5)
symbolic equivalent,
Fr (x.
z)
=A + n
(17.6)
as the Volterra generatrix of first kind, and generatrix of second kind.
A fourth much
if
A
=
1,
as the Volterra
type of operator to which mathematicians have devoted is the operator of the difference transformation,
attention
*po(x)
n(x) \q\(x) u(x
\~
co)
\^ 2 (x)
u(x
\
\
(x)
n(x
where the functions y (x) have a common region of definition. The generatrix corresponding to this transformation is easily t
107
PARTICULAR OPERATORS obtained by replacing u(x generatrix
f
by
r
c
ro)C
w (x)
.
We
thus get for the
(17.7)
Closely associated with the last function
is
the generatrix of the
(/difference operator,
u(x) ]q\(x) u(q\x)
\<i> 2
(x)
u(q 2 x)
u(q n x)
\
f
Replacing u(q x) by its Taylor transformation where
e (0
t
^
.
> u(x),
t
F
q
(x, z)
=
//
o
(x)
+
c/,
r~
(x) c
+^
2
W2 ^ (^) 6

1
(x)
e>^
.
(17.8)
The equations corresponding
F
p
to these operators,
(x,z) >w(.r)/(.c)
i.
e.,
,
are referred to as differential equations, integral equations of Fredholm type, integral equations of Volterra type, difference equations, and qdifference equations respectively. Variants of these types are, of course, common in mathematical literature, but without exception they can be included with suitable restrictions under the general the
ory of the analytic operator. For example, the mixed integral equation the development of which is largely due to A. Kneser, i. e.,*
u(x)
+Y
t
+
x ) u(d)
i=l
f K(x,t) u(t) dt ^a
= f(x)
where the {CJ form a
discrete set of points in the interval included in the general theory under the formulization
,
(a, 6), is
b
K(x,t) Jf
*Belastete Integral gleichungen. 169197.
Rendiconli di Palermo,
vol.
37 (1914), pp.
CHAPTER
III.
THE THEORY OF LINEAR SYSTEMS OF EQUATIONS. Preliminary Remarks. It will become apparent in the subsequent development of our subject that the theory of operators may be envisaged as an aspect of the theory of linear systems of equations, where the number of variables and the number of equations are infinite. Such a system may be conveniently represented in the following notation: 1.
!
t
4"
fl'/ni^i
The matrix
"
V
31
'i2'X
'2
"
"
4" ^"M^ 4"
'
.'"
.
"
*
*
?
i
&mn3'n
'
'
'
~\~
of the system will be designated
A<=\\a if
'
(i.D
.
.
:===
'
fr>>
>
by
\\
and the unit matrix by
where
d i} (Kronecker's
symbol)
is
zero for
i
^ j and unity for = i
/.
The law of multiplication for finite matrices will be assumed to hold in the infinite case provided the infinite series which compose the elements of the product matrix converge. Thus if we have
A ^= the product
may
and
\\a.j\\
B=b
iy 
,
be written
where we assume the existence of the sum
The matrix conjugate symbol
to
A
will be designated
A', that is
and the reciprocal or inverse matrix by A" 1 that ,
108
is,
by the customary
LINEAR SYSTEMS OF EQUATIONS A
where
A
1}
is
109
1
the cofactor of the element
a,,
and
is
the determinant
of A. It is clear, however, that a unique inverse may not exist for A. Definitions and theorems pertaining to this situation will be given in section 9.
In most of the ensuing discussion it will be assumed for convenience that the elements a l} are real, but this is not in general a restrictive condition. Many of the results hold for w hat is called the Hcrmir
tlan matrix,
where we assume that a tj complex conjugate of a }i
=
ct lt
.
The bar over the
designates the
.
Types of Matrices. It will be found that there are several of infinite matrices which are especially useful in the theory of types operators and we shall confine our attention particularly to these. 2.
(A) The secular or normal matrix will refer to the limiting form, as n approaches infinity, of the following: a ln (2.1)
The determinant
where
for

<J,
7
culty that
i
D(n) has
of
=
j
S n we can
and
6 lt
=
write briefly as
1.
It
can be proved without
diffi
the following development:
1
V.
(2.2)
where
r,,
r2
,
,
r n run independently over
*See G. Kowalewski: (1909), pp. 455456.
Eiiifuhruny
in
die
all
the values
from
Dctermtnantentheone.
1 to n.
Leipzig
THE THEORY OF LINEAR OPERATORS
110
It will be useful in another place also to have an explicit expansion of the cofactors of the elements of For this purpose we (n) introduce the determinant in the form
D
ar
,
The cof actor of a sr r
=
rs
^a +
can be expanded thus:
s,
,
D
.
+
V
rs
J
ri ,
a
j
I
#r
rjra
Ct r r
>
s
I 
a
'
r..*
j
r*
,
a rri
,
'
a ''ir
'
...
,
a r/ B3
'
an
.
j
J 1
V
L 2)
I
"
s
n _,
L*
'
//^

^^
I
rirz
"
>
A\ (9 \^4;
\
'
rn ^ 2 
a rn _
i
where n, r 2
,
,
r
rt
_2
^r.^n
2
>
>
^r n .,r n
range independently over
,
all
Another array that will be useful to us (B) matrix, which is the limiting form of the following: n tl U
we
j
n ^12
,
Ooo
the values
is
fc lr
fc 2 ,
the triangular
9
,
,0
,
,0
,
a 33
... ,
,
,n
,
,
a nn
(2.5)
replace the elements {a i; } of the matrix by the special set n 1 and if we abbreviate the de1, 2, {
a
_ ? t
c=
=
D *Kowalewski,
,
n
(a
loc. cit., p. 468.
,
d lf
Oo,
+
..jO*)
;
LINEAR SYSTEMS OF EQUATIONS that
111
is,
an
D
n
(a
,
=
..,a w )
a,, a,,
a n_2
(2.6)
... !
we can series.
establish an important connection with the inversion of power This connection we find in the fact that the series
+ a,x + has for
its
W+
o
,
^
reciprocal the expansion
s
Some
(2.7)
,
interesting conclusions follow
example, since !/{!// (x)} theorem in determinants
= /(#),
we
from
+
(2.8)
}
this simple fact.
For
derive at once the following
:
D
n {l,
It is well
#,/*,
known
.
,
(1) D
W /OO
W }
in the theory of infinite series that if
lim
I
D ^/D n n
and lim
I
\
'/jij'
/)
Dn
1/w
y>
they are equal to each other. Moreover, if either limit exists, it equal to /(>, where o is the radius of convergence of series (2.8). Understanding then that the existence of the limits is implied, we can write
exist, is
lim
i '
D ntl /D
'
n
t= lim n
/<(/>
D
I
1/n tl ]
= a /o n
(2.9)
.
<S)
2 For example, from the expansion cos x* 1 x/2 f # /4 and the fact that I/ (cos x*) has its nearest singularity at x 1
!
2
7i
/4,
we
=
are able to infer that 7T/4
cr=
Km
HlH
d n i/dn
\

5n
1/7* ~
,
where we abbreviate 6n
=D
n [l,
1/6 !,, I/ (2n)
1/2!, 1/4!,
Similarly, since the Bernoulli
numbers
!>
!]
.
are defined by the se
res x/ (e*
x/2
+B
a a;
2
/2
!
(2.10)
THE THEORY OF LINEAR OPERATORS
112
and since (
we
see that r>
&n
is
/
('
i \ n
1;
The third kind of array with which we shall be concerned (C) the Laurent matrix, defined as the extension to infinity of the fol
lowing: "'as
^32
$23
CL22
aft n 
^03
^0
2
^00
&13
O12
10
w
32
_,2l
a22
a^
11
&..J2
^l^
(2.11) 11
ai2
a, 3
21
a2 ,
a^.?
31
<&32
^33
This matrix is closely related to the theory of Laurent series in the theory of functions of a complex variable as will be later pointed out, and its name is due to this fact.* Historically it was one of the first matrices to be employed in the solution of infinite systems of equations, a special matrix of this type being used by G. W. Hill
(18381914) in his theory of the motion of the moon.f If
we make
the particular specialization
we have what
then
is
called the
matrix of an Inform, that
is to say,
the bilinear form
L(x,y)
The elements
~
v
c q p
x,y q
.
(2.12)
of the matrix are obviously the coefficients of the
Laurent series /(*)
=
2
cn z
(2.13)
*See O. Toeplitz: Analytischo Theorie der LFormen. Mathcmatische Anvol. 70 (1911), pp. 351376. tOn the Part of the Motion of the Lunar Perigee which is a Function of the Mean Motions of the Sun and Moon. Acta Mathematica, vol. 8 (1886), pp. 136. First published at Cambridge, Mass, in 1877. See sections 7 and 8, chapter 9.
nalw,
LINEAR SYSTEMS OF EQUATIONS (D)
A
applications
113
fourth array associated with a number of interesting what is called the Jacobi matrix, namely,
is
This matrix
is
tin
a 12
a.2l
a 22
023
aAZ
a 33
0*4
a 43
an
a 43
a54
a.,,
(2.14)
a 5G
the matrix of the Jacobi bilinear form, GO
00
(2.15)
which has been especially studied by 0. Tocplitz.* An interesting connection is established between
this
certain expansions in continued fractions as follows: Let us specialize the elements by writing alu t= a,p a nq
=
=
b a, w associated system of linear equations: >
Then consider the following
matrix and
,
lt
.
(2.16)
We /y
then easily derive the following equations: .
/
fY>
I /v*
\
" ,
.
/vi
v .
>
1
Employing these ratios successively, we obtain the following continued fraction as a formal evaluation of ,TI:
a.2
(2.17)
*Zur Theorie dcr quadratischen Formen von unendlichvielen Veranderlichen, Gottinger Nachrichten (1910), pp. 489506.
THE THEORY OF LINEAR OPERATORS
114
The Convergence of an
3.
infinite determinants, initiated
Determinant. The theory of
Infinite
by the researches of G.
W.
Hill
men
tioned in section 2, was essentially founded in 1886 by H. Poincare, who succeeded in giving an analytical justification to the methods
which Hill had so successfully applied. It is to H. von Koch (18701924), however, that the theory owes most of its development. In a long series of papers, the first published in 1892 and the last in 1922, the significance of infinite determinants and their varied applications to the problems of modern analysis were thoroughly explored. These papers are notable both for the clarity of their style and for their ingenuity of attack upon the complexities of the problem. A bibliography of the contributions of von Koch will be found in Acta Mathematics, vol. 45 (1925), pp. 345847. is the subject of this section, was first proved follow here, however, the exposition of von Koch.t
The theorem which by Poincare;* we
Theorem
1.
The
determinant
infinite
D
<= lim D(n) will exist ttCO
provided both the product of the diagonal elements and the the nondiagonal elements converge absolutely. $
sum
of
00
Proof: Since the hypothesis that
2
!
converges carries with
ft/
tji
it
the convergence of the infinite product
Q=
77
(1
?l
we
+2
!.; D
>
;i
conclude that the product
ii
fl
also converges.
Now
consider the two products
11
j\
1
=1
71
Pn we
set certain values of
If in
D
*Sur
les determinants d'ordre infini. Bulletin de la Soc. Matfienuitique de vol. 14 (1886), pp. 7790. fSur les determinants iniinis et les equations differentiellcs lineaires. Acta Mathematical vol. 16 (1892), pp. 217295; in particular, pp. 219221.
France,
JA
useful sufficiency condition for this convergence
double integral
r oo
~c/o I
/ o
Jo
I
I
di dj
.
is
the existence of the
LINEAK SYSTEMS OF EQUATIONS
Qn
in the development of
term in
D (ri)
115
but the correspondence
,
is
such that no
We
superior in value to its corresponding term in Q. thus see that \D(n)\ Q n Furthermore, we notice that D(n\p) is
<
.
sum of those terms in the development of which vanish when the terms a [i,k D(n}p) (^+1) (w+p)] are replaced by zero. But to each of these terms there corresponds in the development of Q n ^ Q n a term of equal or greater absolute the
D(n) represents
=
ljc
,
value.
Hence we attain the inequality
D(n+p)
We ists for
^
D(n)\
Q, H
(?!
;,
.
are thus able to conclude, since Q M converges, that there exeach positive e a positive integer n' such that D(n~\p)
D (n) < s when n >
ri

and p
any positive
is
integer.
The theorem
is
thus established.
Determinants which satisfy the conditions of the theorem are said to be of normal form. The following corollary is of great importance in the applications of infinite determinants.
Corollary. A determinant of normal form remains convergent if the elements of any row (or column} are replaced by a series of quantities which are all smaller in absolute rahie than a given positive
number. Proof: For simplicity of exposition the
first
let
us replace the elements of
row JL
by the elements m,,
j
m w 2,
a i2
fan, 3,
>
w
,
m

^13
r 
'
'
y
w,
<m
'
j
,
&1M
'
'
'
y
which
the inequality
satisfy
.
Let the new D(n) and D be denoted by D'(n) and D* denote by P' n and Q' n the products obtained respectively. Moreover, in P the factors which correspond to the inn and Q n by suppressing no that term of D'(n) can have a greater modulus dex one. We see than that of the corresponding term in the expansion of m Q' n Hence, reasoning as before, we have values of
.

D' (n+p)
D'(n)
< m QV,
m Q',
4
.
This establishes the corollary. We also note the following propositions, useful in application, which are easily established on the basis of the preceding arguments:
Theorem 1 and the corollary apply with equal validity to (1) the determinant of Laurent type,
D 1=
a l}
, 
i,
j
=
oo
to
 oo
.
THE THEORY OF LINEAR OPERATORS
116
// in a determinant of normal form two rows or two colare interchanged, then the determinant changes sign; the determinant is zero if two roivs or two columns are identical. (2)
umns
A determinant of normal form can be developed according elements of any row or of any column.
(3) to the
PROBLEMS Compute the
1.
B nC=
(
Bernoulli numbers from the determinant
l)n+i (fcrc)!D 2n [l,l/2!,l/3!,l/4!,
Given
2.
first five
D B = D n [l, 1/2!, 1/4!,
show that lim 
Prove that
3.
is
if
D_/D

= lim



D

,
l/(2n)
i/n
~
14
,
!]
<JT*
l/(2n
+ 1)
!]
.
,
.
the determinant
to converge absolutely as ra
oo,
then
it is
both necessary and sufficient that
the series
2 Mi t1
converge absolutely, (von Koch). 4. Show that the following determinant is convergent and compute its numerical value approximately. (L. L. Smail: Theory of Infinite Processes). 1
1/2
1/2 4
1/2* 1
+
1/2
1/2!
1/2 3
1/26
1/25
l/2io
1
+
1/28
1/2"
7
1/2"
1/3!
1/213
1
+
1/4!
.
The Upper Bound of a Dctermijiant (Hadamard's Theorem). Many proofs have been given of the celebrated theorem first proved by J. Hadamard which gives an upper bound to the value of a deter4.
Because of the fundamentally different ideas involved reproduce four of these proofs.
minant. shall
we
The theorem of Hadamard is one of the most thoroughly proved theorems in mathematics, its truth having been established in many ingenious ways. The first general proof of the theorem for complex elements was given by J. Hadamard: Resolution d'une question relative aux determinants. Bull, des Sciences Math., vol. 17 (2nd series) (1893), pp. 240246. See also: Comptes Rendus, vol. 116 (1893), pp. 15001501. It appears, however, that as early as 1867 J. J. Sylvester, in connection with inverse orthogonal matrices, had constructed matrices whose determinants yielded the maximum value case of Hadamard's theorem. Thus
LINEAR SYSTEMS OF EQUATIONS
117
Thoughts on Inverse Orthogonal Matrices, Phil. Mag., vol. 34 (series 4), (1867), pp. 461475. Also Hadamard's theorem was known apparently to Lord Kelvin in 1885 and a proof was communicated to him in 1886 by T. Muir. Proposed as a problem in the Educational Times, the theorem was proved by E. J. Nanson in 1901: A Determinant Inequality, Messenger of Mathematics, vol. 31 (1901), pp. 4850. In his Geometrie der Zahlen, Leipzig (1896), p. 183, H. Minkowski found the theorem as a consequence of the Jacobi transformation of quadratic forms. The literature of this theorem, proofs other than those already mentioned, discussions of the maximum value, etc., include the following items: see:
L.
Amoroso: Sul valore massimo
di speciali determinanti. Giornale di (1910), pp. 305315. G. Barba: Intorno al teorema di Hadamard sui determinanti a valore massimo. Giornale di Mat., vol. 71 (1933), pp. 7086. W. Blaschke: Ein Beweis fur den Determinantensatz Hadamards. Archiv der Math, und Physik, vol. 20 (3rd series ) (1913), pp. 277279. T. Boggio: Nouvelle demonstration du theoreme de M. Hadamard sur les determinants. Bull, des Sciences Math., vol. 35 (2nd series) ( 1911), pp. 113116. M.Cipolla: Sul teorema
vol. 48,
:
,
;
2nd
;
ed.
G. Kowalewski: Einfahrung in die Determinantentheorie. Leipzig (1909), p. 460; Die klasxischen Probleme der Analysis des Unendlichen. Leipzig (1910), p. 378 et seq.; IntegraJgleichungen. Berlin and Leipzig (1930), pp. 106109. T. Kubota: Hadamard's Theorem on the maximum Value of a Determinant. Tohoku Math, journal, vol. 2 (1912), pp. 3738. A. Molinari: Sul teorema di Hadamard. Atti dei Lincei, vol. 22 (5th series) (1913), pp. 1112. E. Pascal: Die Determinanten. Leipzig (1900), 53, pp. 180184. T. Peyovitch: Sur la valeur maxima
THE THEORY OF LINEAR OPERATORS
118
L. Tonelli: Sul teorema di Hadamard relativo al valor maggiorante di un determinante. Giornale di Mat., vol. 47 (1909), pp. 212218. W. Wirtinger: Sur le theoreme de M. Hadamard relatif aux determinants. Bull, des Sciences Math., vol. 31 (2nd series) (19'07), pp. 175179. A German translation of this article is found in Monalshefte fur Math, imd Physik, vol. 18 (1907), pp. 158160.
Proofs of the Hadamard theorem will also be found in any standard text on integral equations. Historical references to the theorem are given by T. Muir: An Upper Limit for the Value of a Determinant. Trans, of the Royal Soc. of South Africa, vol. 1 (1908) pp. 323334, and also by E. Hellinger and 0. Toeplitz: (see Bibliography), pp. 13561357. Bibliographies are found in the last reference and also in T. Muir: The History of Determinants (19001920), London and Glasgow (1930), chap. I (a), pp. 90101. ,
=
^
n. [o i; ], ij If the elements of the determinant D A, then the following inequality holds: satisfy the condition a tj
Theorem
2.
^

D ^ A n ?i"
/2 .
First Proof (Hadamard) Employing the symbol D t= [>>,] for the determinant in order to avoid confusion with the symbol for the absolute value of the elements, a i; ], we replace each element by its conjugate imaginary value and thus attain the new determinant :


U =[]. From the elements of the determinant D, let us now construct a matrix of any p rows (T) and from the corresponding elements of 77, a second matrix (T). Forming the product of these two matrices we then obtain a determinant PP9 where
the elements of which are ~~
~~
"~~
^
i
i^
But the determinant
Pp can be
j
written in the form
PP = 8 PP P^^Q P
(4.1)
,
where
*P,I
Qp
,
*P.PI
We now fix our attention upon the adjoint determinant of the elements of which we shall designate by S tj From the defini.
*It will be convenient in this section to adopt the convention that the product of two matrices is a determinant the elements of which are obtained by the combination of row with row rather than row with column as assumed in the definition of section 1.
LINEAR SYSTEMS OF EQUATIONS
we can immediately
tion of the adjoint
onrl Q &C* PP '= rOpi anu tokh
_/^) ^ ^pi (h)
.
write*
&hh
P
where
p 2
(ll)
is
S
hl
.,
1
p
formed by omitting the
in the note, specialized for
p r
,
,
&Jip
(/I) {,>>
m
=
2,
we
r)
Wp
the complement of s 2 in
u Spp
1
;
1, 2,
.
t>
oo 2
=
t
where Q^ is a determinant of order hth row and feth column from Q If we now consider the minor
and recall the theorem stated have
119
,
Q p From .
this
it
follows that (4.2)
,
where S is the modulus of S v, the last term appearing because the two determinants Spj and Sh are conjugate. We now proceed by induction to prove that Q p is essentially a s lz 2 is negative and P p is negative quantity or zero. Since Q>*\=^ always positive, it follows from (4.2), namely, Q 3 P 1 c^P2 Q 2 1>h
;
\
{>
t


s 3h ,
2 
,
that
Q
3
is also
negative or zero.
By
similar reasoning,
and finally Q p are also negative or zero. Returning now to (4.1) we see that for p ^= 2 we
P___ 2
from which we get Similarly, for
P ^
s^
2
P we
o ^jj
s 22
i>
<j *12
I
22

shall
Q Q 4,
5,
have
2 I
>
.
have
3
=
*The adjoint of a [a t; ] is the determinant A =. [A i;"J, where the A l} are the cof actors or algebraic complements of a l} The fundamental relation between More generally if we define a"A and a is given by A .
1
.
A
**!
and
we
if
N
is
the determinant
shall then
have
M = Na
. I
.
.
A
^l
THE THEORY OF LINEAR OPERATORS
120
P < and hence
P <
g
S
Soo S
in general,
<S
Po
Making use
8
(4.3)
and assuming the
of this fundamental inequality
existence of a value
D
A
Pn
2
a tj
such that
A
^
2n
ri*
,
\
^
A,
we have
^A
D
n
n" /2
finally
.
It is interesting to note that this theorem gives the best upper bound in particular instances since the equality sign may actually and this is the case hold. This we observe is true only when Q n c= due to J. J. Hadamard an cites i when example j. 0, only s^
=
^
1111
Sylvester:*
11
_1
1
111 111
1
1
Similar determinants of 12 and n 20.
n
for
=
=
Second Proof (Wirtinger)
we
:
maximum
value can also be constructed
Making use of the symbol
divide the elements of the ith
row of
D
as well as of the determinant IT by
(Sj)*. Adopting the convenient abbreviations a lj /(s
b
ii3
we
l
b
)
tj
and
^
so that
replace
D
and
D
by the two new determinants
A=[b,
;
]
and A=[F,,]
,
where s 2 8 M )i
Let us
now
consider the problem of finding the
A A of the 2n 2 variables 6 f/ 6^, subject to the lem is solved by the Lagrange rule as follows :f tion
,
*PhiL Mag.,
vol.
(4.5)
.
maximum
value of the func
conditions (4.4).
This prob
34 (1867), pp. 461475. See historical note at the beginning
of this section. fSee E. Goursat (1904), p. 129.
(Hedrick translation)
:
Mathematical Analysis. Boston,
LINEAR SYSTEMS OF EQUATIONS
We
121
differentiate partially
with respect to the variables b ijy b l} and set each resulting expression equal to We shall then obtain the 2n 2 equations
zero.
where we have made the abbreviations
These expressions are clearly the cof actors of the and A respectively.
By
a simple combination
Then
If
since
A
=V
we now form
(4.6), it is clear that
we
we
= An[A
jy
b'
lt
,
elements in A
shall obtain
have, upon recalling (4.4),
the determinant [A M ] which we shall obtain the relation
[A i} A]
and
fe
is
w equal to A i,* and recall
= A*Ai = Xn[F = X w A 7; ]
ty ]
,
or
i= \n n~ a
X
=
m
=
X w and X is obviously impossible, Hence, since X 1, and the desired inequality is established; that is,
AA It then follows
from
is
follows that
.
(4.5) that
D This inequality
<; i
it
~D
g

Sj
s2

seen to coincide with
sn
.
(4.3)
and the theorem follows as
before.
Third Proof (Boggio) This proof is based upon the possibility of determining a transformation of the elements D and D by means of which they will go into determinants of the same value, :
p=B=
[6
l
j
,
zr=g=[&;;i
,
but with elements so related that
2 b A, = r
,
r
^s
,
r,s
=
1, 2,
,
n
(4.7)
.
i=i
Let us write the desired transformation in the form
*M, Bocher: Higher Algebra. (1907),
p. 33.
*
THE THEORY OF LINEAR OPERATORS
122
ri
= &n + 2 m
= l,2,..,n
>fc
6 fc*
t8
we multiply
>
r,t
(4.8)
fc=i
m
For the determination of the
(4.8)
by
b sl
and sum, obtain
ing
which, imposing condition (4.7), gives
from (4.8) that D and B have the same value since each line of from the corresponding line of D by a linear combination of the elements of the preceding line. The same applies to D and B. Hence by virtue of (4.7) we can write
We
B
notice
differs
But from
and
(4.8)
its
companion formula associated with B, we have
_
ri
ri
n
2 *rt b = 2 ti ri
1
__ ft
r,
=1
7j
r
~ 22
'"rs
Sl
+
t^i
(4.8)
r.
t
=l
we get
ri
_
+
*.,<*ri
2W
,*
**,)
fci
ri
b st\ v .(b v rt
and
,si
i
f
(4.7)
n ri __
i_i
V V ^ ^m sl
__
Sl
,S^l
Summing and noting formulas n
A
6 2 ti
o
Vm
A.J fc1
n
__ , rAt
6 .) "fct/ fr
rl
+ V jLt V w ra m rs A.st r
**
Q
6 at qi
t=i si
(4.10)
Since the second term of the righthand that
we
shall
26 and consequently by
which
is
member
have
r.
bn
^ 2 a OH
(4.9),
once more the inequality (4.3).
r
,

is
positive or zero,
it
follows
LINEAR SYSTEMS OF EQUATIONS Formula only
when
m
(4.10) itself is of interest since
0; that
rs
is to
say,
D
when
is
123
it shows that the equal sign holds an orthogonal determinant.*
Fourth Proof (Molinari) The following proof is interesting in that pends entirely upon geometrical intuitions. 3, let us write Limiting ourselves first to the case n :
it
de
~
and observe the fact that
this is the
volume of a tetrahedron with vertices at the
Since this determinant is in(^,2/ 2 ,^.,), (a%,?/ r 3 ), (0,0,0). variant with respect to any orthogonal transformation, let us pass the ^Taxis points
(a 1 ,?/ 1 ,2; 1
),
through the point then becomes
and the XYplane through the point (x z ,y 2 ,z.2 ).
(or 1 ,?/ 1 ,s 1 )
X,
D3
X
X,
Y2
which
is
the volume of a tetrahedron with vertices at the points
(A\,o,o,), (A' 2 ,r 2 ,o), (A' 3 ,y 3 ,^), (0,0,0) But these elements cannot have a value greater than the longest edge of the tetrahedron extending from the origin. If these elements do not exceed A in .
absolute value, this edge
is
not larger than 
D
3

gA
3
A
35
3 3/2
Therefore
.
we have
.
This argument is generalized without difficulty to the determinant D n a ln ), the A'Yplane [%] by passing the Xaxis through the point (a n a 12 o, through (a 21 e& 22 ... 2n ), and the hyperplane (x, ?/,..., f) through the point ,
,
,
The determinant
we
get

Dn
: 
Pn Dn
then becomes '
[A t; ]
,
where
a tj do not exceed
If the elements
hence
,
,
,
A n n w /2
A
A
~ (;
for
i
<
/
.
in absolute value then 
A
l}

^
An^ and
.
Determinants Which Do Not Vanish. It will be useful for us have the following two theorems relating to determinants with positive lower bounds. The proofs given are due to H. von Kochf although the results were known to L. Levy in 1881, to Desplanques in 1887 and to J. Hadamard in 1903.J 5.
in another place to
'See G. Kowalewski, Determinantentlieorie. Leipzig (1909). der Deutschen MathematikerVereinigung, vol.
\Jahresberieht pp. 285291.
22
(1913),
Comptes Rendus, vol. JLevy, Sur la possibilite de Tequilibre electrique. (1881), pp. 706708; Desplanques: Theoreme d'algebre. Journal de Math. Spec., vol. 11 (1887), pp. 1213; J. Hadamard: Lcqons sur la propagation des ondes. Paris (1903), pp. 1314. See also T. Muir: The History of Determinants (190019<20). London and Glasgow (1930), pp. 6869. 93
THE THEORY OF LINEAR OPERATORS
124
3. The determinant A [a ty ] is different from zero the diagonal element is in absolute value larger than the of the absolute values of all the nondiagonal elements in the
Theorem for each
if
sum
i
same row; namely, n
il
+ 2 a^ By assumption, since a^ ^ 0, we can set a >2 k\
a
I
Proof:
k
^
i.
If
I
we then adopt
fc*
If
we make
we then have by
e=
tk /a* %
b lk ,
the abbreviations
^
in the
!
I
fcni
1
we can write A
I
I
,
t
,
i
= ^
fc
k
form
the further abbreviations
the ordinary properties of determinants,
D(b)
=Z?(6 (2) )
,
Multiplying these equations together and removing the factor common to both sides, we obtain
(5.1) n
But by hypothesis
or
more
^2&ifc
Si e
<
1,
and hence
w^e get
generally,
2 k
We
st
6 4fc <> '
^e^ilfr.*
=e
thus reach the result that lim b tk (v)
the definition of
v l fi,
=
voo
D(b
(5.2)
A,= i
i
(v)
)
we
conclude that
limD(b (v) )
=1
.
*See Kowalewski: Determinantentheorie, pp. 1267.
and from
this
and
LINEAR SYSTEMS OF EQUATIONS
Making use
we
of this fact
see
from
125
(5.1) that if
p be chosen
sufficiently large the right member of (5.1) does not vanish and consequently none of the factors of the left member is zero. From this
conclusion the nonvanishing of D(b) in particular is proved and with it the nonvanishing of the original determinant A.*
Theorem
Under
J+.
inequality holds
A ^


an
=2
s,
I
&**;
o* n
o, 2
n
n
2s i=i
where S
the assumptions of theorem 3 the following
:
I
,
^
e*(Ie) 8 '*
<
,
1.
fc=i
Proof: To prove this log
we
0(6)
start with the expansion
=
v i
&<">/*
.
vi i=i
The proof of this essential formula ab initio would require a considerable amount of algebraic manipulation which would be equivalent to developing the wellknown formula of Fredholm for the case where the integrals are replaced by sums:
where we abbreviate n
=
f ^a
f*
Ja
/.&
J
in
which we use the customary notation
If
we
replace this transcendental expansion by the expanded form of the
determinant
^Theorem 3 value of
i
is also true if there exists an index p so that the inequality
we have
Without essential extension of the argument already given, the formula
D(b)D(qb)D(q*b) where q
e 2vi /P
.
D (qP*b)
for every
this follows
from
THE THEORY OF LINEAR OPERATORS
126
according to (2.2), we see the essential equivalence between the algebraic series and D(\). In this situation the integrals defining A v are replaced by
=2
Setting X
=
1
and replacing K(x^y
log
we
of (5.2)
Making use 
D(b)
1
^
1
)
by
6
i;
we obtain the
,
desired formula.
reach the inequality (e
2 s,/2
+
e
2 s,/3 +
2
)
Hence we attain the inequality
from which the general theorem follows as an immediate consequence. The application of these results is made to infinite determinants TI
if
we assume
7")
an
that both 77
6.
2
\
au
\
converge.
iji
i .i
now
and
The Method of the LiouvilleNeumann Series. We proceed methods for solving the set of equations (1.1).
to a discussion of
Three essentially distinct procedures have been developed for this purpose. The first one is based upon the convergence properties of the LiouvilleNeumann series the second, which we shall call the method of segments (methode des reduites), considers the limiting form of the solution of a finite sot of equations when the number of variables becomes infinite; the third is based on the theory of infinite bilinear forms. Historically the first method was suggested by the solutions achieved by J. Liouville (18091882) and C. Neumann (18321925) in the theory of integral equations; the second was essentially em;
ployed by
J.
Fourier (17681830) in the celebrated problem treated by D. Hilbert in connection
in section 8; the third was inaugurated with his treatment of integral equations.
The following theorem tal in all applications of the
due to H. von Koch and LiouvilleNeumann series:
is
is
fundamen
LINEAR SYSTEMS OF EQUATIONS
Theorem
5. If in
127
(1.1) the quantities x\
system
and
6 satisfy the
conditions
x
where
X
and
B
<X
t


b,
,
]
\
a,i
,
j
are finite positive magnitudes, and
if,
further,
we
have
2

^

j^l

d tl
<
,

<
1
,
then there exists one and only one solution of system (1.1) solution can be developed in the LiouvillcNeumann series, x,
=
+ 2 b,,b/ + 2 b,/
&,'
;
2)
+
&/
and
this
(6.1)
,
j
where we employ the abbreviation Proof: In order to prove this
[J=
6/1
+2
we
write
+26.,
&.,&/
Employing the inequality
we
get
* 1

^
B/(l
and the assumption
(5.2)
The convergence
e).
V+
(2)
;
>
of
2

&t; [*j]
enables us
/
to write
2
&i,
^
=2
that
is
&/
&,;
+26
<
2
>
&/
1;
+ = ...
&/
fj
;
j
j
;
to say,
20,^ = 6,

y
In order to prove that this solution is unique we designate by any arbitrary solution of the original equations and then #3, form the sum #1, ^2
q,
Or,)
= + 2 &H *Hh 2 s,
Substituting 6/ for
=^
(p)
(ai)
.r,
we
&.,
get
^ (p) (2
6 i;
.r
y
)
=
a?,
2 &a
;
Making use
of (5.2)
we
(J>>
;
;'
reach the inequality
/
/
THE THEORY OF LINEAR OPERATORS
128
Xi
and hence, since
J.
to the
= lim
tl
(p)
we
(&/),
get
L. Walsh has applied the method of successive approximations LiouvilleNeumann series for the triangular system"
x
+a
t
12
x, f a u
=
x,, 
6,
,
(6.2)
The resulting theorem cannot be derived as a special case under the conditions stated by von Koch, since limitations imposed upon the constants bi are essential to the convergence of the series defining x k .
The
results of
Walsh are stated
in the following
theorem :f
Theorem 6. If there exist positive constants that the coefficients of (6.2) satisfy the conditions 6fc!
00
^MW,
2
a
fc
,l
gP
,
=
fc
6
.
1, 2,
,
6,
M, and
< l/P
b^l
,
then system (6.2) hus one and only one solution for which
9
P
such
,
.r*
<
*It should be observed that the general system (1.1) can be reduced to a diagonal system by means of the following transformation provided the principal do not vanish: (diagonal) minors of a l} \

*ll "12 C
nk~ a nin2" a *,ni^
In this manner
we
i
obtain the diagonal system
t'
1)0
t'.,
l_ "i
1
c .),>
ct
,
_ ~i
i
7
I
co
zt
.
I*

a.y
,
This transformation is due to Th. Kotteritzsch Zeitschrift fur Math, und Physik, vol. 15 (1870), pp. 115, 229268. See also, F. Riesz: Les systemes d'equaiiovs :
lintaires, (1913), Paris, pp. 1112.
fOn the Solution of Linear Equations in Infinitely Many Variables by Successive Approximations. Amer. Journal of Math., vol. 42 (1920), pp. 9196. JThis theorem may be slightly generalized by replacing the second condition with
kj
f r
convergence of the series for
k greater than a all
values of
k.
fixed K, assuming, however, the
LINEAR SYSTEMS OF EQUATIONS
129
Proof: Let us consider the following set of approximations:
xk ^
From
=b
this set
k
k =1,2,...
,
we form
,
the expressions
aW Making use
1
')
+
(6.3)
]
of the conditions imposed by the theorem
we
obtain
the majorant
Af 6 4fc
P M b k+
= Mb /(lPb)
+P M 2
*
*+'f>
+
k
where the symbol less
has than in absolute value."
The
(6.4)
,
customaiy meaning
its
"is
term by term
set of values {x k } is easily seen to furnish a solution of the
we add the equations of (6.3) and double series by columns, we get convergent absolutely original system since, if
sum
the
JC
Zfc
2 a*, jkn
bk
Xj
.
The bound for x k stated in the theorem is obvious from (6.4). The uniqueness of the solution is obtained from a consideration x k n where x k and x^" are solutions of of the differences y k xif
=
'
y
(6.2) under the conditions of the theorem. Obviously the set {y k } furnishes a solution of the homogeneous system: 2/i
+
fc l2 2/2
1/2
We
+a +
l3 1/ 3 
023 2/*

+
1==

employ successive approximation
=
,
,
to solve this system,
obtaining
2
^
(i
i/;
>
,
i
(6.5)
=
i,
1, 2,
thus
THE THEORY OF LINEAR OPERATORS
130
Noting the inequality
yk
^ N X*, X <
\
1/P,
Xg
1,
we
con
clude that
which converges to zero as i approaches infinity. But from (6.5) we (1) t= have y k (i) 0. This result y k and hence we see that y k
=
i/fc
,
establishes the uniqueness of the solution obtained for (6.2). The actual solution obtained by this method is the LiouvilleNeu
mann ful
series previously stated in (6.1).
The following generalization of the theorem just proved is usein application and may be established by arguments essentially
similar to those employed above: 00
Theorem
7.
If system (6.2)
gM
is
+
such that
2
K;i
p
=P
p
for every
value of k, b k b k b < 1/[1 P^D] <*D/P, then system (6.2) has k 1 a unique solution for which x k g , g < 1/[1 f P?/^ )] ^ D/P As in the* previous case the actual solution is obtained as a LiouvilleN'eumann semes. ,
\
^M
.
7. The Method of Segments. The following method scribed as one which defines regions of validity for
lim x
(m) l
= x,
i
,
=
1, 2, 3,
m

,
may
be de
,
n=oo
where the values {x } are the solutions of a set of m linear equations m unknowns. The following theorem was discovered by A. Pellet in 1914* and independently by A. Wintner in 1925,f although the fundamental idea was developed in 1899 by E. LindelofJ in discussing the problem of l
in
the existence of implicit functions:
Theorem
8.
If in the system x
^=
t
c,, i
1, 2,
,
w, where
ki
the constants
c
t
are bounded and the coefficients
arc subject to the
a.k
condition S,
= 2M<1, = l,2,..,cc i
,
(7.1)
*Sur la methode des reduites. Bulletin de la Societe Mathematique, vol. 42 (1914), pp. 4853. fEin Satz iiber unendliche Systeme von linearen Gleichungen. Mathematisclie Zeitschrift, vol. 24 (192519'26) p. 266. See also: Zur Hillschen Theorie der Variation des Mondes. Ibid., pp. 257265; in particular, p. 265 et. seq. ^Demonstration elementaire de 1'existence des fonctions implicites, Bulletin des Sciences Mathematiques, vol. 23, 2nd ser., (1899), pp. 6875. ,
LINEAR SYSTEMS OF EQUATIONS
131
then the x exist and are equal to the limiting form, as (m \ determined by solving the reduced system t
m
*
of
GO
Xi
_2a
ar/m)
t
xk
,
^=
c>
=
i
,
1, 2, 3,
m
,
(7.2)
.
fci
Moreover,
C is
if
sums S
the
numbers
the largest of the
c
lf
t
t
and S the largest of
,
\
then the solutions x are subject
to the limiting condition
Xl
The proof
Proof: the system
A (#
x,
of this theorem depends upon the fact that
x.2 ,
t,
,
xn )
=
i
,
1, 2,
n
.
,
if
(7.3)
,
# M ) are analytic functions of the where the f,(x lf x 2 variables, has a majorant system X, F (X lf X 2  X H ) = 0, where the funcX n ) are positive dominants of /, (# 1( a\, tions Fi(Xi, X 2 n ), >?
,
,
l
,
and if there which X>
,
,
,
,
exists a set of positive values
F (X X t
l9
2
,
X
,
exists a set of solutions aV^,
v ^V
(o)
^> '
j,
i
r **/
o; 2
(o)
are
n) ,
I
j,
,
,
6
 1 JL
,
9 ^,
* * ... *
,
C
^
c,
.
Now
let
,
(0>
n
;>:
for
* ::
7? /t
.
In order to make application to the system x, sider the majorant system,
and
,
X
positive or zero, then there (0) iP n of (7.3) for which
_
;,
,
0)
all
(0)
i
AV
AV
0)
;
t
^^
c
t ,
we
con
us assume the restrictive inequality uO
SA < lfc
1
.
fcl
Then
C'
if
is
the largest of the
numbers \d
,
and
S' the largest of
(SJ
the
sums
X
(0) t
2
^^, it is clear that the set of values
="^.
= CV(1
Hence we are able
SO
AY
0)
:
A% (0)


will satisfy the inequality
to infer the existence of a set of solutions of
the original system.
same reasoning applies to the reduced and hence the method of segments may be employed in
It is also clear that the
system (7.2)
obtaining the solution of the original set of equations. The method of segments is closely related to the well known rule of Cramer, which applies in the ordinary finite case. This rule is gen*The paper by Lindelof gives essentially this theorem. It is also given by Des equations majorantes. Bulletin de la Societe Matliematique, vol. 37
Pellet:
(1909), pp. 93101.
THE THEORY OF LINEAR OPERATORS
132
to obtain the solutions of the successive
employed
erally
segments of
the infinite system. 8. Applications of the Method of Segments. The efficacy of the method of segments in attaining results in the transcendental case from the algebraic case may be illustrated by the following elegant solution of the problem of the elastic plate. This solution is due to H.
W. March.* The problem
of the elastic plate is concerned with the determination of a function V(x,y) representing the deflection of a uniformly
loaded rectangular plate of dimensions (a,b). In the theory of elasshown that V(x,y) satisfies the differential equation
ticityf it is
V*V(x,y)
=A
(8.1)
,
where p is the uniform load, h is half the thickness of the plate, and and
E
plate.
V (x,y)
When the plate is clamped at the edges, following conditions:
V=Q ay
,
=
x
,
=Q
,
=
,
x
=a
x
=a
y
=
dV
=
;
;
,
,
y=b i/
is
subject to the
(8.2,a)
;
=
y
,
=b
.
(8.2,6)
?y
8.r
Let us
now
consider the function
where we abbreviate V*(x,y)
V
2
(x,y)
=Px(xa) y(yb) P = A/8 PCS K/ (A6 + sinh **&)][ (y
,
,
b) sinh X n y
n
+ y sinh L (y X
[ (
x
a)
It is seen that the tial
b)
]
sinl1 /<.#
sin A n x
+2
t6
^/
u
(<
<
a
+ sinh
in
+ ^ sinh
(
/'
x
a ) 3 sin
/'"'^l
function thus defined satisfies both the differen
equation and the boundary conditions
(
8.2,4 ).
The problem,
then,
*The Deflection of a Rectangular Plate Fixed at the Edges. Trans, of the American Math. Soc., vol. 27 (19125), pp. 307318. fA. E. H. Love: Theory of Elasticity, 3rd ed. (1920), p. 496.
LINEAR SYSTEMS OF EQUATIONS
133
determine the coefficients so that the conditions (8.2,&) are also satisfied; we have
is to
= P{ay(yb) +2
(dV/dx)\^
X (yb) [
sinh
i Hy
+ y sinh X (yb) n
2w
= P{
r
(3V /3i/) Uo
b x
6
(xa)
2
<&
sin A n .r
i m a) ] [
a) sinh
(x
fi
mx
n b)]
]
sin p m y} =
ft
+ sinh
+ sinh A
Wn^/(A n &
n
(8.3)
;
+ 2m
+ x sinh
["m&m/ (n^a
//
w(
a)
] }
==
0.
(8.4) It will
the edges x
be seen that these equations are also those which hold on
= a and y t=^b. will
Equation (8.3)
be found to have the solution
r Jo
X
&) sinh
[ (3/
>1 H 7/
+ y sinh X
n
(y
b)
] }
sin
Considerations of uniform convergence show that to interchange the order of integration
equation.
We
/i w
it is
possible this
and summation signs in
thus obtain the system
(8.5)
where we employ the abbreviation
B(A n 6)
= (1 + cosh
w e obtain from
An
6)/U n &
.
by symmetry the system
l
Similarly,
+ sinh Anb)
(8.4)
(8.6) It
we
set
i>
c= >j
a,
^
r/
^
case of a rectangular plate, equations (8.5)
and
we
consider the take the fol(8.6)
that is to say, if
1,
lowing forms: ba
= (87f a*/n* rf) (Sn
if In}
(8n ijV")
^[m 2 [w
2
2
am
6
R(mnr])/ (m
B (w n/ij) / (
2
2
2
tf
+n
2
2 ,
2
)
]
.
THE THEORY OF LINEAR OPERATORS
134
For the square plate, down become
=
1,
bn
2 A nm
bm
,
q
i.e.,
=a
n
the two systems just
,
written
b n =,
K /n*
n
=
1, 3, 5,
??i
=
A
8a3 3 where we abbreviate K The first four equations of
,
=
4
this
# (m?*) / (w ) reduce system explicitly to the (8A) nra 2
2
+m
2
2
.
following: 1.546&1
f0.229&3
0.0656!
+
+ 0.094& + 0.050& + " = 5
K
7
,
(8.7)
+ 1.128& + 0.1146 + = 008 K, = .00292K 0.0061&! + 0.048&3 + 0.081&5 + 1.0916 +
0.016&!
+0.0996 3
3
.
7

.
7
FIGURE
From
the explicit formula for
1
A nm we
A nm .< [8nm A(^ 2
2
have the inequality
+m
2
2
]
)
,
and hence, summing over the odd integers,
2 A nm < 2 8 n m/n (n + m 2
Joo
[8nm
2
2
2
)
A(n +m 2
2
2
)
]
dm
.
Since the value of the integral is exactly unity, it follows at once we may apply theorem 8 to obtain a numerical solution of system Since the righthand members are obviously bounded, the solu(8.7)
that
.
tion will also be bounded.
Applying the method of segments
we
find the values
LINEAR SYSTEMS OF EQUATIONS 0.6475K, &
^_0.0040K,
&5
=
0.0001K, &
0.0003K, 6 lt
135
O.OOlTtf, 6 7
=
=
0.00006K,
0.006K, 
When made
these values are properly substituted and a computation of the displacements of the elastic surface
V(x,y)=V (x,y)+V 1
for
P<= 1,
O.
&
=
1,
the following table
t
is
(x,y)
,
obtained:
A graphical
representation of these values is found in figure 1. In the example just discussed, the conditions assuring a convergent solution of the infinite system of equations were fulfilled. It is interesting to observe that solutions can be obtained by the method of segments in certain special cases where the solutions fail to converge. The following example, originally due to J. Fourier (17681830), illustrates this point* We follow a modification due to F. Riesz.f
Let us solve Laplace's equation 32V
D2V
for the case where V(x,y)
V(0,y) Obviously
=1
,
we can
is
V(x,
subject to the conditions :
yn)
= 2 x*
e (
mi
the constants x m to be determined
*
limV(x,y) X
=0
jT
write V(x, y) as the series QO
V (x,y)
,

m "* cos (2m
1) y
,
from the conditions
See his Theorie analytique de la chaleur. (1822), arts. 171178. systdmes d' equations lineaires a une infinite d'inconnues (1913), pp.
26.
,
THE THEORY OF LINEAR OPERATORS
136
We now
proceed formally. Differentiating this equation an inof times and setting y in each equation, we obtain the following system finite
=
number
:
Clearly the conditions of theorem 8 are method of segments, however, we find
_ 2m
violated.
Applying the
1
ra+fc '
.
m
I
_

824
(4k
2
4 CX>
4k)
It thus follows that
.
3 3 rv)
y2
,
5
"m
n
'
14
.
__ 5
n
2m +1
2m
1
n
and we obtain the wellknown solution V(x,y)
= (4/n) 2
l)
(
w  1 e (2w  1) 'cos(2??i
I)y/(2m
1)
.
mi
we now substitute the values of x m in the original system, it we obtain divergent series in all but the first instance. we if However, employ Borel's method of summation* it is possible If
is
clear that
to
show that all these divergent For example, we shall have
2(2m
l)a? m
c= (4/n)
= (4AO(1 fV' (tt*/2 Jo
*See section
4,
series are
Chapter
5.
3
!
+5 + t*/4
7 !
summable
+
to zero.
LINEAR SYSTEMS OF EQUATIONS
2 (2m
f V' tcostdt
1) *x m t= (4/ji)
Q
137
;
Jo
we admit the validity of Borel summability we see how it is possible to give at least application which we have discussed here. If
uation,
in the present sita rationale to the
9. The Hilbert Theory of Linear Equations in an Infinite Number of Variables. The theory sketched in this section was initiated by D. Hilbert in his classical papers on the theory of integral equations
published between 1904 and 1910 (See Bibliography). These fruitful ideas led immediately to an extensive development by E. Schmidt, 0. Toeplitz, E. Hellinger, I. Schur, and numerous others. A short bibliography of some of the important memoirs will be found in chapter 12.
We shall begin by introducing the concept of Hilbert space. By such a space is meant an infinite array of numbers #,, x 2 x^, $n, which obey ordinary associative and commutative laws and sat,
,


,
isfy the condition
M
where is finite. by assuming that
obvious that no essential restriction
It is
M t=
is
imposed
1.
This concept has been broadened by a number of writers prominent among are J. von Neumann, M. H. Stone, S. Banach, and T. H. Hildebrandt. A complete account of these recent developments will be found in the treatise of Stone: Linear Transformations in Hilbert Space (American Math. Soc. Colloquium Publications, 1932), viii + 622 p. The following postulates are due to von Neumann [See Bibliography: von Neumann (1), p. 14 et seq.]
whom
A
class,
H, of elements
/,
The elements form a
I.
is
g,
called a Hilbert space provided
linear space;
associative operations for the elements
/
+
= /0 = 0,0/ = 0.
i.
and a
e.,
:
there exist commutative and
null element with the properties
/,
There exists a numericallyvalued function (/,#) defined for every pair II. of elements with the properties: (1) (af,g) (/,#), where a is a complex num== (/ 2 ,0) (3) (gj) (4) (/,/) 0, the / 2 g) (/^) ber; (2) (/, (/,,<;)
=
+
,
equal sign holding only
if
/
=
+
;
For every value of n there exists a III. ments of H. IV.
H is separable;
{/ t }, such that for every
i.
g
e.,
in
=
;
^
0.
set of
n
linearly independent ele
there exists a denumerable, infinite set of elements and every positive e there exists an n for which
H
THE THEORY OF LINEAR OPERATORS
138 V. > 0,
H is complete;
m,n
H
i. e., if a sequence satisfies the condition / w {/J of then there exists an element / such that / 0, n /J
> oo,
The Hilbert theoiy
fn Q
\
.
of the inversion of a system of linear equa
tions takes its departure from the related problem of the definition and inversion of the bilinear form y~
A(x,y) t=2a*y x I
i
t
y,
.
i
Definition: The bilinear form A(x,y) is called limited if there exists a positive number M, independent of n, x, and y, such that if arand y both belong: to Hilbert space, we have
The form n
A n (x,y)
^2&u x
t
yj
iji
called the nth
segment (Abschnitt) of A(x,y). for convenience much of the following discussion will Although assume that the coefficients and variables in A (x,y) are real, most of the theorems will apply with equal validity to the Hermitian form is
en
__
=2 a u
H(x,x)
__
%i %i
,
ij=i
where we assume
The bars over the a and the x denote the a yt a of and n complex conjugates x, respectively. In general we shall designate a bilinear form by the symbol A(x,y), the corresponding quadratic form by A(x,x) and the matrix of the form, namely a u by A. In some cases, however, where no be it will convenient to represent bilinear and ambiguity results, a\ tj
.


,
quadratic forms by A. The form conjugate to A(x,y), that is to say, with matrix equal to A', will be designated byA'(x,y) The unit form, that is to say, the form with matrix 7, will be represented by I(x,y). The following theorem is fundamental in the theory of limited .
bilinear
forms
:
n
co
Theorem
9.
If
A (x,y)
=2
,;
x,
y and }
B (x,y)
i;ri
iji
limited forms, then the product
form 00
C(x,y)
=2 CijXtVj ij^i
where C
A B,
that
is,
where
=2 b
,
i}
x, y,
are
LINEAR SYSTEMS OF EQUATIONS
is also
a limited form.
Proof: In order to show this
12 I
where
139
M
This
[
il
it is
2 a* &*J
necessary to prove that
\^M
aMf,
,
JCl
a number independent of n, x and y sum may be written in the form
is
l9
2 <x*a*)( ;=1 2 &*/,)
fc
Lemma.
(9D

il
1
(The Schwarz inequality).
are two sets o/ real numbers, and
u * f uS \1
.
}
Ur?
u lf u2
If
,
and v l9 v 2
,
if
+
+v
and vf
2 2
\
\ v n 2
+
converge, then UiVi{U
and
also converges
h U n V n
\
satisfies the inequality 00
00
I2pf,^ J\ p=l 2Xr\\ /
n
2
Proof: Since 2
(A
^P
+
/*
2 ^P ^ + 2 V + 2 A p=l /i
2
V P)
2
,"
follows that the discriminant
(ittp^p) p=l
2
must be
^ 2
2
^;>
^
,
either negative or zero
ivi^v^o 7>=1

ptl
p=l
?>=!
it
00
2V
/
Jfcl
^
\
;
i.e.,
,
p=l
from which we derive
g JV 2 V J\ 2 V p=i
From
the assumption
hand member the lemma
made
.
p=i
as to the convergence of the left
at once follows.
The Schwarz inequality was given by H. A. Schwarz in his memoir: tiber ein die Flachen kleinsten Flacheninhalts betreffen des Problem der Variationsrechnung. Acta soc. sc. Fennicae, vol. 15 (1885), pp. 315362. This inequality for finite summation, however, was given by J. L. Lagrange for three terms: Nouv. Mem. Acad. Berlin (Oeuvres, vol. 3, p. 662 et seq.) and by A. L. Cauchy: Cours de I'ecole polytech., Analyse algebrique (1812), note 2, theorem 16 (Oeuvres, vol. 3, 2nd ser., p. 373 et seq.) for the general case. The theorem is thus often referred to as the LagrangeCauchy inequality. d* analyse
THE THEORY OF LINEAR OPERATORS
140
We note that the Schwarz inequality holds mutatis mutandis for integrals. Thus, if u(t) and v(t) are real, continuous functions of t in the interval (ab), then the following inequality holds: I
C
C'u(t)v(t)dt[*^ Ja
C
\u(t)\*dt
v(t)\*dt
.
Ja
Ja
This inequality can be generalized from the criteria that a real quadratic positive definite, that is to say, that the quadratic form
form be
In order shall assume no negative values for any values of the variables x x for the quadratic form A to be positive definite, it is both necessary and sufficient that the n principal minors .
t
A
}
=K
shall be positive or zero.
now employ
Let us
the abbreviation
(uv)
From
=/: I
U (t) v(t)
dt
the obvious inequality
where u^t)
,
interval (ab)
^(t) we derive t
,
,
u n (t) form a
set of real continuous functions in the
the positive definite quadratic
form
where we abbreviate
U =(u tl
t
u,)
.
Hence the sequence of determinants
must be
positive or zero.
The determinants
Gram who
U
lt
The Schwarz inequality
U.2
,
employed them
,
Un
are called
is
clearly the special case
Gram
determinants after
in his notable
J. P.
paper: tiber die Entwickelung reeler Funktionen in Reihen mittelst der Methode der kleinsten Quadrate. Journal fur Math., vol. 94 (1883), pp. 4173. The Schwarz formula has also been extended in another direction by O. Holder,
first
who
established the following inequality:*
*Uber einen Mittelwertsatz. Gottinger Nachrichten (1889), pp. 3847.
LINEAR SYSTEMS OF EQUATIONS

2 up v
g V
m p
{
\
Up /oii>}i

The Schwarz inequality
V
Up

+ vp
^(21
} 1/m
m~
the special case
is
7?l
vp

)
m>
,
1
.
pi
Closely related to the Holder inequality
{
2
{
7^1
;;.i
141
H
u,,
')
l
the following:
is
+
1/m
2.
(
2

vp
w ) 1/w

w>
,
1
,
p=l
7>l
which was established by H. Minkowski in 1907* and which kowski inequality.
is
called the
Min
For the case m 2 it follows as an immediate consequence of the Schwarz Thus we have
inequality.
and hence
V .
f
2 M "77
I I
12 I
2
_i_ ** r
from which
it
I I
YM /j
+Y
v pi I
'^p
v
I
^_j
'
I
2
p
I
"
2
wp
1
<
I
I
V
4 2
( ^ Z.J
'
I
!
2
?i

7?
i
Y
I
^_j
I
I
2
v 7?
! I
)*y jLi '
'
I I
vp
2 I I
follows that
V v
.
j
u
I I
+ v p! < 2
'
/;
{

( ^
*
P 1
y '
j
1
2
7^
I

7J
I
5
) '
+ y '
( ^
Pl
^i
I
2
v
I
I
/)
I
5
) /
} *
.
T'7"!
Returning to (9.1) we see that the Schwarz inequality gives as an upper limit to the bilinear form the value
Js
A 1
'A
We must now
show that the
fact that
^a
t)
x y} l
is
limited implies
i/=i
2
that
(
2
,^
.rj
2
gM
form and independent
By
Since
2 ;=i
n
2
(
,
where 3/
is
the upper
bound of the bilinear
of n.
hypothesis
2 2 in
2
i^l
fc=l
&i;
fliy
^.
!/y
I
^^
2 *>
2
2
we may write the lefthand member of this inequality as %i)yj, we may think of it as an ordinary linear form which
i=i
becomes a linear form in an But if we assume that
*
infinite
Diophantische Approximation.
number
Leipzig,
of variables as
(1907),
p.
95.
m

oo.
t
THE THEORY OF LINEAR OPERATORS
142
when XS
+X
2
2
2 A? ^
< 1, it follows that
]
M
2 ,
because, by
1=1
the Schwarz inequality,
=
+
+
^U 2 the equality sign preMoreover, for Xi Ai/V ^.i 2 vails since the upper limit is actually attained.
Hence we get
M
N
It then follows that if is the upper bound of A, and that of N.* B, the upper bound of C will not exceed The two following theorems are also essential in the theory of limited bilinear forms:
Theorem
10.
00
2c ~

a i;
the matrix of a limited
is 
1>
then
;^l
2
%i
converges, where x
2
t
=2
By
&>/
7
the Schwarz inequality
we have
But we have just proved in the preceding theorem that this converges under the hypotheses assumed above. The form
tl.
2^
sum
cr>
(fj
Theorem
if
j=i
il
Proof:
form and
A
00
2
j
If
M
#<
^
1/y
.
limited provided
2 au
2
^ 0?i
;!
tyl
verges.
Proof: The proof follows from two applications of the Schwarz inequality:
(
2
2 i
i/
y,)
=
[
i;_l
2
(
^2
2 ^ ^)yj
ti
;t
./'
iyi
2 1^1
*.*
2
^2(2 ^; ^)
g 2 a/
ji
t=i
2
2^
2
;~i
2
ty^i
We now shall
turn to the problem of solving system (1.1), where we impose the conditions: (a) that the {6,} belong to Hilbert space; oo
(b) that
2
\a*u\
converges;
(c) that the solution {$>} shall also
belong
;=i
to Hilbert space *E. Hellinger and O. Toeplitz: Grundlagen fiir eine Theorie der unendlichen Matrizen. Math. Annalen. vol. 69 (1910), pp. 293001.
LINEAR SYSTEMS OF EQUATIONS Let us
now
143
consider the infinite matrix
AHKIIIf
it
turns out that a second matrix
<=
<&

where i
^
/ is the unit
matrix 
j,
then
larly, if
$> is
W
in
is

1, i
i;
=
j,
and
<5^
= 0,
v=i
,
the backward inverse (or reciprocal) of A.
*P is
Theorem exist for
<5
forward inverse (or reciprocal) of A. Simia matrix such that
A then
which
\\
exists such that 
called the i/ t;

d tj
12.
If
backward and forward inverse matrices both
A, then they are identical.
Thus we have
Theorem
13.
matrix, then this
Y^IV^V
,
If there exists one and only one also the backward inverse.
forward inverse
is
Proof: Let us assume the contrary; namely, that has the properties:
0A = I
<
is
unique, but
,
A
.
Then we have
A$A = A(
=
A
=B
/ Hence we obtain, A, and follows, letting A be any parameter, that we
_/)] Thus
,
(B have
A=I
/)
A
0.
It
then
.
not unique and the theorem follows from the contra
diction.
We next observe that if A has a unique inverse, then a unique solution of system (1.1) will be given formally by 00
x,
= '2
t
.
(9.2)
ll
remains for us to see under what conditions this solution actually exists and satisfies the restrictions imposed by the problem, It
THE THEORY OF LINEAR OPERATORS
144
The following theorem due
to 0. Toeplitz* supplies a set of sufficient
conditions:
Theorem 14 Let A(x,x) be an infinite quadratic form with A which satisfies the following conditions
matrix
:
A(x,x)
is
(2)
A(x,x)
is positive definite;
(3)
The roots of the equation
(1)
limited;
A n !=0
I/I/,
where In and
An
(9.3)
,
are the nth principal minors of
have zero as a limit point. Then a unique inverse exists for
A
which
and
I \
A
,
do not
\
is also limited.
Proof: Before considering the details of the proof, illuminating to discuss the following example Consider the system
it
might be
:
n
#w
=
c
rt
=
n
>
1, 2,
00
Since
2
2 l/^ converges, the matrix of the associated form
is
lim
=
ited,
is clearly not limited. Hence the but the inverse matrix, i d l} x n c= n c n do not necessarily belong to Hilbert space since 
solutions,
,
\\
,
00
00
2n n^l
2
c n 2 is
not necessarily bounded
when 2
cn
2
:g 1.
ni
Equation (9.3) reduces to (/jLl) (fil/2)
and zero
is
(/i
1/3)


1/n)

=
seen to be a limit point for the roots as n +
oo
.
Thus a
limited inverse does not exist.
The details of the proof consist in the explicit construction of the inverse of the limited bilinear form 00
A(x,x)
Y
ah
x, Xj
,
ij=i
which we
shall also
assume
to be positive definite
satisfies condition (3) of the
and whose matrix
theorem.
Let us designate by A n (x,x) the nth segment of A(x f x) and by the matrix of the form. The variables x l9 belonging to Hilbert space, may, without loss of generality, be assumed to satisfy the con
An
ditions:
*Die Jacobische Transformation der quadratischen Formen von unendlichGottinger Nachrichten, (1907), pp. 101109.
vielen Veranderlichen.
LINEAR SYSTEMS OF EQUATIONS
145
Adopting the notation that A (r) is the determinant of the matrix of the rth segment of A (x,x) and A l} {r) is the cofactor of the element a tj in A (r \ we now construct the following transformation due originally to C. G. J. Jacobi (18041851) :* v r c= bS r >u*
+ & % + 6, (r
(r)
2
2
^3 H
h r
b,<
r}
u
r
,
= l,2,3,.,n
(9.5)
,
where we abbreviate b> It
>=A
(r
r r ><
r (r >/[A "A< >'}*
(9.6)
can then be proved that
and since the matrix of the matrix
B
n
=
[
Bn
is
the reciprocal of A A n by the equation *
the reciprocal of
Bn
'1
(B
f
n
we
tt ,
Bn = A n
B' n Similarly, if
is
related to
is
t

form
this
6 0)
seo that
(9.7)
.
Z? n ,
)^=A
we have n
the equation (9.8)
.
It is now necessary to anticipate a theorem established in section 2, chapter 12. The roots of equation (9.3) we shall call the characteristic numbers associated with the matrix A n From (2.22) .
and inite
(2.23) of chapter 12
we know
that
if
A n (x,x)
a positive def
is
form we have
Max A n (x,x) MaxA,,
where
M
1
M
n
MinA n (x,x)=m n
,
^) =l/m
,MmA
w
n *(x,x)
(9.9)
;
*=l/M n
,
(9.10)
m
n and n are respectively the largest and the smallest of the roots of equation (9.3). This theorem applies equally well to the infinite form A(x,x).
Employing this fact and noting condition (3) of the theorem, we are able to show that A (n} > for all values of n. If this were not the Ar case and if one determinant, let us say A u) were zero, then // / x x would vanish for ^ of would Hence a set 0. r values, Xi, 2 exist in Hilbert space for which we should have A r (x,x) 0. Setall we should then have values of x to other zero, ting equal A r+tn (x,x). t= 0, m 0, 1, 2, An oo. Hence the equation I /n would have /n as a limit point, contrary to the original ,
=
=
= =
\
,
,
=
,
,
assumption. *t)ber eine elementare Transformation eines in Bezug auf jedes von zwei VariablenSystemen linearen und homogenen Ausdrucks. Journal fur Math., vol. 53 (1857), pp. 265270. Also Gesammelte Werke, vol. 3 (1884), pp. 583590.
THE THEORY OF LINEAR OPERATORS
146
this we conclude that the Jacob! transformation (9.5) the coefficients b 0) are real. Moreover from (9.10) that exists and
From
t
we
obtain
B
n
(x,x)
From
^
\/Max B' n (3c,x)B n (x,x)
this
we
= Max A
conclude that B(x,x)
is

n
limited, hence B'(x,x) is
limited, and from theorem 9 above, their product is also limited. In similar manner, employing condition (1) of the theorem, it l is easily proved that B (X,X), [B~ (x,x)] and their product are also f
I
limited forms.
The following two equations
A
A**=B*(B')*B'B
complete the proof of the theorem. In the argument just set down
was shown that a unique limited inverse existed for a quadratic form of a special type, namely, one that was positive definite. It is possible from this conclusion, however, to derive both necessary and sufficient conditions that a general quadratic form have a unique inverse. These conditions are it
set forth in the following theorem:
Theorem 15. A real limited quadratic form A(x,x) possesses a limited forward inverse if ami only if A' A does not have zero for the limit point of its characteristic numbers. The form also possesses a limited backward inverse if and o
We
its characteristic
shall first
numbers.
show that the condition
there exists a forward inverse X(x,x) with matrix A *= / (x,x) that is to say,
have
X
is
necessary.
If
then
we
\X
l}
, \\
,
2
2
ffi
=2
7=1
2 Xi, x 2 a* #*) *
<
ti
;i
fci
From
the Schwarz inequality and the fact that the variables beHilbert space, we then obtain long to OO
00
S*, 2 i=l that
^ [2 y=l
OO
GO
(S*i, *) 2 il
;l
.
fc=l
is,
[Max (XX')](A'A)
From is
OO
2 (2 <* **)*]'
the assumption that
a positive number
;
hence
it
X
exists,
^1
.
that Max (X X') cannot be zero and
we know
follows that A'
A
LINEAR SYSTEMS OF EQUATIONS
147
thus the spectrum of characteristic numbers cannot have zero for a limit point.
To show that
if
that the condition is sufficient, we need merely observe A(x,x) is a given real, limited quadratic form, then the form 00
A'
GO
A = 2 (2,.a,) y1
s
l~l
positive definite and limited. Hence it may be discussed under the conditions of theorem 14. If condition (3) of that theorem is fulfilled, then there must exist a unique inverse, Y, which is also limited. is
Hence we have
this observe that Y A' is the desired inverse of A.* conclude by noting that theorem 15 is immediately extended to limited bilinear forms. It is also applicable to Hermitian forms
and from
We
A
provided we substitute in the theorem.
A' and A'
A
respectively for
A A'
and A'
A
PROBLEMS! 1.
Given the bilinear form 00
A(x,y)
=2
M
P 2/,
,
/>'/=!
where the variables belong fc,
to Hilbert space.
If the series
= 2IVl
=2I,J

i\
rl
converge and
if
^
kp
K,
m^
f
where
/*,
and
*
are constants, then the upper
/*
most equal to V*/* 2. Show that the form considered in problem examining the special case
bound of A(x,y)
at
is
.
a pq 3.
is
If
A (x,y)
is
(log p log
a limited bilinear form, and
also limited provided the
power f(x)
converges absolutely for x
=a
not necessarily limited by
1 is
if
[
.
aM
\
<
a,
then show that
series
= c^x
f c 2 x*
.
*Another demonstration of theorem 15 has been given by E. Hilb: tiber die Auflosung von Gleichungen mit unendlichvielen Unbekannten. Sitzungsberichte der Phys.Med. Sozietat, Erlanqen, vol. 40, (1908), pp. 8489. See also F. Riesz: Les syst6mes (liquations lineaires. Paris (1913), pp. 8994. fThe problems in this list are due to I. Schur: Bemerkungen zur Theorie der beschrankten Bilinearformen mit unendlichvielen Veranderlichen. Journal fur Mathematik, vol. 140 (1911), pp. 128.
THE THEORY OF LINEAR OPERATORS
148 4.
Show
that the two forms
pql
t
fl
p
A
are limited provided the bilinear form
\t\>
provided
\apq
defined in problem 1
(.<,?/)
is
limited
and
. \
Let fp (t) and g p (t), p 5. 1, integrable square over the interval a
2, 3,
^
t
,
^ r
<^
i
&,
be a set of arbitrary functions of such that
&
Ja
Ja
Now
ai>q
*
>/
compute the constants
"
l
(
J. and construct the
If the form bound of B(x,y) 6.
bilinear
form
A (#,?/) is
2
a
^ //(/
/;
at most equal to
^
nas the upper bound
l
/z
(/*
+
a,
then the upper
^).
and ^(x,x) are two positive definite Hermitian forms and if respectively the largest and smallest characteristic numbers of A(x,x)
If A(x,x)
a, a' Sire
that /?, ft' the largest and smallest characteristic numbers of B(x,x), prove the characteristic numbers of the form
and
lie
between 7.
a'
f
p and
Prove that the
ft
.
maximum
where 2' nieans the term p 8.
Prove that the
is at most equal to not an integer. 9.
is
q
maximum
^ when
X
is
value of the forms
is
omitted from the sum,
is
at most equal to
if
.
value of the forms
an integer and
is
at most
Prove that the form
limited and that its upper bound
is
at most equal to
irj\ sin
X
^ when \
X
is
LINEAR SYSTEMS OF EQUATIONS
149
10. Extension of the Foregoing Theory to Holder Space. Although the purpose of the present volume will be served by limiting the variables to Hilbert space, it is instructive to consider the more general problem of solving system (1.1) where the unknown quantities are
assumed
to belong to
Holder space, that
where they satisfy
to say,
is
the condition
2
x*
il
V^M
,
p>
i
(io.i)
.
M
The
positive constant may be set equal to 1 without limiting the generality of the problem. The quantity p is called the exponent of the space. It will be convenient also to introduce the ratio q
=
P/(p
3),
from which we have
!+!=!. P Q
(10.2)
= =
Since p and q are separated by 2, except when p 2, we q 2. that 1 hence that and assume < q 2, p may The related problems of solving the set of equations (1.1) and of obtaining the reciprocal of the bilinear form
^
^
A(a?,)=2^a? i;=l
1
tf ;
(10.3)
,
under the assumption that the x, and the y belong to Holder spaces of exponents p and q respectively, have been extensively studied by F. Riesz.* Further contributions to the theory have been made by St. Bobrf and L. W. Cohen.$ t
The principal tools employed in these investigations have been the Holder and Minkowski inequalities, which we have defined in the preceding section. A typical application is found in the following the
orem due
to E.
Landau
:
Theorem 16. If an infinite set of variables {x er space of exponent p, then the linear form
t
}
belongs to a Hold
2 a, x, ti
converges for
all
values of x if and only
Holder space of index
t
if
the set [a l ] belongs to a
q.
*Les systemes d' equations lineaires, (Joe. cit.) chapter 3. fEine Verallgemeinerung des v. Kochschen Satzes tiber die absolute Konvergenz der unendlichen Determinanten. Mathematische Zeitschrift, vol. 10, (1921), ,
pp. 111.
on a System of Equations with Infinitely Many Unknowns. BulleAmerican Math. Soc., vol. 36 (1930), pp. 563572.
JA Note tin of the
tiber einen Konvergenzsatz.
Gottinger Nachrichten, (1#07), pp. 2527.
THE THEORY OF LINEAR OPERATORS
150
The proof of this theorem is obtained as an obvious derivation from the Holder inequality. St. Bobr has extended von Koch's theorem on the convergence of an infinite determinant (see section 3) as follows:
Theorem
together with
17.
The determinant
all its
minors, converges absolutely provided 00
H an
(1)
the product
(2)
there exists a
converges:
number
oo
^
q
such that the double series
2,
oo
2/\ V 71
I
.Zi /=!
I
n l
il / ; f~ J
.
>
I
MO 4^ \L\).t)
>
cow^r^/es.
The proof
of this theorem
attained by an argument similar to
is
3. The details will be omitted here. Cohen in his study considered the problem of solving system of that system are replaced by (1.1) in which the coefficients a to for a system, which has the following detersay, <$; an that is
that employed in section
l}
+
,
minant:
The quantities {&,} which comprise the right hand members of the system are assumed to belong to a Holder space of exponent p and the coefficients {a l} } are subject to the single condition:
2 [2 K, /i
i^
1
']^
1
'
<m
1
,
<
p
^2
,
(10.5)
1=1
where
m is finite.
A
system equivalent to (1.1)
Prom
(10.5)
it
is first
follows that lim

a,,

constructed as follows: 0. Hence there exists a
=
i=oo
integer
?'
such that the following inequality holds: Vz
The
<
I/ (1
original system
+
is
<
2
when i >
,
i
.
then replaced by the following equivalent
one: 00
%i
+2
Ci;
Xj cr= 6i
,/
;
+c
i3
t=z
d
i
(6 l j{~ aij)
eit=dibi
,
,
/=!
(10.6)
where we write di
t= 1 for
i
^
to
,
rfi
e= 1/(1
+ aH
)
for
i
>
i
.
LINEAR SYSTEMS OF EQUATIONS
=
Since the di are bounded and the c li that the infinite product 77(<5 ri 4" c a) an d the 00
OO
V f V 2< X 2<
I/*.
li;
I
/
>
for
i
>
t"
,
it
follows
sum
> = 9^
n 4
la\i/(
151
both converge. Hence the determinant of the system fulfills the conditions of St. Bobr's theorem. The solution of the equivalent system is then attained by Cramer's rule, since the minors converge absolutely, and the condition that the (bi) are in a Holder space of exponent p is sufficient to establish the convergence of the solution thus found. The following theorem may then be stated:
ties
Theorem 18. If the determinant A does not vanish, if the quanti(6J of (1.1) belong to a Holder space of exponent p, and if the
a solution of the original system may be found by solving by Cramer's ride the equiva^ lent system (10.6). The solution ivill belong to a Holder space of excoefficients {a^} satisfy the inequality (10.5), then
ponent
p.
It is instructive to
examine the situation with respect
to the
transposed system on
a,
+ 2tt;i* =
&.
(10.7)
>
where the quantities {a i} } are subject to the inequality (10.5). The pertinent theorem is immediately derived from a consideration of the following inequality established by an obvious application of the Minkowski inequality: 00
00
V / V 2* \ 2* i=\
i=i
00
00
I
I
* n Q/(i)\M) a>j '
,

j
\
*=i
I
1
n tj a
.
IP(PI)\(PI) \
]
;=i
Noting that inequality (10.5) imposes upon the coefficients of the transposed system an identical inequality in which p is replaced by q, we immediately derive the following:
Theorem 19. The statement of theorem 18 also applies to the transposed system (10.7) provided the (6J are assumed to belong to a Holder space of exponent q. The solution will belong to a Holder space of exponent q.
The theorems which have been given above
afford sufficient con
We
ditions for the existence of a solution of system (1.1). shall conclude with a statement of the theorem of Riesz which specifies both a necessary and a sufficient condition for the existence of a solution:
THE THEORY OF LINEAR OPERATORS
152
Theorem 20. If in system (1.1) the coefficients arc subject to the condition that the following series converge: 00
5
1 I
Ip/fp 1 )
a
(i r

i 2
)
then in order for the system to have a solution belonging to a Holder space of exponent p > 1, it is both necessary and sufficient that there such that exists a positive quantity
m
I
I
for
n "V IM n 2^ nil ^i
all positive
\
\
< wi ~
integers
rfl/
oo 1 f ^s \ j^i
n
X
n and for
1
*
w? lll i n ^ij ,
all
P/(Pl) \ (Pi)/P
values
For a proof of this theorem the reader Les systemes d equations lineaires of Riesz. y
/
\
m
is
l
.
referred to chapter 3 of
CHAPTER
IV.
OPERATIONAL MULTIPLICATION AND INVERSION Algebra and Operators. Fundamental
1.
any theory of opera
to
tors is the law of composition or operational multiplication. The problem that confronts us in the establishment of such a law is essentially different from the equivalent problem in linear algebra. It seems
worth while
to indicate briefly the nature of this difference. of a linear algebra is based upon three concepts
The construction
:
c n9 generally the existence of a set of unitary elements, c l9 c? multiof scalar existence of a field (A) finite in number; (2) the is number the of from the which units, general pliers, independent constructed by the addition of scalar products of the form i.e.,
(1)
,
9
A&
9
(3) the existence of a multiplication table for the unitary elements; that is to say,
where the multipliers belong
to (A). n
n
The product
of
two such numbers,
A

,r

t
c,
and y
= J^B
3
c J9 is
ii
11
then found through use of the linearity postulate to be
The problem of operational multiplication, on the other hand, is * n z~ n z~ based (1) upon an infinite set of primary elements, 2 z m+n Z' 1 1, z z which form a group, z m z n and (2) upon a class of functions, (Ai (x) }, the combination of which with the elementary operators is not independent of them. It is this dependence that so greatly complicates the problem of the multiplication of 
,
,
y
,,
9
,
=
,
,
operators.
In spite of these essential differences, however, there does exist particularly where the class of functions (A (x) } is the class of constants, a formal analogy between operational symbols and the symbols of algebra. The source of this formal resemblance will be discussed in the ensuing pages. in
many cases,
t
153
THE THEORY OF LINEAR OPERATORS
154
Let us for the sake of ready reference designate the differential operator of infinite order by the symbol
A(x,z)
=a
2 (x) f ai (ff)3f a ,(x)z
+ .......
,
and the polar operator by the symbol
It will also
be convenient to refer to these occasionally as generaand second kind respectively.
trix operators of first
The Generalized Formula of Leibnitz. It was first proved by 2. Leibnitz* that the nth derivative of a product is given by
*uv t=vu (n}
zn
+ nv'u
+ n(n
(n 1}
~2
I)v"u (R >/2\
(n)
\v
f
u
.
(2.1)
This formula is capable of an extraordinary generalization, bebut a special case of the following one:f ing
F (x, z)
>
uv
= vF
(x, z)
u +
+ v"F
" s
v'F' s (x, z) >
(x,z) >u/2l
+
u/1
!
......
(2.2)
,
n
n
where we have used the abbreviation F~ (n) (x,z)
d F(x, z) /dz Under proper convergence assumptions, formula (2.2) is obviously obtained from (2.1) if F(x, z) is a generatrix operator of first kind. That it also applies to integral operators will be apparent from a consideration of the expansion
c
x
r
*
Jo
Jo
Jo
=
*
r
t)*u(t) (v(x)
f\x
.
(xt)v'(x)
Jo
.j.
= v (x)
z
n
(xt)
2
v" (x)/2l
V (x) n z' 
*u
n (n+l) (n+2)v'" z~ n
,
1
1
z~
}dt/
(ri
*u + n (n+1) v"
M > w/3 + !
.
1) z"
!
2
 u/2
!
(2.3)
Since this is equivalent to formula (2.2) applied to the function we see that it also applies, with proper convergence assumptions,
to the general polar operator. Moreover it will be observed that
formula (2.2) also holds when Z~P log z. not an integer, and F(x, z) // The first case follows from a slight generalization of (2.3), (n 1) being replaced by F (n) The second case is established as follows
we
specialize F (x, z)
^
/*
=
,
!
:
.
*In a letter to Johann Bernoulli and later in his Symbolismus memorabilia etc.
fFor
this see
Hargreave: London Phil
Trail*.
(1848).
OPERATIONAL MULTIPLICATION AND INVERSION
By
definition
155
we have X
t)^{\og(x
(x
y(ti)}u(t)v(t)dt/r(n)
t)
/
\
'V
f
f\
/ X
i 7.
X (_!)/!} d/r(/<) =
ij
OO
( \
1
^\/ ^
'v* ( *v \
i~\ ^
**
/
0
v(/')}
J"{log(a;
(*)
00
X 2^ = V OH~1) (n)
{
(xt)P~* (_!)"/!} dt/r(p)
(a;)
n_o
But since y(/0
1/j"
>
^ 00
^ f'O'+l)//
1
>
.
we can
write C*
w^ c= f
2:^ log z >
l
(.r )
^' log 2
I
/ o
or
^
(log
1
(iv
i
+2
wi
where we write
//(/i+2)
But then
F
it
(r?)
...
is seen,
=
(n)
F^
(/i+w
by
1) H1 //(//+!)
that
explicit calculation
equals the coefficient of
i;
(M)



if
+
2)
(/(+/i
=
.
z v log z F(z) which was the ,
(,T)/n!,
desired result.
trix obtained tor,
F(x,
role in
Proceeding from the generalable to derive the product genera
Boiirlet's Operational Product.
3.
ized formula of Leibnitz
z).
much
we
are
now
when an operator X(x, z) operates upon a second operaThis formula is due to Bourlet and plays a fundamental of the theory that follows.
We
shall designate the product of X(x,z) into F(x,z) by means of the symbol X(x,z) > F(x,z), or occasionally by [XF] (x,z). We first prove the following fundamental theorem:
Theorem 1. If X(x,z) andF(x,z) are both operators of kind (or both of second kind), then the operational product f
first
THE THEORY OF LINEAR OPERATORS
156
can be expanded formally into
X~ F
= F X f
[X F]
O
4
n
(dF/dx)
F/fcr) (d*X/te)/n\
+
(3.1)
.
Proof: Let us assume first that both X and F have Taylor's expansions in z about the origin and let us denote them by
X(x,z)
^X
(}
+X
(x)z
+X
(x) tF 1 (x)z
+F
(x)
l
2
(x)z*
+
,
(3.2)
F(x,z)=F It is
then clear that
we
shall
[XF] *u(x) t=X(#,z)

2
(x)z*
+

.
have (Fo(x) u(x)
+F,(x)
u'(x)
or making use of the generalized Leibnitz formula (2.2),
L
{F
<
><>
(a;)
.
J0
Designating the differentiation of u(x) by placing side of the arrow,
we
z*
on the
left
obtain
f 1O
LO
get
l
{6'A"(x,2)/()^} <0
we
(z*F ;_u
'JO
(SF/dx)
We have thus established (3.1) for operators of first kind. But the formula has equal validity for operators of second kind because
OPERATIONAL MULTIPLICATION AND INVERSION
157
polar operators are similarly included under the Leibnitz formula (2.2). The proof just given may be used with suitable modifications provided X(x,z) and F(x,z) are assumed to have Taylor's expansions in 1/z instead of z. From the result of theorem 1 it is easily seen that, in general, the product of two operators is not commutative, since
However, in the case of constant coefficients, that where X n (x) and F n (x) of (3.2) are constants, we have
X*F = F*X = X(z)
F(z)
to say,
is
,
and hence we are able
to conclude that two linear uniform operators with constant coefficients are commutative. It will be found upon examination that the expansion (3.1) also holds when the functions contain terms of the form z~ v log z. An explicit statement of this corollary follows :
S(x,z) and T(x,z) are operators the symbolic v multiplication of which is given by the Bourlct product, then z~ n w z log z S(x, z) and z~v log T(x, z) are also such operators. Corollary:
If
Proof: If we abbreviate y(z) our problem is to show that
=
zr v log* z
and Q(Z)
2
)
To prove
this
S(x,z)
we expand
=
!
+ .]
log
z,
.
in the series *
T n (x)z"
,
co
<s>
can then write
A(x,z)
^
(
S) >
QTtt(x ) z 
it
But
and T(x,z)
S n (x)z n and T(x,z) n^
We
C(.r,,~)
/2
z~v>
since
n
=
we have shown
(
/")

*eT n (x)z
.
s~,
in section 2 the validity of the Leibnitz <S~)
rule for
z
n ,
in particular,
^ S
and hence formally for

general,
we
n
(x)
S)
are able to write ^
r
_/>
A (x,z) n
*For the generality
L mL ~
jr
,
(d
MT
n
/dx m )
M
(d
m )
n Q z /m
I
u
of these expansions see section 14, chapter 2.
n ,
in
THE THEORY OF LINEAR OPERATORS
158
A(x,z)
co
=
m
(d
m
m
%
)
(d
mT
n
m
n
/dx )QZ /ml
r>
m ?
)
(d
m
)n
T/dx )/ml
.
mo
We thus
and the corollary
attain the desired equation
is
proved.
PROBLEMS 1.
If
R
=x
e~ z ,
show that
Rm = _. 2.
Prove that for the operator of problem
Rm 3.
If
P
=z x(l
y
e~ ),
Pn where 4.
mz
^
Prove that
where F(x)
>
Rm = Rm
Rni+n
+ (p
_^_
m
)
?
1.
F (P)  R^ = RM ^ F (P + m)
a polynomial and
is
1,
show that
the operator of problem
7? is
Rn

t
P and #
,
are the operators defined in the pre
ceding problems. 5.
If
F(x)
is
a polynomial and
F(PR) =F(P)
if
~
F n (x)
J\(
(1
e~ z ) n
=
0, 1, 2,
,
n,
F(x), prove that
2
6. Prove that every linear difference equation with x can be expressed in the form
where fr (x), r

coefficients rational in
are polynomials and /(as)
is
a
known
function.
[The operators employed in the six problems just given are due to G. Boole, p and TT respectively. For a systematic account of these operators see L. M. MilneThomson: On Boole's Operational Solution of Linear Finite Difference Equations. Proc. Cambridge Phil. Soc., vol. 38 (1932), pp. 311318; also MilneThomson: The Calculus of Finite Differences, (See Bibliography),
who designated them by
chap. 14.] 7.
in the 8.
Express the difference equation
form suggested by problem
6.
Prove the identity
where J n (x)
is
the nth Bessel function.
OPERATIONAL MULTIPLICATION AND INVERSION Prove the identity
9.
2)>M >
where
159
Y n (x)
is
< cos
the Bessel function of second kind defined by
Yn (x)
[J n (x)
(The identities given in problems 8 and 9 are due to J. C. Hargreave [See Bibliography: (Hargreave (2)] and to H. M. Macdonald: Note on Bessel Functions. Proc. London Math. Soc., vol. 29 (18971898), pp. 110115. See also G. N. Watson: Theory of Bessel Functions. Cambridge (19'22), pp. 170171.)
The Algebra of Functions
of Composition. In the preceding expression of unusual generality for the comand F. If we now specialize our field position of two operators to include only the class of operators defined by Volterra and Fredholm integrals we can construct an algebra of functions of composition which is fundamental to the theory of integral equations. To exhibit the basis of this functional algebra let us define two operators by means of Volterra integrals 4.
section
we found an
X
:
V*(x,z) +u(x)
=
V
=
(x,s)
s
(*:)
Jf*f(x,t) a
J(' a
u(t) dt
,
g(x,t) u(t)dt
.
Forming the composition of these operators and employing the formula of Dirichlet* for the interchange of the order of integration, we shall then have \\(x,z)
*{V
2
(x,z)
?<(*)}
=
(*f(x,t)
f g(t,y) u(y)
dt
I
v(%>y)dy
dy
Ja
Ja
^
Ja
where we abbreviate y(x, y)
CV \
f(x,
t)
g(t, y) dt.
Jx It is thus seen that the composition of two operators defined by Volterra integrals is characterized by a function of composition, y(x,y). A similar situation prevails for operators defined by Fredholm integrals, namely, operators of the form
F(x,z) +u(x)
=
ff(x,t) u(t) Ja
*See section
6,
chapter
2.
dt.
THE THEORY OF LINEAR OPERATORS
160
In order to discover the analogies which this operational product possesses with the ordinary rules of algebra, Volterra was led to the development of an algebra of functions of composition, the main features of which we shall now present. It is clear that the operation
^(x,y)=^
f(x,t)g(t,y)dt
I
Obvious limitations as will, in general, define a new function y(x, y) to integrability and the region of definition must, of course, be imposed. Since, however, we are now concerned only with the formal as.
pects of composition, these important considerations will be disregarded for the present. Volterra has called y(x,y) a function of composition of first kind for the case where a(y) x, and a function of comy, ft(x) position of second kind when a(y) 3(x) =b, where a and b a, are both constants. The functions f(x, t) and g(t, y) are called per mutable provided
=
=
they satisfy the relation f*
f(x,t)g(t,y)dt=
a(x,t)f(t,y)dt
I
J$(
permutability of first and second kinds being defined for the cases a and a(y) b respectively. x a(y) c= y fi(x) p(x) It is with permutability of first kind that we shall be concerned here.
=

,
,
,
Following the notation of Volterra,
we employ
the symbols
In cases where no ambiguity will result, this symbol will be modified by placing the asterisk betwleen the two functions the composition of which is studied. In this capacity the asterisk plays a role analogous to multiplication. Thus we can represent the composition of /
and g by the symbol v(s,2/) It will is
=/ * g
be seen directly from the definition that the composition
associative
;
i.e.,
f(g h)
= (fg)h
f
or
/
* (g * h)
Thus defining two new functions /
* g
,
.
we have
= (/ * g)
* h
.
g ^ h and y(x, y)
=
OPERATIONAL MULTIPLICATION AND INVERSION
f*
(g
* h)
ry
=
161
f(x,s)
Jx
ry
=
s*y
ds
Jx
\f(x, s) g(s,
t)
h(t,y) dt
.
Js
Interchanging' the order of integration by use of Dirichlet's formula, we get
rf(x,s)g(s,t)h(t,y)d8 JX
= That composition
/*
f yd,
is
h(t, y) dt
=
(/
# g) * h
.
also distributive,
(flr
+ fc)=/
immediately evident from the distributive nature of integration. In order to establish a formal analogy with the processes of algebra it is necessary to impose the limitation of permutability upon the functions entering the domain of our calculations so that the commutative law may also hold, namely,
is
/
We
*g
=g
# /
.
prove the veiy important fact that functions itself will be members of a permutable set. This is seen to be a consequence of the next theorem. shall
first
formed by successive compositions of a function with
Theorem a
fg t 5i
x
2.
i b,
If f(x,y) is a function integrable in the triangle
and
if f\(x,y) is the
h(x, y) then
we
shall
=
f /(a,
hth iterated function,
t)
:
assume that
,
have 'i(x.t) fi.ttt.'u) dt
Proof
fh i(t, y) dt
We it
.
i
=
l,
2,

see by definition that this is true for k and establish it f or is true for i
=
proof then follows by induction. Hence we have, by assumption,
,
i
i
h
1.
= We now = k + The l.
1.
THE THEORY OF LINEAR OPERATORS
162
MX,
y)
Applying
=
f h(x,
t)
f "/(t, s) ^^(s, y) ds dt
to this iterated integral the Dirichlet
interchange of the order of integration,
Mx,V)
=
f "/**! JX
=
f fkn
(*,!/)
.
formula for the
we have
f(t,s) dtds J(fk(x,t) X
(x, t) /Vfc.! (t,
y) dt
,
Jx
which
seen to establish the theorem for the subscript k\l. have assumed in this, however, that the theorem is also true h 1 and this fact must now be independently established.
is
We
=
for
i
We
again employ induction and consider the iterated integral
,
(*,t)/2
=
(/,!/) dt
f/(M)
f/Oi',0
tlsdt
.
Applying the Dirichlet formula to interchange the order of we have
inte
gration,
(.r,l/)
=
f /(,!/) /ar
py f*(x,s) f(s,y) ds.
JX It
then follows that
By
we can
write
successive repetitions of this
argument we
establish the gen
eral result,
fn(x,y) If
we
we
=
/(.r,t)
/,,!
(*,!/) c/^
cast theorem 2 into the
see that
it is
i
(^,
f(t,y) dt
.
symbolism of permutable functions
equivalent to the index law
_ fm+n *
f
where
m and n are
tn ^j<
f
n

integers and denote the
in generating the functions.
number
of iterations used
OPERATIONAL MULTIPLICATION AND INVERSION
163
Another step in building the algebra is to give meaning to the symbol /. By this we mean an element which is defined by the equations
f*f = f*f = f(x,y)
/
,
= 0.= 1
.
obvious that /
plays the role of unity in the algebra its foris not possible. Thus the symbol cannot be regarded as a function of x and y since this would imply that it satisfied the two integral equations It is
;
mal determination, however,
1J
f(x,y)=
dt ( f(x,t) f(t,y)
,
Jx
(4.1)
y)=
J("f(x,t) X
f(t,y)
ilt
.
an integral equation
in y with f(t,y) as the the of the inversion of a Volterra kernal, theory integral that a solution of (4.1) is possible in general only when f(x,x) f(y,y) 7^0,* But these conditions are incompatible
(4.1) as
Regarding
known
we know from
=0,
with each other. In order to avoid this difficulty we shall regard / 1 merely as an element which combines with any other function to produce that
~
function.
Another symbol, / 1 of similar nature, may be defined as an element which in composition with f(x, y) plays the role of / That ,
.
is to
say
/*/*
=/
.
Following the model established by E. V. Huntington,f G. C.
Evans has formulated these
results into a set of postulates which dealgebra of permutable functions.J Thus the postulates of addition may be stated as follows fine the
:
A I. A 2.
/
+
If f
A
If
.
A 4. *See:
exists in the system.
(f+g)+h = f+(g+h)
A3i. 32
<7
g
\
g
}
h
+ /=h+ /
If af e=
A
f
Survey
ag
of
,
then
we have g
,
then
we have g
=h
,
h
.
.
a being a positive integer, we conclude that
Methods for the Inversion
Indiana University Study, Nos. bra.
.
of Integrals of Volterra Type,
76, 77, p. 9.
fThe Fundamental Law of Addition and Multiplication in Elementary AlgeAnnals of Mathematics, vol. 8 (1906), pp. 144. JSopra 1'algebra delle funzioni permutabili. Memorie delta reale accademia
del Lincei, vol. 8 (1911), pp. 695710.
THE THEORY OF LINEAR OPERATORS
164
Similarly the postulates of multiplication (composition) become:
= g =
Ml.
/
M2.
(/*#) *
M
If /
3i.
ft
ft
/
* #
=
/
g * /
=
ft
* (0*
*
ft)
.
ft,
where
/ is a nonzero function,
then
/,
where
/ is a nonzero function,
then
.
M3 ft
# g exists in the system.
2.
If
1B
/
*
.
M4 M4
2.
* (g
(tf
+
+
ft)
+/* */ =
ft)
M5. /*flr=0#/
* g
ft
.
.
.
Since these postulates correspond to the postulates of ordinary it follows that the identities of the latter, in so far as they depend upon these postulates, are carried over to permutable functions. We thus have the index law, the binomial theorem for integral coefficients, the factorization law for polynomials, etc., holding in the algebra,
algebra of permutable functions. 5.
A
Selected Problems In the Algebra of Permutable Functions.
few problems
power inherent
will illustrate the
in the notation
of the algebra of permutable functions.
Example and bounded
Given that K(x,y) is a known function integrable an interval (a&), let us solve the equation
1.
in
k(x,y)=K(x,y)
+M*K
.
Writing this in the symbolic form
k *
we may proceed by
1
=
The function k(x, y) Example
2. ,
let
*
lf Afc
algebraic methods.
k * (lo _;[ K)
a region (a&)
A'
is
=K #
1
*
K
,
Thus we get ,
referred to as the resolvent kernel of K(x, y)
.
Given a function g(x, y) integrable and bounded in us find the solution of the equation
OPERATIONAL MULTIPLICATION AND INVERSION This equation
or, in algebraic
be written in the form
may
g*
= /* l + /V2! + /V3! +
1
... ,
symbols, as
g *
The
165
=
l
1
e'
.
from algebraic analogy
solution
will then
be
/* P or
/
=g
y)
(:r,
The convergence of
2
X y)
(
,
f
.
hand member of this equation, integrable and bounded in the
the right
under the assumption that g(jc,y)
/2
is
from the general inequality
interval (ab), follows
G (ba) n /nl tt
0"
^ G
where \g(x,y)\
Example /V L,(.T,
T/)
ent of
k,
is
*
If
3.

<,
in (ab)
fc, (.r,
?/)
.
is
the resolvent of fc,
,
the resolvent kernel of
K~
K(x, y), prove that
K(x,y) and the resolv
is
.*
Because of the relationship between the kernels, we have
(5.1)
Noting the permutability of the functions involved and use of identities (5.1), we have
(K #
fc.)
* (K * ^
k,
= _K

c=/<:
c=JS:
=K*
fc 2 )
(K
*
(K
* (K
^'
k,)
fc t )
fc t
k,) \k
Since this result
may
K
=
* (K *
[fci
=K
*
ft a
(fc,
making
)]
K)~K
*
*
(fc x
*
A: 2 )
^ (K * k 2 )
+ fe) =K
* (K
2
f^
*
fc 2
.
be written in the form fc t
*
fc 2
K
2
*
(k,
*
fc 2 )
,
the truth of the theorem is demonstrated. *This theorem, which is due to Evans, ^s stated in slightly different form by His equation connecting the resolvent with its kernel is k \ K rzz K*k K + K*k Evans' statement whereas the definition adopted in this study is k
him.
=
*
concludes that k*k (1911), pp. 453460.
is
the resolvent of
K
2 .
See:
,
.
AM
del Lincei, vol.
20
(2)
THE THEORY OF LINEAR OPERATORS
166
PROBLEMS. If
1.
is
fcj
the resolvent kernel of K> and k 2
K
the resolvent of
is
prove
t
*
that (&J 4 k 2 )/2 is the resolvent of K~. (1911), pp. 688694.]
+
k
If
2.
Ki
and k
k^
+
k.
+ K2 = K.*k and if K and K are K + k = K*k where K = K + K2
= #!*&! and k
able with each other, prove that
kfk*
Atti (ki Lined, vol. 20 (2)
[Evans:
2,
2
permut
l
[Evans: Atti dei Lincei,
.
2
l
,
vol.
20 (2)
K^K.,
(1911), pp. 688
694.] 3. is
Prove that the function a solution of the equation
V(\;
x,
=
y)
\f
+
X 2 /V2
!!+ \
n
*/nl
'
f'
The Calculation
6.
Function. functions
which
of a Function
,
t)
V(/i;
t, j/
Permutable with a Given
One is
of the principal problems in the algebra of permutable that presented by the calculation of a function X(x, y)
permutable with a given function F(x,y). In section 10, chapter 2, we have solved this problem for the case where F(x, y) 1. We now turn to the general case, which may be is
=
stated as follows
We
:
seek a function X(x,y) such that
X*F=F*X where F(x y) 9
is
assumed
F(x, x) If
F(x,y)
is
sufficient to derive
=1
,
to satisfy the canonical conditions
(dF/dx) y = 9
,
= (dF/dx)
y=f
*=
.
(6.1)
not in this form, the following transformation is from F(x y) a function of the canonical type: 9
b(y,) F[.m(x
L
)
9
m(y
l
)^
,
(6.2)
where we use the abbreviations
e
b(x)
=x,
m(y
l
)=y
,
m'(x
}
)
= I/F(x, x)
For example, suppose that we have F(x,y) z=x\~y. We then It follows from the fact b(x) ==ar 1/2 /2 compute a(x) =x~ l/2 .
,
that that
we
get
J*x or in the present instance, x l
F(x, x) dx
c=x 2
.
We
,
thus obtain
OPERATIONAL MULTIPLICATION AND INVERSION
167
and the new function becomes
F(x,y)
(xv*+tf
is seen to be in canonical form. Since the composition of I \(x if y
which
with
t
XT.
is
(x l9 2/0
=
(#0 &
(i/O
1 .)
X[w (#0 w (3/0 ] ,
equivalent to
=
/
ft
(#06
'/i
F[??t(a; 1
(i/O /
Xj.
6(i/0
we
see that the transformation defined above conserves composition.
if Xi(#i, i/O is determined we can compute X(x, y) by the transformation inverse to (6.2). Assuming that the function F (x, y) is canonical and that in addition the derivative 3 2 F/dx dy exists and is continuous, we now seek to determine the function X(x y) permutable with F(x y) We introduce the new function
Hence
,
9
9
=F*X = X*F
Taking derivatives with respect
to x
.
and y we obtain the equa
tions
r JX
(a,t)
X(t,y) dt
,
(6.3)
ty/ty == X(x, y) If
we employ
t)
F'
y) dt
y (t,
.
the abbreviations
A = F'x + F f, it is
+ J X (*,
r= F' v
+ F'/ + F +
 F +
F;3
_ F +
  
,
,
clear that the solution of the integral equations
(6.3)
may be
written
(6.4)
THE THEORY OF LINEAR OPERATORS
168
= a^/BT/
X (x, y) Integrating the
first of
r {fy
(a;, t)
Jr
/dt} / 2
(t,
these equations by parts
y) dt
we
.
obtain
f fv(x,t)
=
where we use the abbreviation / 12 Since
X(x,y)
^dcp/dx
dl
,
d
=
by
(6.1), this reduces to
P/i2(s,*)
.
(6.5)
Jat
Similarly, the second equation
X(x,y) *=<XF/dy
becomes
(\(x,t)
=
where we abbreviate /21 df z /dx But we can demonstrate that / 12 J.

=/
21
Peres in his notable exploration of this
breviation
H = d*F/fixdy, we see from
/ 21
(Mj) dt
a fact
,
field.*
If
(6.6)
,
first
proved by
we make
the ab
(6.1) that
and
Fy'(x,y)= Consequently
from which
If
and
we
(6.6)
it
we have
follows that
substitute the value / 12
=/ = 2i
/(a:, ?/)
in equations (6.5) first, wie obtain
and subtract the second equation from the
as the integrodifferential equation for the determination of y(x, y)
=
{
s) f (s, y)
(s,
y) f (x, s)
}
ds
:
.
Jx (6.7) If
we
recall that the solution of the partial differential equation
*Sur (1915).
les fonctions
+
?
= p (x, y)
permutables de premiere espece de M. Vito Volterra. Paris
OPERATIONAL MULTIPLICATION AND INVERSION
may
169
be written in the form
r JH
where
2ti
=y
2v
x,
= y\x,
and q\ (y x) is an arbitrary funccan be written as the following integral
then equation (6.7) equation tion,
}
:
Jr
dt
f
^
u
r(tu, s)
{
f(s, t~\u)
Jtll
.1
,)*}
(
(6.8)
.
u upon the first Making the transformation s ^~ { t r \ t { u upon the second, we may then write >*
and
integral
s
= To (y
+
x)
dt
^(_r_]_f__ H
Setting
fp ()
x)
(y
,?+ f
(t
+70 /(*?/,
,
1)
f
(r+t
,
^0
^ M
C 2U q(s) ds, where q(s)
=
t/r
>'+^+?f)}
I
is
(6.9)
.
an arbitrary func
^0
tion,
we
solve (6.9)
by successive approximations: f*'2H
=
q(s) ds
x)
,
*^0 f*2u
r*v
<Ti(x,y)
~Vo(y
+
x)
dt
(r)
Ju
[f(r\t
u,t}u)
Jo
r
Ju
q(s) ds
.
Interchanging the order of integration in the last two integrals,
we have
=
/ 2U I
f*
Q(s) ds
JQ
+
I
V
/ 2
Ju
It
r*2U
Q(S) ds
rf*
Jo
{f(r\tu, t+u) J&
f(tu,r+t+u)]dr
=
(7(8) t/0
d+
f "() *^0
THE THEORY OP LINEAR OPERATORS
170
Continuing this process we are able to put the solution of equation (6.8) in the
form
+A
q(s){A (s;x,y) J2U
(s;x,y)
l
\}ds
+A
2
(s;
x,y)
(6.10)
,
where
4
and
=1
=
A^s.x.y)
,
V
dt
(
Jf{f( s
Ju
r
+tu,t+u)
in genei'al,
=
A n (s;x, y)
'
dt
(
J(a
Js
'(A*.* (s;
A nWhen obtain
all
tu,
(s; i{u
l
r+t
)/(r+i
r, t\u)
f(t
u,
t+u) r)} dr
u, t\~ii
this value of y>(x, y) is introduced into equation (6.6)
the functions
permutable with F(x, y)
X(x,y)
;
that
.
we
is to
say '
X (x, y)
=
q
+
(yr)
Q (s) A'u (s; x, y) d*
(
JQ
f J(s) A(s;x, J
r/(t,y)dt Jf
t)
ds
,
00
where
A (s;
x,
y)
J^A,(s;
x,
y)
.
t=0
The second tion,
may
integral, by an interchange of the order of integrabe written in the form y
p/00
ds
Jo
J(*tt
A(s;x,t)f(t,y)dt
.
Hence the desired function X(x, y) becomes '
"
=q(yx) +
X(x,y)
g(x) [A'u (s;
x,
y)
y
A(s;x,t)t(t,y)df]ds J(*+X
As an example
let
us indicate the method of calculating the func2 I/O /8
=
2 tion permutable with F(x,y) 1 (x Since this is already in canonical
d
2
F/dxdy
=
xy.
From
sequence of functions
H
#
.
this
we
*
= xy
form we
calculate
without excessive
find
:
1
ff
2
(x'
y
2
2
)
/4 2
!
,
H
difficulty
c=r
the
OPERATIONAL MULTIPLICATION AND INVERSION
H * 1 # // * 1 * H = xy(x* H *1*H *I* H *1 * tf Hence
=
/ (x,y)
[xij
+ xy (x
4
2
y
)
2
4

!
4
)
/4
2
y~) /4
.ry (.r 
2
2
2
f
/4
y
=
]
)
4!

2
.r//
171
,
!
2
cosh
{
y~) /2)
(.i:
.
The solution of equation (6.8) then proceeds by the method indicated above. The first two approximations are
A
= 1 and A,
(s ;jc,y)
= 2xS S y
s
.,/
where we employ the abbreviation S f
+ ZyS^,^/ (s2u)
(s2u)
= sinh{
(x
7. The Transformation of Peres. In the an equation of the form,
2u) f/2}.
we
last section
derived
" *
q(t) y(t;
f
^0
which determined
all
functions permutable with a given function
We
shall abbreviate the right hand member of this equation F(x,y)> the by symbol Q(q) and shall seek to discover values of *r(t',x,y) such
that the equation (7.1)
is satisfied.
transformation X(x,y) Q(q) which satisfies (7.1) is a transformation of Peres to conserve and is called said composition since it was J.Peres who first indicated its important role in the the
Any
ory of permutable functions.
We
shall
prove the following theorem
:
Theorem 3. If n(x,y) is an arbitrary function finite and integrable in a region (ab) and if m(x,y) is defined in terms of it by the ,
series
m (x,y) =
n (xjj)
+n
2
n
(x,y)

l
(x,y)
~\
,
then the transformation &(q) defined by the equation
Q(q) is
=(1
+ m)
* q *
(1
+ n)
a transformation of Peres.
From
the definition of m(x,y) and n(x,y)
it is
clear that they
satisfy the equation (1
+ m)
*
(1
+ n) = (1 + n)
*
(1
+ m) = 1
.
THE THEORY OF LINEAR OPERATORS
172
Hence we have
Q(p) *
Q(q)=
+ m) + m)
(1
= (1 Theorem then
4
ij>(t;x,y) is
V'(t'> x >y)
+ n) X (1 + m) * (1 + n) =
* p X
(1
* p *
q
*
q*
(1
+ n)
If m(x,y) and n(x,y) arc defined as in theorem given by the equation rut
0+
n(x}t,y) {m(x,y
3,
J
m(x,
s)
n(s f
t,
y)
ds
.
(7.2)
In order to prove this theorem we first explicitly evaluate the terms in equation (7.1). After some rather long but straightforward transformations, equation (7.1) reduces to the form f* ?/ r
f*u
x~r
p(r)dr
y(r\x,y
,y)
s)
^0
*^()
f*U8
y(r\x,t) y(s;t,y Jjnr
From
we
this
derive the relation
y(r{s;x,y)
= ^(s;x\r y) +
s)
ij>(r;x,y
f
J' S
v (r;x,t)
l' J$r
v(s;t,y)dt
(7.31
,
.
which defines the desired function y (;#,?/). and r by x and employing the abbreviation Replacing x by
y(# ;0,y)
= n(x,y), we reduce this equation to
rv8
n(x\s,y)
=y(s;x,y) +n(x,y
s)
+
Jx
n(x,t) y>(s;t,y)dt
.
Since this is a Volterra equation of second kind in which the unfunction is y>(s;x,y), its solution is obtained by the ordinary methods and is found to be
known
n(x,y
y)
+ By means
s)
rys
[n(t+s,y)
JX
n(t,y
s)}m(x,t)dt.
of the identity
m(x,y) +n(x,y) this equation is
+
Cv
m(x,t) n(t,y)
dt^O
,
*x
immediately seen to be equivalent to (7.2).
OPERATIONAL MULTIPLICATION AND INVERSION
173
The Permutability of Functions Permutable With a Given 8. Function. By means of the theory of the transformation of Peres we are able to prove the following useful theorem :
Theorem
5. All functions permutable with a given function are F(x,y) penmitable with each other.
The proof tion of Peres
of this theorem for functions defined by a transformaimmediate. Thus let us suppose that we have
is
X(x,y)=f>(p)
,
Y(x,y)=Q(q)
.
From
the fact that p(y x) is permutable with q(y they belong to the group of the closed cycle, it follows that
X
*
Y = Q(p)
# Q(q)
x), since
= Q(p *q)=Q(q # p)
Hence, in order to prove our theorem, it is necessary to show that all functions permutable with a given function can be defined by a transformation of Peres. As a matter of fact we shall prove that there exists a function y(t;x,y) which forms the nucleus of a transformation of Peres such that the function can be expressed in the
form
It is then clear that given by the formula
all
functions permutable with F(x,y) are
X(x,y)=Q(p)
We
first
consider the equation
F(x,y)
whose solution f(x,V)
=
.
=! + !*/ * 1 + 1 is
easily
\H
found
* / *
1
+
*
/
*
H
*
!*# + ...]
1


,
form
in the
+ H * l*H + H
*
1
*
,
(8.1)
where we use the abbreviation
H=
i~F / ex dy
.
In order that F(x,y) should be in the desired form sary that we should have
it is
neces
THE THEORY OF LINEAR OPERATORS
174
y(t;x,y)dt^=
I
vo
dr
\
f( x +s,y+sr)ds
J
j
Jo
r)dt
f(s+t,y+s
(8,x,t)
,
where /*
oo
^
B (s ;*,*)
J
n^l
in
g
Oo f*
/*
g"
J
g2
d Sl fti *
J
/. 2
*
.
.
.
*
=
/.,
n (x,,)
which we use the abbreviation f (x,y) f(x\s,y\s) But it is not difficulty to prove that B(s;x,y) thus defined s
,
.
is
a
solution of the equation
r
B(s;x,y)=
Jo
s
f(x+r,y+r)dr V
+ We
f'dr J(x B(r',x,t)f(r+t,y+r)dt. Jo
have thus established the relationship
y(t\x,y)=B(t\x,yt) It
(8.2)
(8.3)
.
remains for us to prove that the function y(t;x,y) thus de
fined is the nucleus of a transformation of Peres.
Returning to equation (7.3) and making* in r, y *== \ s, we have
=
variables x
the change of
it
ij
(8.4) If
we
r
let
= = s
y(0;^;) which proves that If
we
reduces
0, this
v ; (0;a',?/)
to,
+ ^f \'(0;,t)
=
abbreviate y(r;g,t])
iy(0;
0.
= B(r\s]
we can
r)
write (8.4) in
the form
B (r+s
;
I
r,
i]
r)
=B +
(s
;
f
,
;)
f ^(r;
+ B (r r,
5
r,
?')
B(ij;
;
r)
;
i;)dt
t,
.
^
Forming the derivative of then letting s
B/ (r
;
I
0, r,
we
yr)
this equation
with respect to
s
get
= B/ (0
;
f,
*;)
,t
r) B/(0;t,>j)ett
,
(8.5)
and
OPERATIONAL MULTIPLICATION AND INVERSION
where we mean by
B
175
'
the derivative with respect to the first variable. where /(f, ??) is an arbi, ??) /(, n ) trary function, and make the transformation x $ r, r, y ?= y
Let us t
=
now
f r.
8
=
write B/(0;
=
Equation (8.5) then reduces to
f"s(r;,s) f( s +r,y+r)ds
.
Js
Integrating both sides of this equation with respect to the first variable we obtain finally
i=
f(x+r,y+r)dr
\
Jo
H
+
(' Jo
dr
( B(r;x,s)f(s+r,y+r)ds J jc
with (8.1), this equation with slight change in seen to be equivalent to (8.2), which establishes the desired identification of y(f ;.r,?/) in (8.3) with the nucleus of a transformation of Peres. It is interesting to note that the permutability of functions permutable with a given function was first conjectured by Volterra and later proved by E. Vessiot.* Volterra and Peres also established the theorem, the proof the the latter being reproduced above.f
If f(x,y) notation
is identified
is
The proof of Vessiot is elegant because of its simplicity. pends essentially upon the fact that the function X(x,y)
=a
fl
F(x,y)
+ a F + a,> + 2
2



It de
(8.6)
,
where the
a> are arbitrary constants subject only to the restriction that the series converges uniformly, is permutable with F(x,y) and with any other function of the same form.
Since (8.6), because of the infinite number of arbitrary parameters upon which it depends, has the same degree of generality as the closed
form
X(x,y)=q(yx) we
conclude that (8.7)
is
+
f *"*(*) y(t\x,y) dt Jo
also permutable with
,
(8.7)
any other function
similarly defined. *Sur les femotions permutables et les groups continus de transformations fonctionnelles lineaires. Comptes Rendus, vol. 154 (1918), pp. 682684. f Volterra: Teoria della potenze, del logaritmi e delle funzioni di composizione. Memorie dell' accademia dei Lincei, vol. 11 (1916), pp. 167269. Peres: Sur certaines transformations fonctionnelles et leur application a la theorie des fonctions permutables. Annales de Vecole normale superieure, vol. 36 (3) (1919), pp. 3750.
THE THEORY OF LINEAR OPERATORS
176
9. Permutable Functions of Second Kind. The problem presented by permutability of second kind is essentially different from that of permutability of first kind and does not possess the intimate connection with the theory of operators, at least as developed in this book, which is exhibited by the latter. Permutability of second kind has aspects of a theory of linear substitutions and can be most effectively discussed from its algebraic basis. For this reason we shall give only a casual resume of one or two results which are most di
rectly useful in the theory of operators.
Let us ciative
note that the operation of second kind is both assoWhen we consider its commutativity, howmust, as in the case of composition of first kind, impose con
and
ever, we ditions.
We
first
distributive.
Theorem a ^x $
by
theorem
shall first prove the
a
5;
6.
y ^
If b,
:
F(x,y) is any function inteyrable in the square then the function "
*
)
i$
t=F(x,y){
F(x,y) dx
F(r,y)dy}/(ba)
permutable imth unity. Proof: The proof
immediate from the equation
rb
fb
Ja
is
&(x,y)dx
Ja
b
f
F(x,y) dydx/(b

J
a)
nb
=
^a
a
<&(x,y)dy
.
The next theorem shows the connection of composition of second kind with the theory of linear substitutions. Theorem
7.
If ^ve define n
A
/I
V
\
/
/ /y n
I
\
ji\Ji,ij)

7
ft
Uij
i^
+ }
i
'V'\/Y 10/1 \ju ) ijj \y )
ty = l
and n
B(x,y)
where fi(x) and (ab), and if
tfj(y)
J^ b,j fi(x)(jj(y)
,
are functions integrable over the interval
y) dy
dx<=c
t]
,
then in order that A(x,y) be permutable with B(x,y)
and
sufficient that
we have
ACB=:BAC
,
,
it Is
necessary
OPERATIONAL MULTIPLICATION AND INVERSION
177
where we abbreviate A
I
I
~
I
Z>
I
I
M
I.
I
M
/nr
M
Proof: Forming the composition of A(x,t) with B(t,y)
f A(x,t) B(t,y)dt= J a,
a l}
b kl
we have
f,(x) g,(t) fk (t)
"(a
ijkl
y
n
^
&ij
\rj QI \y )
Dki /i
(f'jK
.
Similarly the composition of B(x,t) with A(t,y) yields 6
B(x,t ,t)A(
J If
we form
the matrix
where n
we
see that
M
is
identically equal to the matrix product, that
is,
M=ACB where Similarly, the matrix N = = a .
\v kj
Vk }
is
2j
I
\
,
bki Cii
l}
,
equal to
Since
A (x,y)
both necessary and sufficient for the permutability of that JV, we derive the theorem.
it is
with
M=
B (x,y)
10. TT^e Inversion of
Operators (Bourlet's Theory).
We
next con
method of finding the inverse of an operator
sider Bourlet's
S(u) =/Or)
,
(10.1)
where f(x) is an arbitrary function and 5 an operator of generatrix F(x,z). This is formally equivalent to the problem of solving an arbitraiy linear functional equation and it is with the formal aspects that
we
shall
always possible
S (u) where f(x)
=u is
is
That the inversion from the following example:
mainly be concerned.
easily seen
xu'/l
!
+ x*u"/2
not a constant.
!
xW/3 H
= / 0*0
,
is
not
THE THEORY OF LINEAR OPERATORS
178 Since S(u)
=
=a
u(Q)
constant,
it is
=
will exist except for the special case f(x)
clear that
no solutions However,
a constant.
the equation
=a
S(u) will obviously
be satisfied with the function
a
u(x)
+ xg(x)
,
=
where g(x) is any function analytic in the neighborhood of x 0. The problem of inverting the equation (10.1) is clearly that of finding an operator of generatrix X(x,z) such that
X (x,z) because
it is
*
F (x,z)
= [X
F]
=1
(x,z)
,
at once evident that
u(x) =X(x,z) >/(#) will be
a formal solution of the original equation.
Definition:
The function X(x
f
z)
is called
the resolvent genera
trix of the original equation.
As an elementary example,
consider the integral equation ,
Referring to section function is
6,
chapter
Substituting this in (3.1)
/(0)=0 2,
we
see that the generatrix
we immediately
X*F = X(le^) /z +
er
.
obtain ***
which, for the determination of X(x,z), must be set equal to 1. Choosing X(x,Q) as some arbitrary function of x, i.e., fj(x), solve for X(x,z) and thus obtain
Hence the desired
solution of the integral equation will be
Since by hypothesis /(O)
=
0,
the last term vanishes.
we
OPERATIONAL MULTIPLICATION AND INVERSION
179
We now
consider the essential feature of the Bourlet theory, concerned with the number of zeros of the generatrix function regarded as a function of z. This we shall state in the following
which
is
theorem
:
Theorem
8.
If the generatrix function associated with the lin
uniform, and complete operator S has exactly m zeros in z in the finite plane, where m is zero or a positive integer, then the equation ear,
S(u) will
have
m
form F(x,z)

(10.2)
solutions provided the generatrix function
=
e~ (x
~c
^
is
not of the
.
Proof: Case 1. Suppose that F(x,z) has no ,>zero in the finite Then, since F(x,z), by theorem 3, chapter 2, is at most of genus one, it must be of the form plane.
F(x,z)
We
^a(
can then write equation (10.2) as
F(x,z) *u
= a(x)e
(f > y
>u
= a(x) u(g
{
x)
^0
.
(10.3)
It is now clear that if g(x)\x= constant, equation (10.2) not will have a solution other than u(x) ?= 0. But if g(x) ] x c, where c is a constant, then any function c will be a solution. But this solution u(x) which vanishes for x corresponds to the equation
=
ei*
c
>*+u(x)
=0
,
which we have already excluded. Case
2.
Suppose that F(#,z)has m zeros in the finite plane. is at most of genus one, it must be of the form
Since the generatrix
F(x
9
z)
=e
ff(
* )s
P(x,z)
,
where P(x,z) is a polynomial in z of degree m. But from (3.1) we know that
Hence we can choose the function X(x
X(x from which we derive
z) so that
f
+ g,z) = P(x z) 9
,
THE THEORY .OF LINEAR OPERATORS
180
where h(x)
is
the inverse function of x f g(x)
;
i.e.,
h(x)
satisfies
the equation
h(x) It follows
from
+ g[h(x)]=x
this that F(x,z)
is
.
the generatrix of the sym
bolic product of the operators with generatrices e (x}z and P[h(x>) ,z\. Thus w<e see that the solution of equation (10.2) is reduced to finding the solutions of the ordinary equation of infinite order m.
P\h(x)
9
z\
*u(x)
.
manner we establish the theorem. be noticed that we have not treated the case where the generatrix function has an infinite number of zeros in the plane. This problem was studied by Bourlet who made the following obserIn this It will
vation:*
a le prouver pour le cas ou le nombre des zeros est en montrant infini, que, dans ce cas (en supposant, bien entendu, la transmutation inverse complete), le nombre des constantes arbitraires est aussi infini. La chose parait vraisemblable et il serait tres interessant de la prouver, car elle montrerait que les transformations (ou substitutions) sont les seules transmutations additives, uniformes, completes, telles que la transmutation inverse soit de meme "II resterait
nature.
"Je
n'ai,
malheureusement, pas encore pu prouver cette proposiElle est evident dans le cas des co
tion dans toute sa generalite. efficients constants."
The theorem thus conjectured by Bourlet
unfortunately not
is
true, as he himself pointed out in a subsequent paper by means of an explicit example furnished to him by S. Pincherle.t What some of these difficulties are will appear in subsequent chapters of this book. As an example illustrating theorem 8 let us consider the integral
equation
u(x)
r
i
(x
"o
By
a simple calculation
Expanding the function
_
u(t) dt
+ te /z + ix/z s
in parentheses as
*See Bibliography: Bourlet (1), p. 184. fSee Bibliography Bourlet (2), p. 337.
.
the generatrix function
ie s /z
obtain
i
t)
we compute
z
we
+
2
A/z
2
)
a power series in
.
z,
181
OPERATIONAL MULTIPLICATION AND INVERSION
;A)
= [(l
i*
/*)
+
(x~lx~
For the determination of the principal numbers, the coefficients are set equal to zero, that
ij)
(1
In order that these equations determinant
first
two
is,
ix =
may be
>
we equate
consistent,
the
I
to zero,
and hence define the principal numbers as the roots of the
equation ,l(;t)
==1
^
^/12
=
.
The generatrix function may then be written
Since the second factor is independent of r, the solution of the integral equation is reduced to the solution of the differential equation
that
is to
say, u ( x)
= ax +
b.
Another example showing the general efficacy of this theorem in application will be found in section 6 of chapter 9 w here the Euler r
differential equation of infinite order
is
discussed.
The Method of Successive Substitutions. A number of im11. portant applications, notably in the theory of integral equations, have been made of the method of successive substitutions. In order to describe its formal aspects, let us discuss the inversion of the linear functional equation
S(u)
where S
is
functions.
+
,
(11.1)
a general linear operator and '/>(.T) and f(x) are known we indicate by S 1 the reciprocal of S, that is to say,
If
THE THEORY OF LINEAR OPERATORS
182 S'1 
S t=
two ways u(x)
then the solution of (11.1) can be expressed formally in
1,
:
= S
1
* /(a?)
S
1


+
> /]
'
[?
(l/V) [S > (!/
frS'
S
{


/)]

* (/A/0 }]
,
(11.2)
.
(11.3)
The proof that both of these expansions furnish formal of equation
(11.1)
members with
?{S^f
and similarly
solutions
immediately obtained by operating on both
Thus we get
S.
S(u)=f
is
in the first case

S l
in the second case,

s
S
(/A/0
*
{ (
It will be noticed that in this formal development we have omitted reference to the complementary function, which takes account of the solution of the homogeneous equation
S(u) +y(.r) w(a)=0 If
we
let c l (x)
(11.4)
.
be any solution of the equation
S^c
U)=0
l
(11.5)
,
then the corresponding complementary solution of (11.4) tained formally from the series u, (x) *= c (x) l
S" > (c lV ) 1
+S
1
may
be ob

which we may designate by the functional symbol Because of the linearity of the operators, it
F(c,,f/). is
seen that
if
we
write ni
A,c,(x)
>c(x)
,
i~i
where the
A
l
are constants and the (c (x)} form a complete set of
solutions of (11.5), then
v
OPERATIONAL MULTIPLICATION AND INVERSION
= F(c,
U (x) is
183
the complementary function of (11.1). As an example, let us consider the inversion of the equation (xz If
we
write S
2
xz'
+a
z
\1
f z,
)
we
=
> u (x)
1
introduce
.
Jo "^
[>(.s.r):/o/l
we
2
a*
readily find that
/^'^J
[_t
If
as the lower limit of this integral,
co
,
we can
write
where C is Euler's constant. Noting that
we then
=
logs >/(*)
=
/(.r)
lim fllog ar=o /
u (x)
(11.2)
=
3
a;
we
+
C]
/(Od
=
0,
2
/3
a;
obtain 4
/ (3
5)
Noting that the equation (xz z Ci
f)
(.r
find
Employing
tions
0,
(x)
A,
c2
(x)
t=rr
/?
log
.r,
.
get
4
2
/ (2
2

we then
.r
The general
+ o;V (3 5 7) + ^) > n(x) = has 2
4)
a;
/ (2
solution of the differential equation
be
u(x)
=u
(x) \Ui(x) {u 2 (x)
.

4
is
6)
the solu
2
thus found to
THE THEORY OF LINEAR OPERATORS
184
This example is selected from a number used by W. 0. Fennel] [See Bibliography: Pennell (1)] to illustrate a method of symbolic division, which depends essentially upon the algorithms given in formulas (11.2) and (11.3). Since this method has intrinsic interest,
we
shall illustrate it
by inverting the simple equation (z\~ x) >
l/(z
device
+
ar),
to
which meaning
is
.
symbolically written u(x) given by the formal operational
In PennelFs notation the solution =a=
1
u(x) is
:
z
+x
1
x z /z
l/z

+ x*/(3z)
(a)
(b)
A
solution of the
7
form
=^ x
+
x ?'/(l 3) x*/(l 35) thus attained. Since the division other?/(,r)

x /(l 3 5 7) fis wise follows ordinary algebraic rules, it seems necessary only to call attention to the fact that line (b) is obtained by performing the indicated integration of line (a). It will also be noticed that the inversion is the one which would be attained by an application of for
mula

(11.2).
If the positions of x and z arc reversed in the division sible to obtain a second solution, as follows:
*
J
1
Z
l/.r 
1
l/x l/x
+ l/.r +
3) /x* f
(135) /V
2 4
3,/x
3/V
4
3/V
(1
35)
/.r
fi
notice that the solution thus obtained, n(x) 5
pos
2
l/x*
We
(1
it is
7
+
^
l/x
\
l/x
3
It is, is completely divergent. f (13 5) /x ( (l3)/.r however, summable along the imaginary axis and we have the asymptotic expansion >
"
!J iu(xi) co e
r*co
e~ lt
Jx
*
dt
OPERATIONAL MULTIPLICATION AND INVERSION
185
A
deeper insight into these difficulties will be gained from the discussion in later chapters.
PROBLEMS Writing the equation
1.
form
in the
u(x)
where S(z)
=
(1/6) (z
2
= (1/6) +
+
e kr
S(z) > u(x)
,
az), solve by the method of successive substitu
tions.
Solve the equation
2.
(xt)
x
u(x)
Jo
u(t)dt
.
by the method of successive substitutions. Obtain by the method of successive substitutions the solution of the
o.
equation
u(x)
a
+
bx
X
(x
I
t)
u(t)dt
.
Jo
Show u(x)
that the solution
m VXa
2
2 h &
VX 12.
given by

sm(VXx
Some Further
shall describe
is
,
jc)
where
,
c
= arc bin.VXa
Properties of the Resolvent Generatrix. We specific properties of the resolvent gen
below a few
eratrix which are useful in application. (a)
//
we assume that F(x
f
form
z) is of the
F(x,z)^l
Mj(x,z)
(12.1)
,
then the following function,
X(x,z)
=I + ig(x,z) + WUv:)
+/'(/">
(*,;:)
+.
,
(12.2)
where y (m)
means the generatrix of the nth power of the operseen to be a formal solution of the equation
(x,z)
tor, is easily
X(x,z)
To prove
this
we
^F(.r^)=l
substitute (12.1)
.
and (12.2) inequation (12.3)
and thus obtain
X > F = 1 + A (gg) + A
2
[#
<
2
>
(12.3)
g*
(dg/dx)
THE THEORY OF LINEAR OPERATORS
186
_L
.
.
.
Since
we have by
(m)
= gg
g
(x,z)
(m
definition
^
the multiplier of X m is seen to vanish for every value of m, and (12.2) is thus the formal solution of equation (12.3).
Let X(x,z) be expanded
(b)
X(x.z) =boU)
Then X(x,z)
x
l
,
i
a power series in
in
+M^)~+ Ms)s + 2
"
z,
(12.4)
.
a positive integer or zero, is a solution of the
equation
^x
F(x,z) >n(x)
l .
Designating this solution by ii (x) we see from equation (14.2) of chapter 2 that the coefficients of z r in (12.4) and hence X(x,z) itself are formally expansible in terms of a set of special solutions Since of F(x,z) * u(x) *= f(x), namely, ?(#). i 0, 1, 2, 3, t
the coefficients of (12.4) are b,(x) the identity r
(d
X/dz
r
)
.
=~ii r (x)
rxn.'L
=
=
r
(d
X/dz

r
\.o)/r\
+ rb'Dxt'ii.n^l^
,
.
we
obtain
x r Uo
.
(12.5)
of the first studies based upon the functions u (x) was made y 1) > u(x) t=g(x). by A. Hurwitz* who considered the equation (e In this case the functions u (x) are given explicitly by M,(X) B t+1 (o:)/(i+l), where B n (x) is the nth Bernoulli polynomial ob
One
t
=
t
tained from the expansion,
If
+ ckUi(x) +
g(x) =^ a f
di.T
+
<*&~
>zU(x)
+
+
9
,
is
=
then the function u(x) the formal solution of the
original equation. I.
to
SheflferJ
has extended this idea for the operator,
which corresponds the resolvent,
*Sur 1 'integrate finie d'une fonction entiere. Act a Mathemaiica, (189697), pp. 285312. fSee Davis: Tables of the Higher Mathematical Functions, vol. 2. $See Bibliography. Sheffer: (1), p. 351.
vol.
20
OPERATIONAL MULTIPLICATION AND INVERSION
=
X(x,z)
f*
(e**/z)
c?*X(z)Y(t)dt
187
,
JGQ
where we employ the abbreviation xr/
vr,j.\
\
Y(t)^e
X(z)
:
$~
Sheffer proved that the generating function,
f(s,x)=
u<(x)s*/il=
is uniformily convergent provided s 5 o < where # is the absolute value of the smallest zero of B (z) Assuming that the symbols 2 and X(x,z) may be interchanged and noting the fact that X(x,z) ~^e* f =k X(x,s)e* we easily derive Sheffer's result: ,
.
,
c*'X(s)Y(t)dt/s It is occasionally useful to
(c)
know how
.
the equation
F(x,z) +u(.r)=f(ff)
may
(12.6)
be reduced to a differentia] equation of infinite order with con
stant coefficients,
*u(x) <=g(x)
,
where we write cp
(z)
To make operators, of
member
= ^ +
2
this transformation
wMch
X
Hh
we
first
?n
+

.
define the following set of
(x,z), the resolvent generatrix, is a special
:
By means of the Bourlet operational product these operators are seen to be related in the following manner :
X
n
(x,z)
Xy (X,
Z)
t=z n *X (foX
,
+ CflXi + X
2 \
,
(12.7)
THE THEORY OF LINEAR OPERATORS
188
It is obvious that (12.7) transforms (12.6) into a differential equation of infinite order with constant coefficients, namely,
y(z) >u(x) =X
(12.8)
.
13. The Inversion of Operators by Infinite Differentiation. A method of inverting operators which possesses useful applications
may
be described as follows: Let us assume that we wish to solve the equation
F(x,z) >u(x)*=f(r)
(13.1)
.
Operating on both sides successively with the following system of equations
z,
z'~,
we
z*, etc.,
obtain
:
+ 2z dF/dx + 3 FAv (z*F + 3s 3F/UT + 3z3 F/3.r 2
2
(z*F
2
2
= z  f(x) +VF/dx*}  w(a;) = s
>
} 2
2
(x)
(13.2)
,
3
These equations taken together with the original one form a system which may now be solved algebraically for u(x) in terms of f(x) and its derivatives. The resulting operator is the resolvent generatrix for the original equation, assuming, naturally, that the sys
tem can be solved under some existential criteria. A more explicit specification of this method may be obtained
we
specialize [l
F(x
+ Mff)]
t
if
z) as the operator
+ai(x)z
+ az(x)z*\ \a
n
(x)z
n
+
.
(13.3)
The system which we wish )
to invert
+fli(.r) u'(x)
+a
+
then becomes the z
(x) u"(x)
a M x(a;) u
f ollowting
:
+
(x)
+ = f(x) ..
(13.4)
,
OPERATIONAL MULTIPLICATION AND INVERSION
189
we designate by D(n,x) the determinant of the first n equaof this system after we have suppressed terms of order greater tions than n 1, we can give its explicit form as follows If
:
D(n,x)=
(13.5)
2a' n _2 nl
2, ro
we
If
by
D
1
represent the cofactors of the elements of the first column D n (n,x), the formal segmentary resolvent D 2 (n,x), ,
(n x) t
9
operator in the sense of the method of segments as developed in section 7, chapter 3, may be written
X
n
[ZM'M?)
(x,z)
+zD
2
(n,x)
\
l
\z*
D*(n (13.6)
The
limit
X(x,z)
=\imX n (x,z) / CO
is
obviously the formal resolvent operator for the equation (13.1). sufficient condition for the existence of X(x 9 z) is ^eadily ob
A
tained
from theorem
8 of chapter
3.
If
we make
use of the abbrevia
tion
and interpret
a 00 to
K it is
clear
mean
(
upon summation of the absolute values of the terms of n
(13.4) that the
the
sums
sums
S
t
=2
\a
ik
of theorem 8 become in this case
:
a
,
(n
a a 4(
^
'
c*
'
dx
n (n
1) a"/2
!
+ = e~*
x
(e
THE THEORY OF LINEAR OPERATORS
190
Hence we have the following result The resolvent operator X(x,z) corresponding to (13.3) can be obtained by the method of segments, provided :
dn
e~ x
(e
<
x
1
n
,
=

1, 2,
exists
and

ajc
14 The Permutability of Linear Differential Operators. We have spoken in previous sections of the theory of permutable functions in
the sense of the definitions of composition. It is now of interest to consider the problem of permutability for linear differential operators, the ensuing discussion being derived mainly from the investigations of J. L. Burchnall and T. W. Chaundy (See BibliograpJvy) .* We shall consider two linear differential operators, P(z) and and Q is of order n, subject to the conQ(z), where P is of order dition that they shall be permutable, namely that
m
Let us
first
Example Q(z)
is
consider some examples.
We
1.
shall
Computing the A(z)
Then
if
alter )mut,
A H=
0,
we
shall
P
A and B Now since
.
is
of first degree
and
is,
1
(z)
,
of
Q*P = p'z +
==P*Q
These equations are
where
assume that P(?)
of second degree; that
P and '
Q,
(23
?
we
get
+ ft)a'
have
easily integrated
= 2a + A
,
and one obtains (14.1)
are arbitrary constants.
we may write with the
help of (14.1)
*The reader is also referred London (1927), pp. 128132.
to E. L. Ince:
Ordinary Differential Equations.
OPERATIONAL MULTIPLICATION AND INVERSION
191
Since any Q written in this form is obviously permutable with see that this equation expresses both a necessary and a sufficiP, ent condition for the permutability of P and Q, when P is a linear and Q a quadratic function of z.
we
Example
2.
As a second example consider
the two quadratic
operators
Computing the alternant, we obtain
+
(af
From
+ 2
a") z
2/?'
a'/
the permutability condition
f
a'
,
a/
+
This system of equations lowing relationships
/J
+ /' =
2(3'
may
+ ad' +
a'/
we
0,
+
2/?'
(5"
yfi'
obtain
+
" ,
be integrated and yields the
fol
;
a
VzAa where
^4,
B, and
C
2
+B
+ Aa'=^ 2Aft +C
z >
(P
Q z> (Q
,
are arbitrary constants.
Let us now consider the operators P / is the identical operator / > f(x) f(x). inant, E, between the equations
P
(14.2)
,
HI and Q KI, where We now form the elim
+ az + p H = HI) == 2 + az + a'^ + ^ + ^ KI=z* + yz\6 K = KI) ==z* + yz + fz + dz + HI=zz
3
,
3
2
7f^
=
Kz
=
,
,
2
^'
,
and thus obtain
of the relations given in (14.2), we are able to elimthe functions of x and thus to obtain the following equation between the constants and
Making use
inate
all
H
K
:
THE THEORY OF LINEAR OPERATORS
192
H
2HK + K*+
2
44
(2B
2
)
H
2BK + B
2AC *=
2
.
(14.3)
Q > u = Ku, fiC
it
H
One thus reaches the
in (14.3). p<2>
=
> it takes account of the fact that P Hu and and is evident that P and Q may be substituted for
now
one
If
_ 2P 
Q
f Q<*> 4.
identity
4A
(25
2Q + D
P
2
)
2
2AC
0.
The results obtained in the preceding examples are capable of the following generalization :
Theorem 9. If P and Q are permutable m and n respectively,
differential operators of
order
P Q=Q
>
P
,
then they will satisfy an operational identity of the form
where f(H,K) Proof
is
a polynomial of degree n in
H and m in K.
Consider the differential equation
:
HI) u(x)
(P
=
(14.4)
,
H
is an arbitrary constant. Let the identical operator and um be a fundamental set of solutions of this equation. Let us also assume that Q is an operator permutable with P so that if Ui is any member of the fundamental set we shall have
where /
is
u lf

^2,
,
HI) *Q}>tti={Q> (P
{(P
Q
Hence
>
sively equal to
u
t
(x)
is
a solution of (14.4) and by setting we obtain the relations
i
Qv
?
where
when
K is <5
i
i;
=
is j,
u^O
.
i
succes
1, 2, 3, etc.,
/~i
If
HI)}*
/) I
iif)n
___
i
i
ni ft i" '^m'2'^2 ~
ft /} / tl '>witt'i

any constant which
I
'
*
*
_!_ ~
fl
ft
I
^nnn^m
satisfies the equation,
the Kronecker symbol which is zero when then there exists a solution u(x) such that
Q
>
u
= Ku
.
i
^
j
and
1
OPERATIONAL MULTIPLICATION AND INVERSION
193
K
m
Since there are obviously such values of for each value of find for every a set of systems of the form
H
H, we can
which have a common
m
% =
(P
HI)
(Q
KI) >u
,
(14.5)
=Q
,
solution.
H
for Similarly for every value of K, there exists n values of which the equations of (14.5) have a common solution. Thus between the numbers and there exists an algebraic relation of the form
H
K
f(H,K) which
of degree
is
Also, since
P
Let us
,
in H and m in K. u t= //w and Q ^ u^=^ Ku,
n 
/(P,Q)
The order of
0
w=/(H,A')
=0
this equation is clearly equal to
now show
follows that
it
(14.6)
.
mn.
that the equation
/(P,Q)=0
(14.7)
,
holds identically. Let us first observe that every solution common to (14.5) is also a solution of (14.6). Now let {H } be a set of k distinct numbers to which corresponds the set of k solutions {C7J common to the system have been inserted. (14.5) in which the corresponding values of The functions U i9 E/ 2 U k are linearly independent for if they were not then there would exist a set of constants {c,} such that t
K
,
p<2>
,
Operating on the lefthand member of this equation with P, ... pikD we then obtain the system
f
f
y
dff!' tf x
But
+ c H. 'U* Hh c*H 'U = 2
k
2
since the values of
H
l
k
,
j
=
1, 2,
,
k
1
.
are distinct, the determinant
does not vanish and consequently the values of the c l are zero. there Since this conclusion holds for any choice of the values of will exist an infinite number of linearly distinct functions Ui, U 2 which satisfy (14.7) But since this equation is of order mn, it cannot possess more than mn such solutions and hence must hold identically. The actual determination of the equation (14.7) is most easily accomplished by forming the eliminant for the set of equations
H
,
.
zr
z *
=
,
r
(Q _ KI) =
,
s
 (P
HI)
= =
0, 1,
,
n
0, 1,
,
m
1
I
,
,
THE THEORY OF LINEAR OPERATORS
194
PROBLEMS. Prove that the operator z + (x) is permutable with the 1. 22 4. p( x )z + y(x) if and only if z 2 4 p(x)z f 7(x) (x) [z where A and B are constants. (#) 4 B] [z +
~
+
+
S(x)z
operator A] >
j
,
2.
Show
that the operators
are permutable. 3.
P^z
If
2
4
a(x)z
+
and
(x)
13
=z + 3
Q
y(x)z
+
e(flu)
,
show
that
where F(x,
=
z)
+
3')s~
(2y'
+ ti6' + 5" + "_/?'5_ p"y
4 (2s'
4.
(25' 4
3
a
2
f
=3
+A
p"
333 show that
+
3113 n/J
 n+ 16
a"
a" y
3/3"
3a")z 2
y 7
)
55
4
ae'
.
3
<*2
8
1
2'
3y8'
f
5=:
,
y"
'5
2/3'y
For the operators of problem y
+
y'
a" 4
+
a'
>
/3
4
A
provided
B
+A B
13
0,
,
2
\
4
8
Q_>p
Q
2
4
/3'
P
C
42 a
+
where A, B and C are arbitrary constants. 5.
P
If
and Q are the operators of problem p<3)
15.
A
^
Q(2)
2,
show that
.
Class of Noji
quantum theory
of nonpermutable operators, the interchange of factors being subject to the equation
QP where
c is
PQ^c
(15.1)
,
a constant. said in the
As we have
first chapter, such operators were first studied by Charles Graves as early as 1857. They were introduced into the quantum theory by W. Heisenberg and have been extensively studied among others by P. A. M. Dirac, N. H. McCoy, W. O. Ker
mack and W. H. McCrea.
(See Bibliography) obvious that in multiplication the respective positions of the operators is of primary importance. Following a suggestion due to McCoy we shall say that an operator is in normal form when all the factors involving Q alone follow all the factors involving P alone. This definition is arbitrary since we might as easily have had the factors involving Q precede the factors involving P. However, if F(P,Q) is an operator in normal form, then one can easily show that It is
.
OPERATIONAL MULTIPLICATION AND INVERSION
F(Q, low
is its equivalent in which the factors involving Q.
P)
all
195
the factors involving
all
P
fol
For example, we have
C
)P=QP
cP
2
.
One of the simplest examples of nonpermutable operators is furnished by the specialization: Q *= x, P We see that z, c *= 1. t=^ fi & where is the operator introduced in section 12, chapter 2. QP We shall first prove the following theorem, due to Charles Graves. 9
Theorem 10. If F(Q) and G(P) are functions expansible as power series about the origin and if Q and P are nonper mutable operators which belong to the class defined by (15.1), then we have the following formal expansion:
F(Q)G(P)=G(P)F(Q)
Proof:
We
first
+ cG'(P)F'(Q) +
1
G"(P)F"(Q)
observe that
(QP)P= and hence in general
From
We
this
shall
we immediately
now show
Q n G (P) =G(P) Q
derive
that
+ c G' (P)
W C,
Q""
hc n
_j
+c
1
G
(
a
G" (P) M C 2 Q w
>(P)
w
Cw
Q"i G(P)
= Q G(P)Q + cQ G'(P)
M
C, Q"
which by (15.3) becomes
+ c G'' (P)
]
nd Q"
1
+c
2
[G" (P) Q
(15.4)
,
where n C r is the rth binomial coefficient. Using induction we assume that this expansion compute
is
1
2
+
true and then
,
THE THEORY OF LINEAR OPERATORS
196
=
for
Since by (15.3) the expansion was true for n 1, it must hold other positive values of n. The expansion (15.2) is derived in an obvious manner from
all
(15.4). this theorem, one
Using
Qs pa e$ e p
=p =
3
may
obtain immediately the following:
Q
g c p2
2
__
e ep C Q
Q
+6C
2
p
^
c
.
In the application of nonpermutable operators the normal form of functions of Q P is of importance. We shall first prove the
+
following theorem:
[See Bibliography:
Kermack and McCrea
(2),
p. 223.]
Theorem (P
11. If
we adopt
the notation
+ Q) = P + nCi P M Q + ^ pn2 Q n
(n)
2
_
1_
Qn
^
then the expansion
(P is
+ Q)
t=
(P+Q) (P+Q)


(P+Q)

given by
6
Proof:
We
first
+.
.
(15.5)
observe that
which may be derived immediately from theorem 10 or proved by direct multiplication and use of (15.1). Now assume that (15.5) is true and then consider
= P(P+Q)
(M)
2!/l!
+ (P+Q)
(n  2)
(4!/2!) n C 4
+
(P+Q)
Q+
c
(n^2)
(P+Q) <
1
>]
[P(P+Q)<*> (Jl 4)
Q+
c
(7z_4)
(P+Q) (w
 3)
OPERATIONAL MULTIPLICATION AND INVERSION
197
2r! r!
But from the identity (2r+2)
2l>!
(2r+2)
!
!
clear that the above expansion reduces to (15.5) with n replaced is obviously true for n 1, we may use
it is
by wfl. Hence since (15.1) induction to prove
it
From theorem in
P
y
Theorem and if
12. If
where (P}~Q)
rem
11, then
n
for
11
all
we
higher values of
n.
derive the following general proposition:
F(P)
is
a function developable as a power series
(n) have the same significance as in theohave formally
and (P\Q)
we
shall
F(P+Q)
)
]
+
(15.6)

Proof: Making use of the results of theorem 11,
F(P+Q)
2^n(P+Q) (n)
nO
oo
+ This tity
sum
is
(Vac)
2
+
(W3F no
4?
n
n C 2 (P+Qy^ i
'^> n C (P+Q) 2 F n ^ ^ w^o t
get
91
oo
en
we
+



.
seen to reduce formally to (15.6) through the iden
THE THEORY OF LINEAR OPERATORS
198
Under the assumptions
Corollary:
of theorem 12,
we
Jutve
It is a curious fact that nonpermutable operators of the kind just discussed are basic to Dirac's expression of the quantum theory. The connection is made through an extension to quantum mechanics of the theory of contact transformations as it appears in the founda
tions of classical dynamics. It is obviously impossible to sketch adequately the theory of contact transformations in a brief section, but the formal connection between such transformations and the quantum theory may be indi
For the basic theory of contact transformations and
cated.
their cen
tral position in analytical mechanics, the reader is referred to E. T.
Whittaker: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge, (1904), chap. 11, and E. Cartan: Leqons sur les invariants integraux, Paris (1922). An account of the extension of this theory to the phenomena of quantum mechanics will be found in Dirac's treatise [see Bibliography: Dirac, chapter 5] and in
WeyPs theory
of groups [see Bibliography: Weyl (1)].* Let (#i, Pn) be a set of 2n variables, which ,,
,
,
Qn P ;
lf
Po,
,
P
n)
by means
2n equations. Then form
of a set of
equations are such that the differential
dF = 2 [P dQ l
l
if
the
p dqj l
1=1
is a perfect differential when it is expressed in terms of p q and their differentials, the transformation is called a contact (or canont
ical)
,
t
transformation.
For example, the equations
define a contact transformation since
we employ
If
which
is
the abbreviation [u, y], called a Poisson bracket,
defined as follows:
q,
then
it
3p>
dp. cq
can be proved that the following* conditions must be satisfied
*The reader is also referred to an excellent work by E. Bloch: L'ancicmie et la nmivelle theorie des quanta. Paris (1930), 417 p., in particular chapters 11 and 18.
OPERATIONAL MULTIPLICATION AND INVERSION
199
provided the variables (Q lf Q 2 Q; Pi, P 2 Pn) and (q l9 q 2 are connected a contact transformation: Qn', Pi, P2> pn ) by ,
,
>
,
,
,
,
[Pi,P,]=0
[Q
,
f
,QJ=0
[Q,,PJ=d,,
,
.
(15.7)
In order to preserve the formal aspects of molar dynamics so that "classical mechanics may be regarded as the limiting case of quantum mechanics", Dirac was led to introduce operators p and q, which satisfied equations (15.7). Hence the basic commutation t*ule of Heisenberg appeared in the form
pq
qp
where h >= 6.547
=
c
[
p]
c ==
,
ih/ (2n)
,
27
10~ erg sees, is Planck's constant. In the Heisenberg theory the operators p and q are matrices and hence the third condition of (15.7) became [q, p~]
where
=/
,
/ is the unit matrix.
PROBLEMS 1.
Prove that
if / is
a function of
P and
Q, partial derivatives
may
be de
fined as follows:
dQ df
The following problems were communicated 2.
+ Q) =exp 2 c 11 [
Prove that
+ P)
exp (Q 4.
author by N. H. McCoy:
Prove that exp (P*
3.
to the
If
<(P)
is
= ^ exp
any polynomial
mn [
2
in P,
c
1
1
show that
expO(P) +Q]z=exp [A(P)]expQ where we abbreviate
5.
Prove that
if
a
is
a constant, then cr
e dPQ
=L
2 ^n ^n Q
n
/nl
,
THE THEORY OF LINEAR OPERATORS
200
where we abbreviate
6.
Establish the identity n
7.
8.
e^ 2 +Q 2
zn (sec 2c)* e^
Express
e p2+
' 1
k c nH(
tan
2<")
_
p2 c< r " 1 l0 ^ (0a 2c)pg
x
c^
'
1
tan
2(>
)3 2
.
in normal form.
16. Special Examples Illustrating the Application of Operational Processes. We shall conclude this chapter with three examples which will illustrate the usefulness of symbolic processes in applied prob
lems. These examples might be multiplied many times, as one readily surmises from the fact that operational methods are essentially invoked in any problem which is formulated in terms of a functional
The Heaviside
an outin an The discussed standing example application. problems this section were selected mainly to illustrate the varied fields to which symbolic methods may be applied. equation.
calculus, developed in chapter 7, is
of such
Development of the Disturbing Function of Planetary Motion is
One of the most striking examples of the usefulness of operators found in the extraordinary complexities of the problem of develop
ing the coefficients of the disturbing function of planetary motion. This application, the essentials of which have general interest, was first given by S. Newcomb in 1880 ;* the procedure was later recast in a more satisfactory form by K. P. Williams.f Suppose that we have two planets moving in elliptic orbits with semimajor axes a and a', eccentricities s and e', and perihelia at angular distances w and w' from the selected node. Let r and r', r < r', be the radii vectors, / and /' the true anomalies (the angular distances of the planets from their respective perihelia measured at the common focus in the sun), and V and Vf the true angular distances of the planets from their common node, that is,
customary to represent the mean values of V and V' by A and X', and the mean values of / and /' by g and g'. Thus in the discussion It is
*A Method of Developing the Perturbative Function of Planetary Motion. American Journal of Mathematics, vol. 3 (1880), pp. 193209. See also: Astronomical Papers of the American Ephcmeris, vol. 3 (1891), pp. 1200; vol. 5 (1895), pp. 148; 301378.
fThe Symbolic Development of the Disturbing Function. of Mathematics, vol. 51 (1929), pp. 109122.
American Journal
OPERATIONAL MULTIPLICATION AND INVERSION of circular orbits the quantities F, F', A, A', #, #',
and
a, a' respectively.
=
/,
We
and
201
r9 are replaced
by employ the further where / is the acute angle
f,
r,
shall also
abbreviations a = a/a' and osin i/kj, between the planes of the two orbits. If we represent by A the distance apart of the planets at a given time t e= t 09 then this function may be written
A where cos
W
is
2 9* (r f r
*==
2rr' cos W)
* ,
given by
cosTF^
a2 )
(1
cos(F
+
F')
2 cr
cos(F+
F')
.
The disturbing function may then be written
R
a
J
'
(r/r
^\
/2 )
cos IF
.
P
fVP
7
NODE Since, however, the second term by certain well
efficients of the first
may be absorbed into known modifications it
ficient to consider
the cois
suf
only the development of the first term. The problem of the disturbing function, then, is that of the explicit expansion of the reciprocal distance, J \ in terms of some one of the parameters involved. It is obviously not desirable for us to carry out in detail any one of these expansions since such expansions are of technical interest to the astronomers and may be found explicitly given in the references cited above. However, the operational method by which the obvious computational difficulties were modified is of general interest and merits description here. The problem just described may be stated more generally as follows: Let us consider a function, F(A,B), in which the parameters A and B are power series in a third variable, /t, that is,
A=
+^h+a
2
h 2 /2
1
f
a,
h*/3
!
+
,
THE THEORY OF LINEAR OPERATORS
202 If is
=
we now employ the abbreviations, p d/dx and q =: d/dy, it we may write the Taylor's expansion of F(A,B) about
clear that
the origin in the symbolic form
F(A,B) *=e*w*F(x,y) Let us
now adopt
*,IM>

the following abbreviations: &
= Ap + B q
e<=G(h)
,
Expanding the function G(h) as a power
G(h)t=G(0)
,
series,
+
we
seek an expression for the coefficients in terms of the elementary (r) operators r Let us designate G (0) by D r We then note that .
.
and hence, from the expansion of Leibnitz, that
G (w+1) (/0=2
(t+1
nC r
>G (w 
r)
(fc)
.
r_o
Letting h
=
in this expression
their operational equivalents,
4
=D
D n t=D
3
n
t
$i
+ 3 Do +
1l
we
and replacing the derivatives by
obtain
f) 2
C D n i & l
2
+
n
C
2
D^2 03
Hence we may write D! A
where
+D
2
7tV2!
JD 3
(Z)^) is the symbolic representation of the expression within
the brackets.
From
this it follows that
F(A,B)=G(h)
OPERATIONAL MULTIPLICATION AND INVERSION
203
= e* e<> *F(x, y)
F(A,B)
We may illustrate the application of this symbolic method in the 0. We then write simplest case, where both orbits are circles and o
=
where we abbreviate
A
=
or,
B
2 a cos (A
The development according
We
have # 1=
=
0, #!
to a
2 cos
can
(^
/I')
now be A')
r/
.
effected as follows:
=
y q,
where we abbre
viate
From D!
these values
we then compute
=
t=
v q
D
,
D Then from
4
2
=^
v
q
q* f
2
f
12
2 p 2 i>
q
2J
2
3
^^
J i/
+ 12 p
^
2
6 yp
f
.
,
the development
=1
we
D
,
%U+
I/)
+1
(*
+
2
I/)
readily compute
^ COS 1 = ]/,[! + cos 2(X >F^v _> F = ^o v + 1& = 4 [5cos 3 >
D! )
1/0
(A
''
A')
,
2
2
3
J5 3
#
F=
4
_,
F
(35 ^16
^
v
120
^ + 48)
/I')]
(A
A')
>
+ 3 cos (*
=o [35 cos 4 (A f 20 cos 2
be observed that, if we abbreviate pressions just attained may be written It will
where
Pn (cos
is
the nth Legendrian polynomial.
I')
(/I
^
A
*') ]
,
A')
+ 9] A',
.
the ex
THE THEORY OF LINEAR OPERATORS
204
Hence we have attained the classical expansion for the reciprobetween the two planets. The general case is the extension of these results to the more complex problem. In order to show the efficacy of the method in the general proband that both orbits are elliptical. lem, let us now assume that cal of the distance
^
or
be sufficient for our purpose to neglect the ellipticity of the outer orbit and to obtain the development of the disturbing function in terms of the coefficient s. In order to attain this development let us first write It will
a' Jo"
2
1
+a
2
f a*
A m cos m (V
B m cos Cm
A)
w+1)/'
(m
1) A]
cos [(/wf 2)A'
(m
2)/l]
[ (
A m E my C m etc. are functions of and a.* If V and I are replaced respectively by itf + the coefficients are represented by y*N, we may
where if
2
m _ j}
o
,
,
N cos (r let
iv'
2
w
and
g'
+
then write
+ w + vg' + /(
fi
g)
.
In order to avoid notation extraneous to the problem in hand, us further abbreviate this expression by writing
2
a' /Jo 1
Va
N cos pg
.
Introducing, now, the eccentricity of the inner planet, we replaced (i by r and g by /. Newcomb found it more convenient to use the logarithms of r and so we shall write ,
log a ~f
log r
f*=9
A
+B
where we have
A^=a m wi
we
If
we employ
c
=e
(m)
im
s m /m\
B^
,
ft
mi
a3
*For these et seq.
= =
.
the abbreviations
+ e~
lm<J
s
,
(m)
=e
im<J
find typical values of the coefficients to <*!
m m e /ml
V2
c(l)
2
,
e~ imy
,
i
=V
1
>
be
(3/4)c(2)
+i/2
,
(17/8)c(3)
explicit values see
Newcomb: Astronomical Papers,
vol. 5, p.
339
OPERATIONAL MULTIPLICATION AND INVERSION fa
,
= We
=
(13/4) is(3)
(5/4) is (2)
+
205
,
(3/4) is(l),...
note also the following combining forms of the symbols
c(m) and s(m)
:
= c(wfH) +c(m c(m s(??i)s(ft) = c(m~n) c(m)c(n)
n)
,
n)
.
Referring, now, to the general theory we specialize the operaby identifying the x and y of p and q with log a and # respectively and the a, and ft, with the coefficients just written down. and q only on cos ,u g, we have Noting that p operates only on for the explicit form of 9^ the following: tors $
t
N
$1 >
1/2
N cos
In similar
^2 
//
#
=
$i >
manner we
% N cos
.a
flr
N c (/O
find for
= (c(/*+2) P
+ c(//
N cos ^ (3/4)p + (5/4)/i] >
7^ 2
[
2)
l/^
[_(3/4)p
/<
(5/4) /i]} >
N
The other primary operators can be similarly constructed and from these, by ordinary multiplication and subsequent operation upon 2V c(//), the values of D,  N\ c(fi) may be computed. Thus one will have
c (ft)
= [c(//+2) + [/i i
+
2){(l/4)p [/i
(3/4)]p+ 2
(3/4)
]
p
+ /r
(5/4)
,,}]
>
N
.
The operators of Newcomb, represented by the symbol 77/, are Their efficacy is the coefficients of c (/*+.?) in the expansion DJs\ cos u g immediately evident when we see that each single term i/o of the expansion of a' Jo" 1 is expanded into a series with terms of the .
N
form
THE THEORY OF LINEAR OPERATORS
206

[77/
V2 AT]
e
9
COS
(pg
+ jg)
.
The complexities of the expansion when the second eccentricity introduced are similarly resolved. For further details the reader is referred to the original papers. is
The Propagation of Electromagnetic Waves
The equation of Maxwell for the propagation of electromagnetic disturbances in vacuo are conveniently written in the form i\
ax
77T
T
O
= curlAf
a
,
E e=
E E
fl
ft
o t
where
71
=
curl
8 t
M
M
E
(16.1)
t=and (A/,, M, ) are respectively the vectors of the electric and magnetic forces, x is the dielectric constant, // the magnetic permeability, and a I/c, where c is the ve
(E x
,
V9
z)
y,
:
=
locity of light.
and by div
By
E <=
curl
E we mean
the vector
the equation
Similar definitions hold for curl
M and div Af t=
0.
L. Silberstein (see Bibliography) has attained an elegant operational solution of the first two equations of (16.1) in the following
manner: Let us assume solutions of the form
(16.2)
where the that
is,
powers of t are functions of x, y, z. assume further that 3/at is permutable with the
coefficients of the
We now
curl,
that a
,
:
curl
= curl .
dt
Hence, substituting (16.2) in the readily obtain the following relations
a .
dt first
equations of (16.1),
between the
coefficients:
we
OPERATIONAL MULTIPLICATION AND INVERSION
_ (_1) V w ^ (_i)^ V m+i
<2mi)
2m
2i
M
(2M > err
(_ 1)^1
(
a ^) Cur l<2mD
curl
(2M1
>
*
E
V 2> CUrl (2w) >
^M
,
(m
=
207
1, 2,


OO )
,
;
M
,
where we employ the abbreviation v2
= (a
2
x fn) 1
.
be substituted in (16.2), the following symbolic then obtained:
If these values
solution
is
M (*//*) *{(<>* curl)
It is
^
obvious that a further simplification
may
be achieved by
writing
E
t
= (cos
(
vt curl )}
>#<>+
(//A)
}
>M
(sin(?^curl)}
,
(16.3)
M = (cos (vt curl) t
}
^
Mo
}
(*///)
(sin (v< curl)
}
> EO
.
be seen from the permutability of the curl with itself, that this operator may be handled as if it were an algebraic quantity. Hence a calculus in which the curl is replaced by the letter p would have essentially all the formal properties of the Heaviside calculus. We could, for example, invert equations (16.3) and thus obtain It will
E
n
= {cos (vt curl)} +Et
Mo ==
(cos (vt curl)
}

M
1
Oe/*) } t
f
(*//<)
M
{sin (vt curl)
}
~>
(sin (vt curl)
}
>E
t
t
,
.
This striking property of the curl has been employed by E. P. to obtain the solution of the first two form. in of integral (16.1) equations
Northrop (see Bibliography)
The Propagation of a Population By Fission In his well known volume on Natural Inheritance, London (1889), Sir Francis Galton proposed the problem of determining the
THE THEORY OF LINEAR OPERATORS
208
probable number of surnames, out of an
initial total of
N in a
stable
population, which would become extinct in r generations. A mathematical solution for this problem was attained by H. W. Watson. More recently T. H. Rawles (see Bibliography) has applied operational methods to a closely related problem which we shall discuss here. This problem may be stated as follows: Suppose that a population consists of N individuals divided into two classes, which we designate by x and y, and suppose that in the P initial state there exist P members of the x class and Q *= N members of the y class. Let us further suppose that each time a mem
ber of the population dies another divides so that the population reis to determine the probable distribu
mains constant. The problem
members in the rth generation. obvious that in the first generation
tion of the It is
we
shall
have the
fol
lowing possible cases:
A
member of (a) divides (b) a member divides; (c) a member divides (d) a member ;
;
member of the y member of the x member of the y and a member of the x
the x class dies and a
class
of the x class dies and a of the y class dies and a
class
of the y class dies
class
class
divides.
the
If the respective probabilities are assumed to be determined by in each class, then it is clear that the respec
number of members
tive probabilities, designated will
by p a
,
etc.,
pi
for the four contingencies
be the following:
P **
Q '
AflV
T
Let us
call
pbb
P Pl ^ WNl
~_ N
Pdd ~~
'
NI
the function So
=x
the initial state of the population,
We
Q Ql ^ _ 77^1'
'
p y<>
and
let
us define the operator
then observe that
A
>
S
=p
a
x p* y*"
+
(p b
+p
c)
x p y*
+p
d
x p " y*

1
determines the probable distribution of the population in the K N ~ K in the function generation. Hence the coefficient of X y
first
S r t=A >So r
K
will give the probability for the distribution of members of the and members of the y class in the rth generation.
N K
class
Now
consider the function
x
OPERATIONAL MULTIPLICATION AND INVERSION
209
which converges uniformly for x, y ^ 1 and e < I. It is, in fact, a homogeneous polynomial of degree N in x, y with coefficients which are functions of
that
a,
is,
2 = n2A n (c)*ir
(16.4)
.
o
It is
immediately seen from the definitions that
2
satisfies the
following equation
2
<>
J2 = So
(16.5)
.
2
can be found by the inversion of this equation, then the probable distribution of the population in the rth generation will be given by the coefficient of e r If
.
We
accordingly seek an explicit determination for the functions A n (e). Hence substituting (16.4) in (16.5) and equating coefficients, we obtain the system
An
i{A^(n l)(w+l) + A n+1 (n+l) (m
\A 1)
}
tl
[n(nl) +wi(w_l)]
=
d Pn
=
(n
,
0, 1,



,
N)
,
(16.6)
where we abbreviate i Kronecker symbol. The determinant of
e/[N(N
1)],
N
m
n,
and
d/.,,
is
the
this system, which we represent by D(h), can be factored into rational factors, and will be found to reduce to the following product
=//
D(A)
lq k )
(1
,
Ku
where we abbreviate q k
(N\~k
1)
(N
Jc).
Solving system (16.6) by Cramer's rule for
A where
a
(e)
=C
8
(l)/D(l)
A
s
(e),
we
obtain
(16.7)
,
the determinant D(k) in which the s column has been C, (/) the righthand members of (16.6), that is, by d, P replaced by Expanding the righthand member of (16.7) into a series of partial fractions, we obtain the following explicit solution s
is
.
lq k )
(16.8)
,
fc_i
where we abbreviate
which C' a (A) denotes C,(A) with the factor moved. in
1
i
N(Nl)
re
THE THEORY OF LINEAR OPERATORS
210
Expanding the partial fractions of (16.8) into series in I and replacing A by e/[N(N 1)], we obtain the desired expression for
A
8
(e),
namely, N
CO
2
20fc,(r)e'
(16.9)
,
where we abbreviate aks
The quantity *
=2
(r)
&*3 (r)
the probability that the population, originally in the state of P of P of the y class, has s members of the x class the x class and and 2V s members of the y class in the rth generation. is
N
PROBLEMS Assuming that two planets are moving in circular orbits, which are inclined at an angle / to one another, compute three terms of the disturbing function as a power series in the ratio of their radii. 1.
2.
Compute the value
of
D
Evaluate the equations the equations 3.
4.
N c(p).
> 3
(IG.fJ)
on the assumption that the vectors satisfy
Determine the probability, E(r), that
different, the
members of the
of the original members.
Hint: Let
P
=
I
in a population of
N
members,
all
rth generation will be descendents of a single one
(Rawles).
and note that
is
=N
N
times the a^(r) since E (r) is descended from a particular one of the
E (r)
probability that the rth generation original members. Show that
f
and hence that
fcD(2v
N(N1)
fc)r J
Prove that the population tends to a state in which descended from one of the original members. (Rawles) 5.
Hint: Show that lim E(r)
=
all
the
members are
1.
rjo 6. Prove that the mean number of generations for a population to reach the state in which it consists of descendents of one of the original members is
(N
1)2.
(Rawles) 00
Hint: Compute
2
r[E(r)
E(r
1)]
.
CHAPTER V GRADES DEFINED BY SPECIAL OPERATORS 1. Definition. In much of the work which has preceded we have been concerned w)ith formal aspects of our subject. It is now our purpose to introduce a concept which is deeply imbedded in the difficult convergence problem of the theory of operators, i. e., the concept of the grade (Stnfe) of a function defined by a linear operator.* By the grade of the function f(x) as it is related to the operator
S we
shall
mean
the limit,
L = lim sup L n N^J where we
define,
As an example
let
us consider the Volterra transformation, K(x,t)
It is well
known
that
ed in the interval a ^ x ^
if b,
n
\S (f)\
where
F
,
e
f(x) is a function integrable and boundthen the following inequality holds :
^ F Kn (b
a)
n
/n
!
,
K
the maximum value of f(x) in the interval and the value of K(x,t) in its triangle of definition. f are thus able to infer that L >= and hence that the series, is
maximum
We
PS'(/) is
an entire function of Similarly, if
we
A of
+A SM/) J
M S(/)
H
n
genus not greater than one.
consider the Fredholm transformation
S= Jr K(x,t)
e*> 3 dt
,
a
we know
that the series
*The term grade as a translation of the original designation Stufe was suggested to the author by J. D. Tamarkin. The limit L has also been called degree and exponential value, the latter being used by I. Sheffer [See Bibliography: Sheffer (3)]. fFor this inequality see the author's study: The Theory of the Volterra Integral Equation of Second Kind. Indiana University Studies, (1930), p. 9.
211
THE THEORY OF LINEAR OPERATORS
212
where h is the smallest zero of D(l)* converges only when A < A We are thus able immediately to infer that L 1//U Of special concern to us will be the elementary operators 1/2 and ,
=
.
Clearly the first of these is a special case of the Voltorra transformation and hence for it we have the simple result that L For the second operator, however, the matter is quite different as we can
z.
=
see
from the following
We
special cases
(1)
f(x)=e'
(2)
f(x)
(3)
f(x) *=l/x
:
,
= a polynomial
,
L=a
;
L=
;
L=oo
,
more thoroughly
shall discuss these results
.
.
in the next section.
2. The Grade of an Unlimitcdly Differ entiable Function. 0. Perron in 1921 (see Bibliography) stated several theorems relating to the grades of unlimitedly diiferentiable functions, by which we mean
the limit,
L = lim sup L n where we
define,
Ln
?=
(n)
/
(x)
,
1/rt . 
From tion
it is
the examples which we have given in the preceding secclear that L may be either finite or infinite. In the former
case f(x) has been called a function of exponential type.f If L is infinite or zero we shall indicate the mode of approach of
Ln
to these values
where or
is
by means of the customary symbol L n
a positive function that approaches
oo
(p(n) in the second.
O[tp(n)~\, in the first case
It will appear that the analytic properties of f (x) can be charmeans of the function (p(n). To show this
acterized in a measure by let
us expand f(x) about the point x
f(x)
where
cn
=C
=
f
Q
\CI(X
(n)
(a)/n\
a) .
+c
2
(x
=
a)
a, 2
\c n (x

The assumption that f(x)
a)
n )
,
an entire func
is
tion is equivalent to the assumption that
limF n =/ (n)
(a)/n!
^
Stirling's formula, n\ / 2n ^^ n comes, lim n co e L n / (2n)
Using
F
n^
'
nn
~
0,
1/w
e"
tl
==0
.
(2nn)
}
,
this condition be
from which we conclude that
*See chapter 1, section 10. fThis term is due to G. Polya: Analytische Fortsetzung und konvexe Kurven. MatJiematische Annalen, vol. 89 (1923), pp. 179191. Functions of exponential type have been specially studied by R. D. Carmichael Functions of Exponential Type, Bulletin Amer. Math. Soc., vol. 40 (1934), pp. 241261. :
GRADES DEFINED BY OPERATORS
Ln
f(x) is an entire function, then lower order than n.
if
=
213
[?>(%)],
where
y(ri) is of
=
a f(x) is an analytic function expansible about x a has the circumference it on of which within a circle of radius Q essential from or we have branch point, pole, singularity Cauchy's if
Similarly
theorem,
Making use
we
infer that
Ln
of Stirling's formula
=
(n) also note that if f(x) center a, and if \f(x)\
We and
and noting the constant
limit
.
is
M
<
analytic within a circle of radius r on the circumference of the circle,
then by Cauchy's integral formula r
~ 2ni ?I
we
T
'
(n}
\f
is
^dt (
have
shall
This
I
J
(a)\
< n\M/rn
(2.1)
.
known
The
as Canchy's inequality. class of quasianalytic functions is characterized similarly
Z
by the fact that
/(
~L/w(n)
is
We obtain the
divergent.
class of
fcth
7o
Denjoy, for example, if we have L 0(n log n logon logk n) where log.n n, log.?? log log log log log n> etc.* In order to sharpen our criterion for the case of entire functions
=
let
us recall the definitions of Laguerre and Borel. If in the function,
/(,?:)
Q(x)
is
f e * (} II
(1
X/d
tl
)
a polynomial of degree q
+
+
V~ / " a n Z
and k
is
'
'
xk /k
(l
k
n
,
the smallest integer for
00
which the
series,
2
1 1/l^ul^ converges, then p, the larger of the two
Wl
numbers
q and k,
is
called the
genus of the entire function f(x). The en
smallest value Q for which the series
2 l/Kl
p+ e ,
>
0,
converges,
is
called the real order of the function /(#).
From
Poincare's theorem that the quantity / '(0) P[(n ) h l)]/^! tends toward zero as n  oo, provided f(x) is a 4 l)/(p of function genus p and fc is any positive number, we are able to infor that fer every entire function of genus p, L n [r(?OL where (
+
=
*See A. Denjoy: Comptes Rendus, vol. 173 (1921), p. 1329; zinsky: Annals of Mathematics, vol. 30 (192829), pp. 526546.
W.
J.
Trjint
THE THEORY OF LINEAR OPERATORS
214 s= n (p ~ l)/p
*
For functions of genus zero for any finite value of a.f comes a (Ln) A theorem which applies to the function,
n
criterion be
this
n
g (x)
= Tl
x/a n )
(1
+ *'/ 2fl
2
+ * k/k a *
+
(2.2)
,
tt^l
can be derived from the known inequality,
where
a is the real order,
Replacing n\ by that
Ln =
If / (x) is
(2)
(L n )

n
.
an entire function of genus
n (p~ l)/p
L
=O
\fp
(n)
]
,
f(x) is an entire function of genus zero, then the limit for any finite value of a.
n l  1/a e^~ l/a
cr
If f(x) is
(4)
If f(x)
(5) 7t
then
.
1/<7 .
an analytic function with a regular singularity
in the finite plane, then
log
p,
:
is
Q
<
e.
If f(x) is an entire function of form (2.2) and a e= $ ) e, the real order of f(x), then w*e have the inequality,
(3)
where
ti
augmented by
approximation we are able to infer
,
(1)
n
of the function g(x),
/a e~^ /(J a 1/fr [y (n) ] where
where
co
Q,
its Stirling's
n
logo
is
cp
= n.
(n)
quasianalytic of
logfc n,
where
n
logo
Denjoy
= = log log log
class k, then
= log log n, Iog
3
n
n, etc.
PROBLEMS 1.
Determine the grades of the following functions: cosh x, log x, x x r(#),
2.
Discuss the grade of the function
,
n(x)
=
I
e**g(t)dt
,
Ja where a and 3.
6
are finite and g(t)
is
a function of finite grade.
Discuss the grade of the function
where g(t)
is
a function of
finite
grade.
*See fi Borel: Lemons sur les fonctions evtiercs. Paris, (1921), Chapter III; also Poincare": Bull, de la Soci&te Mathdmatique de France, vol. 11 (188283), pp. 136144. fFor an independent proof of this see Ritt: [Bibliography, Ritt (1)], p. 34.
GRADES DEFINED BY OPERATORS 4.
Let 6
be a set of complex numbers such that
b v 62 ,
,
lim(6 n )i/=0 If then
215
.
we have
prove that F(z)
> f(oc)
a regular analytic function in the entire region of [See G. Polya: Bibliography, p. 191.]
is
existence of the analytic function f(x).
Prove the following theorem:
5.
If in the integral
=
u(x)
L
f e** Y(t)dt JL
a of Y(t), then the grade of u(x) is a Cauchy circuit about the pole t if L is a Pochhammer circuit (see section 4, chapter 8) about the a and t b, then the grade of u(x) is equal to the larger of the two numbers a and 6 if L is a Laurent circuit (see section 7, chapter 2), then the grade of u(x) is infinite. is
equal to a; two points t
;
6.
If the function
f(x) is
of finite grade
=/ + fa/11 + / *V2! + />V3! +
show that the
g,

2
ff
series
=/ + /a* + /,** + /
(*)
converges uniformly in the circle
<
\x\
1/q
3
*3
+
'
'
'
.
3. Functions of Finite Grade. In many of the important applications of the theory of operators it is convenient to assume that the functions considered are of finite grade which we shall designate by
the letter
We
q.
Theorem f
(m)
(x)
,
and
1.
its
Theorem
below a number of theorems relating to four of which are essentially due to Perron.
shall state
such functions, the
first
is
If f(x)
mth
of grade
integral
I
1
Ja 2.
q,
then
iJO
3.
.
n
\d
If f^(x)
q 2 =^ q
greater than q 2
derivative, q.
/ a
If f(x) is of grade q, then f(ax) is of grade \a\q.
lim
Theorem where
mth
m f(t) dt are of grade
n n n (n Proof: Since d f(ax)/dx i=a f >(ax),
tively,
its
f(ax)/dx
and
a\q
follows that .
'
f 2 (x) are of grades
then a f l (x) l
n l/n
it
THE THEORY OF LINEAR OPERATORS
216
Proof: Since fi(x) and f 2 (x) are of grades q and q 2 there will exist a positive function M(n), such that lim l
,
qz
>
=
1,
for which the following inequalities hold
From
this
we
:
obtain the inequality,
and hence establish the theorem.
Theorem
4.
is
at
fi(x)f 2 (x)
Proof:
If fi(x)
is
of grade q
most of grade
q,
+
the rule of Leibnitz
By
qz
and f(x)
of grade q 2 then
.
we have
where C w
is the mth binomial coefficient. Since f l and / 2 are of grades q^ and q 2 respectively, they satisfy the inequalities stated in the proof of theorem 3. rt
Hence we
get,
and from this inequality the theorem is at once derived. That the grade q l ^2 may actually be attained by the product sin x, f 2 === cos x where function is seen from the example, f q l t= q 2 z= 1. Then we have fJ 2 1= sin a? cos x y sin 2a;, for which
+
l
q
=
9
=
2.
Theorem 5. If f(x) is of grade q, then there will M, s and n all independent of n, such that
for
all
exist positive
f
constants
,
values of
n
greater than
n
f .
Proof: Since by hypothesis lim sup n
(w)
/
(o;)
1/n
^
q,
there exists
j>
a positive function M(n), lim [M(n)] 3/rt
t==
1,
such that
HJ)
(n)
\f
(x)\
^M(n)q n
.
= 1 + e/q > + s/q) value of M will dominate M(n)
But since lim [M(l MLO
"]
1//l
a properly chosen nf The theorem is a consequence of this .
1,
this function for
when n exceeds
fact.
Theorem 6. If f(x) is a function of grade q and if F(z) is an operator with constant coefficients which possesses a Laurent expanwithin an annuhts that contains z sion about z q within it or
=
=
GRADES DEFINED BY OPERATORS
217
its interior boundary, then the function F(z) > f(x) a function of grade q.
upon
Proof: Expanding F(z) in
we
scribed
F(z)
is
at most
Laurent series in the annulus de
its
shall have,
= (a + a,z +
+
2 z*
+
)
1
(friz
+ &2*r + 2
Operating upon the function F(z) > f(z) with
we
the absolute value of the result
z
and taking
tl
obtain,
+f(x)}\
+&2/
^
(x
2
t)
+&
ll+3
(w
(*_i)Y2!

>)
+
(3.1)
H
H6

/l
..
From theorem 1 we know that if x be restricted to some interval we can find a sequence of positive numbers M n n = 0, 1, 2, such that lim [M n ] 1/H *= 1, for which the following inequality holds: (ab)
,
M
,
M
Let us assume, moreover, that n +i so that the sequence n ^ no null and thus has elements. Under this is nondecreasing {M n } condition we know that lim [M n ] 1/n 1 implies the limit lim n+p /M n
=
M
H~JJ
nij^
every value of p. Replacing the various terms of (3.1) by these dominating values and using the abbreviation in p (n) we then get, n + p /M n 1 for
M
,
\z{F(z)*f(x)} where we abbreviate, r
m
r
(n)q
r
72
,
()
=
\b r
\m. r (n) q~
r ,
r=i
rO on
Jx a
From

&*,

(*
t)>f(t) dt/ml
.
niO
the condition, lim n=
m
r
(n)
=
1,
which holds for every value
j)
K
r are able to conclude that m r (n) is dominated by where q* is a constant independent of n and r, and d is an arbitrarily small positive quantity. Since, moreover, q is an interior point of the annulus of convergence of the Laurent expansion of F (z) the series
of
r,
we
K
,
,
THE THEORY OF LINEAR OPERATORS
218
we have the inequality, h(ri) ^ 1^ for all values of Moreover we obtain from n ^ n \, the inequality m_ r ^ r < n, and hence we conclude that
converges and
^
for
all
M
M
n. 1,
values of w.
Finally, since lim
6,
=0 and since f(t)
is
bounded in
(a,b)
,
we
r'jo
can write,
where ]/(*) < F and &, < B. Taking the nth root of the
and noting the inequality, [A n
we
tive,
obtain, lim
n
inequality,
> {F(z)
\z
A
< A f B, for and B posi> /(#)} 1/rt ^ g, from which the
f J5"]
1M
theorem follows as an immediate consequence. A result of more extensive character than the one just proved may be established for operators introduced by differential equations of Laplace type. This result follows :
Theorem
7.
If
an operator F(x,z) can be represented
in the
form,
F(x
9
z)
=**
f
e rt
Y(t) dt
,
then the function defined by F(x,z)  f(x) exists and is of grade q provided f(x) is of grade q and Y(z) is analytic throughout the interior of the circle @ R, where R is greater than q.
~
Proof: Since f(x) is of grade q it is dominated by a function of the type P(x) e qx where P(x) is an entire function of genus zero. We shall, therefore/ consider the operation F(x,z) * P(x) e qx Let us first write the identity, .
F(x,z) 
(uv) =.i(F(x,z) *r
+
?{'F'(z,;:)
 v
+ u"F"(x,z)
(3.2)
>v/2l\
,
where the differentiation refers to the variable z. P(x) and v <= eqx we Specializing this formula by setting u
=
get f)
+P'F/(or,9)
P"F "(x,q)/2 (l
!
+ } e<*.
(3.3)
GRADES DEFINED BY OPERATORS
D
= er*(Dx)
n
~
d/dq we where we set,
Employing the abbreviation F(x,q)
> I(x,q),
dt
Substituting this in (3.3)
we
219
also note that
Dn
*
(3.4)
.
obtain,
on
F(x,z)
^Pe =^P^(x)(Dx) n *I(x,q)/n\ n=o (lx
.
(3.5)
But we
D
from
> I(x,q) (3.4) that in and Y(q), general,
see
D n +I(x,q) Making use
&<*{
of this
(D+x)
it ux"
tt
we
1
= e^(D\x)^^ Y (q)
(D+x)
Y(q)
}
2
> /(
(3.6)
.
are able to derive,
nx

1
= e*T (g), D
+
"~ 2
+ n ()irI)x
2
(D+x)
"
/2
!
l)M"7(^g)
(
(l)
n
x n l(x,q)
.
But since Y(q) is analytic its uth derivative, by (4) section 2, n 1/n 1. Hence the function dominated by n a nl where lim [M n ] n n n > "x is dominated a;) } 1) F(^) by nA nl where ( ~]/(D { \_D A is suitably chosen. But since P(x) is an entire function of genus zero we can find a set of values m such that \P (n) (x) < m n /Q n 1 From this we see that where Q is arbitrary and lim ['W w ]/ 1//l rtLTl K the series expansion of F(x,z) > P(j?)e' P(0) I(x,q) is domi
=
M
is
M
+
,
tl
\
nated by the major ant, 00
V jLJ
M
n
i,,
(A/Q)
,
Q >A
,
lim (M,,m n )
'/
=1
.
y)
n=rO
We
are thus able to conclude that (3.5) is uniformly convergent. In order now to show that F(x,z)  Pc qx is a function of grade q we observe the identity,
71=0
and write
(3.5) in the form,
F(x,z)
*P
THE THEORY OF LINEAR OPERATORS
220
Making use
P(D)
of (3.6)
> I(x,q)
we
achieve the further simplification,
^=P(Q)I(x,q)
=
where we write Q(z) \P(z) Expanding Q(D^~x) as a
we
l f e'
*Q(D\x) > Y(q)
,
P(0)]/z. series in
D
and operating upon Y(q)
get,
Q(D+x)

Y(q)
^Q(x)Y(q)
+ Q'(x) Y'(q) + 
Let us
now
Q
(wl)
(x)
operate upon this equation with
Y (m) (q)/m\\zn
.
and discuss the
result.
+ QOHO
(
x)
y (m) (q)/m \\
.
is analytic throughout the interior of the circle m dominated 1, m a ml q $ a < R, lim [M m R, by M m >0. Moreover, since Q(x) is an entire function of genus zero, which has Q (n) (x) is dominated by a sequence of positive values for all finite values of A. We can also the property that lim A"^ n
Since Y(z)
Y (nn (q)
M
is
l/>>l
~\
,
/
=
M
assume without
=
/
loss of generality that
iy n
^
^
/ltl
.
Hence we attain
the inequality, oo
S,,(a)
s 2] ftf^/'^a" m 0
.
Making use of the limiting property of the sequence {y n } we see that this series converges and hence furnishes a Weierstrass majorant for the function S n (;r) for every value of n and for a* in any finite interval.
Furthermore, from the assumption
?/
M
^
;
w+1
>
But from the limiting property of the sequence a positive quantity C, independent of n, such that y n
Hence we can
0,
we
{y,,}
obtain,
there exists
< C/A n A > ,
a.
write,
M
From the limitations upon m this series is seen to converge and the majorant thus obtained establishes the fact that S (x)=Q(D+x) is

a function of genus zero in the variable x.
GRADES DEFINED BY OPERATORS
Combining these
results
we can
221
write (3.5) in the form,
r F(x,z) *Pe" ^P(0)I(x,q) +e'"S(x)
where S(x)
a function of genus zero.
is
We now
(3.7)
,
need the lemma
Lemma. The grade The proof
is
:
of I(x,q) docs not exceed
q.
immediately attained by the use of the Schwarz
in
equality, (see section 9, chapter 3) l
dt
u*(t) J("
("u(t) v(t)dt JQ
v*(t)\dt^
(
^o
o
Forming the nih derivative of I(x,q),
we
=e
specialize u(t) (n)
\I
rt
Y(t), v(t)
(x,q)\^{
2xt
\e
('
Y
t
2
Jo
n
and thus obtain,
,
(t)\dty q
Since the integral exists by hypothesis sired inequality, lim [/ (M) (x,q) ] 1/w q
^
n
(2n+I)^
we
2 .
obtain at once the de
.
Employing this lemma and making use of theorems 3 and 4 we are able to conclude that the grade of the function defined by (3.7) is q.
Corollary
1.
If F(x,z)
is
the operator,
F(x,z) =e~*z
r e^Y(t)dt v
,
a.
then the function F(x,z) /(.T), where f(x) is of grade q, grade not larger than the larger of the two numbers \a\ and q.
The proof
is
immediate
if
we
is
of
write
and note that *
is
= /(O)
The result of the lemma proved above combined with theorem 6 a statement of the corollary.
THE THEORY OF LINEAR OPERATORS
222
If F(x,z) is defined to be the operator of theorem 6 a if P(x) function of genus zero, then F(x,z)  P(x) is a zero. function of genus
Corollary
and
2.
is
4.
ficulty
Asymptotic Expansions. Perhaps the most characteristic difwhich one encounters in a practical application of formal oper
ators to the solution of linear equations is the frequent intrusion of divergent series into the calculation. We have already encountered such a difficulty in section 11 of the preceding chapter and many others will appear in the ensuing pages. It is impossible to make an exhaustive investigation here into the fascinating but profound problems presented by divergent series. We shall find it necessary, however, to be familiar with certain aspects of what are generally referred to as semiconvergent or asymptotic series.
An
asymptotic expansion of a function f(x)
a series of the
is
form,
a
+ ajx + a /x 2
limx n \f(x) the
sum &
1
S n (x)\=0
a n /x n

,
(nfixed)
,
'
a?loo
is
}
satisfies the condition that
which, although divergent,
where S n (x)
2
+ &2/# +
+a
2
(4.1)
,
n
/x If this condition, which was first stated by H. Poincare in 1886,* is satisfied we say that the series is the asymptotic expansion of f(x) and we represent it by the symbol, f(x)
CND
a
+
f
0,1
ai/x
/x
+a
2 2 /a
+ .a
n
/x
n
.
n \
(4.2)
.
The first example of an asymptotic expansion encountered by mathematicians, and certainly one of the most interesting, is that associated with the gamma function, c?x**r(x)/(2n)* OD 1 f i/i2x
+ 1/288.T
2
139/518400
3
.f
The characteristic property of asymptotic series which brought them into prominence in astronomical investigations long before they were regarded with even moderate favor by mathematicians, was the fact that for large values of x they converged in general with great rapidity to the value of the function. The error was noticed to be a
function both of the magnitude of x and of the number of terms used. Hence for values of n which did not carry one too far into the divergent expansion and for correspondingly large values of x, the error Ics integrates irregulieres des equations lineaires. Acta Mathematica, (1886), pp. 295344; in particular, pp. 295303. fThis expansion to O(l/x s ) is given in Davis: Tables ol the Higher Mathematical Functions, vol. 1, p. 180.
*Sur
vol. 8
GRADES DEFINED BY OPERATORS
223
was
in many practical applications extremely small. It has become almost a rule in the computation of tables of functions to seek first the asymptotic series. One of the most important clews to the nature of asymptotic series is the Borel theorem on continuation, which we state as follows: a n are singularities of the function, If a l9 a2 ,
,
u(x)
=
/^oo
e f
(4.3)
,
'O
where
(z)
t=
2"
An
z n /nl
and lim A n
1/n 
exists,
then u(x)
is
analytic
throughout the interior of the polygon containing the origin and formed by drawing through a perpendiculars to the lines joining the t
singular points to the origin. The proof of this theorem will not be given here since
it is to
be
found in numerous places.*
The
summation of divergent
application of the theorem to the
series is at once apparent from the following consideration. If the expansion of y(xt) is placed in (4.3) and account taken of the inPOO n e~ t t dt I n\ , the following series is obtained : tegral identity,
=
*J o
00
u(x)=J^A n
xn
(4.4)
.
nQ
It is clear that this expansion converges for .r < a, where a the smallest of the moduli of the singular points, x a,,, a2 etc., but diverges outside of (and possibly on) the circle of radius a. Hence, in general, equation (4.3) furnishes a functional equivalent for series (4.4) in regions where it is divergent. For example the series,
=
is
II ( tt' \^
is
'V* t/v
\ J
1 J
r~ I
w
'7*
.
!
0*^ v
I
divergent on and exterior to the unit
I
,
...
I
circle.
,
However,
its
Borel
equivalent,
u(x)
=
f #*(! +
tx
+ t*x*/2l + .)
dt
*^o
(J/L
,
/; converges in the half plane, R(x)
<
1,
where R(x) designates the
real part of x. *E. Borel: Lemons sur les Series Divergentes. Paris, (1901), chapter 4; Whittaker and Watson: Modern Analysis, 2nd ed., Cambridge, (1920), p. 141; Davis: Tables of the Higher Mathematical Functions, vol. 1 (1933), pp. 4547.
THE THEORY OF LINEAR OPERATORS
224
Although the Borel theorem applies only to functions for which
we encounter
no simple way appears for defining
a blemin general the
region of summability.
An
example
is
found in the
u (x)
which
is totally
=1
a;
classical series,
+ 2 \x*
3 \X A
+
,
divergent.
The equivalent Borel
integral,
exists throughout the half plane, R(JC) In this book we shall be concerned
>
0.
mainly with asymptotic series which appear in expansions of a Laplace integral of the following general form,
f JL
F(x) where
L
c*^
D(t) dt
(4.5)
,
path is the positive usually effective either in determining the region of summability of the asymptotic series or, if the integral representation is assumed as given, in determining the
real axis
is
a path in the complex plane. to o>, the Borel theorem
from
If the
is
form of the asymptotic representation of the
integral.
In order to effect this equivalence, let us make the transformation s #, where we abbreviate y .= ,(/(0). We shall then g(t) obtain (4.5) in the following form :
F(x)
=
=.c~^
(
'e**S(s)ds
(4.6)
,
V (g~ !>+{/] ) /g'(y O+0] ) in which where we write, 9(s) means the inverse of the function g(x). If we now assume that (s) is a function of the form, }
~
l
,
l
g~ (x)
()
where
is any function for which the integral mal expansion,
exists,
then (4.6) has the for
GRADES DEFINED BY OPERATORS cc
e*F(x)
A sv
more general expansion may be obtained by writing, e(s)
still
logs
=
Noting the integral,
(s).
Too e*
s/i
log s
cte
=
[F(/z
+
l)ArMH][*(ji +1)
Jo
where *(/*
225
+
!)
r'(/*
+
1) / r
+
(/*
1)
,
we may
_
write,
eo*F(x) 1)
Other generalizations are obtained by taking the nih derivative with respect to n of the Borel integral.
Returning now to equation (4.5), we may broaden our inquiry by seeking conditions under which the path may be so chosen that the integral has an asymptotic expansion. This problem w&s apparently first suggested by G. F. B. Riemann (18261866) and the solution indicated in a posthumous paper.* The details of its development are due to a notable contribution made by P. Debye in 1909, who applied it to the study of the asymptotic development of Bessel functions.f In equation (4.5) let the path L be so chosen that the following conditions are satisfied. First, R{g(t)} shall change as rapidly as possible along L; second, as
have the
limit,
Km R{g (t) }
t
*=
approaches infinity along oo,
where R{g (t)
}
L we
means the
shall
real part
f=oo
ofg(t). t
In order to see the significance of these conditions } iv, and thus obtain,
=u
g(t)
As
is
well
=R(u,v) \H(u,v)
known both R(u,v) and
let
us write
.
I(u,v)
satisfy Laplace's
equation,
+
3
=
*Sulh> svolgimento del quoziente di due serie ipergeometriche in frazione continua infmita. A fragment edited by H. A. Schwarz from Riemann's: Gesa/mmelte Werke, 2nd ed. (1892), pp. 424430. fNaherungsformeln fur die Zylinderfunktionen fiir grosse Werte des Arguments und unbeschrankt veranderliche Werte des Index. Mathematische Annalen, vol. 67 (1909), pp. 535558.
THE THEORY OF LINEAR OPERATORS
226
and the CauchyRiemann conditions,
aR
di
dR
di
(4 8) *
In order to determine the path along which R{g(t)} shall change as rapidly as possible, we set the derivatives of R(u,v) equal to zero,
^
=n ^ = Tdv
3/2
/
=
Then from the CauchyRiemann equations we 3//9v
=
0,
shall have,
and hence derive,
= dnThus we
x
(49)
fill
t=
.
dv
find that the desired
I(u,v)
where k
path
t=k
L
the one for which,
is
(4.10)
,
is to be determined from the solution of equations (4.9). If consider the equation R R(ii,v), we note that the points Ro derived from (4.9) are not points of maxima or minima in MO, v view of equation (4.7), but are saddle points or passes on the sur
=
we now ,
face. Hence the curve (4.10) is a curve on the surface which passes through one of the saddle points and along which R g(t) makes its most rapid change.
=
t a Returning now to equation (4.5), let us designate by t saddle point and let us effect the transformation: s G g(t) where we write G g (t ) Then obviously since g'(t) 0, we shall
=
haves(^ It is
)
=s'(t
)
~ =
.
,
=0.
then possible for us to write,
s= (tty {0 +
fi r
1
(t_t
+g,(ttr [>}
)
.
(4.11)
=
=
T we shall have dt/ds Employing the transformation s T*dt/dT from which we proceed to evaluate dt/dT in terms of T. To accomplish this we write, 2
,
1/2
dt/dT and then evalute the
^a T
coefficients
0.= where the integration
is
,, < 1/2*1)
n
n
(4.12)
,
by means of Cauchy's theorem, C dt
dT ,
Jjy
around the zeros of the Tplane.
GRADES DEFINED BY OPERATORS In the
plane this
an
is
equivalent
= (1/2*1)
227
to,
f [s()] i(n+1) dt
,
(4.13)
the values of a n thus being determined from the coefficients of equation (4.11). The following explicit values will be found useful in the application of this theory:
+ Gg^/Oo
203/00
5
3
2
3
40 1 /0
7
)
+
2
3/<7o
=
(4.14)
,
2
J
/
2
=
T2 s* are substituted two values of T, 7\ f s (4.12), two expressions for d/ds are obtained as follows: If the
}
,
Returning to the original integral we are now able to write
it
in
in
the form,
= er r
F(x)
{
f
6 #(*,)
fvj
^ ds +
e**d(t 2 )
J
ds
/v
where L x and L 2 are the two branches of the path L in the splane, these branches being separated by the saddle point. Since s is real
+
L 2 it is seen that the integral can, in general, be over the path Lj expanded by the Borel theorem. As an example let us consider the expansion of the integral, ,
JfL Setting g(t)
= cosh
t,
we
g'(t)
from which we get f
c=
0,
we then
t
=
2*
,
08
=
t
fa
^
obtain the definitive equation,
t=sinh
= imi, n =
have,
We then compute
e x cosh
the values,
0,
= 1,
,
2,
.
Selecting the value
THE THEORY OF LINEAR OPERATORS
228
The path of integration about the point
seen to be the axis of reals and from oo, it is clearly the circuit from infinity
is
R[gW]
the condition lim
=
tn
Hence we
0.
e 18 {
x F(x) *=e f
get,
(dt 2 /ds)
(dt,/ds)
}
ds
,
JQ
%"
er*
1
S'
I
(a f a 2 s
Jo
,
When F(x)
If
(4.13),
the Borel integral
is
+as
2
4
\
ds
)
.
employed, this becomes,
more terms
of this series are desired,
we
return to equation
which in the present instance becomes, a w =(l/2:7iO
(coshi JfL
I)""*
1
'
dt
,
r7o
2!
J where we have written n
+ %>Vi
= 2m.
'
=
Since the path is about s 0, it is merely the coefficients of s m in the de0of the function 2* ( 1 f 1/2 5 )' velopment in power series about s The reader will notice that F(x) is twice the Bessel function
clear that the values of
cr.
2w are
=
1
PROBLEMS 1.
Derive the expansion 00 7
et*
2.
dt
~
1 e*
135
13
1
2
(
1
Determine the asymptotic expansion of the
gamma
function by consider
ing the integral
F(x)
Show
I
e~ s
x 1 ds i=
x^ 1
e*
Jc
I
Jc
e~* f
=
t 1, and that the path of integratwo branches (0,1) and (l,oo). Expansion Related to Stirling's Formula, derived
that the saddle point
is
the point
tion is the axis of reals consisting of the
[See G. N. Watson: An by the Method of Steepest Descents. Quarterly Jcnimal of Math., vol. 48 (1920), pp. 118. Also H. T. Davis: Tables of the Higlwr Mathematical Functions, vol. 1, (1933), pp. 5657.]
GRADES DEFINED BY OPERATORS Prove that
3.
 2

log x
where
229
Bn
is
Show
5W (^)
+
or
C
oo
(
e**\
J
is
e
,
x* n
defined by
t
I]
hUt.
e
}l
is
the
sum
<"'
a:
dt<*>
t\
f
112! 3! j
x2
or
/:
a*
x^
of n terms of the series, prove that
SH (*)<
/(*)
I/ (2in)
also prove that
lim *{7(*) 5.
B" o 2n
that
f^ J If
l)^
(
number and ^(x)
the nth Bernoulli
t(x)= log 4.
^
2x
(*)}[
=0
if
x
>
2n, then
;
.
Find the asymptotic form of the solution of the equation /00
U(X
(l/2a*)
)
+
V2
e<*<>
u(t)dt
.
J* 6.
Given the equation
=log x
&u(x)
show that the asymptotic form co
u(x) If k 7.
=V
2
log
k
_}
,
of the solution to a first approximation
(x
1
log x
)
i
show that u(x) =.
2vr,
Derive the following expansions
r^
cos
=
t
dt
I
J
(O
1
i
A
sin x
+
_ (A cos x f
B
(
Z>
cos
a;)
(*')*
J"
^
f
sin
J
1
f
_ dt z^
sin c)
x
where we have
1357
13 (2x)2
135
1
Hint
:
Consider the integral '00
and make the transformation
t
=
(2aj)*
135 79
,
,
is
given by
THE THEORY OF LINEAR OPERATORS
230
The Summability of Differential Operators with Constant Coefficients. In theorem 6 of section 3 we considered the case where F(z), a differential operator with constant coefficients, operated upon 5.
a function of finite grade. It is clear, however, that this restriction excludes from consideration a large class of problems of which the equation which defines the psi function, y (x) may be cited as typical :
,
u(x
~\
u(x)
1)
l/x
(5.1)
,
or in terms of a differential operator,
Since l/x
is
>
1)
(c~
u(x)
= l/x
.
a function of infinite grade it an operation upon
ticipated that, in general,
function of infinite grade. of equation (5.1), namely series
As
is
tj>(x),
is
certainly to be an
it
will yield another
known, the principal solution appears as the following divergent well
:
'jn
y>(x) co log x
(l)"*B n /2n x 2n
l/2x

,
(5.2)
71=1
where
B
n are the Bernoulli numbers (see section 12, chapter 2). This series, however, is summable by the method of Borel and we are thus able to obtain the following integral equivalent of (5.2) :
e
The theorem
situation that
l
)
+l/t}dt
here revealed
is
R(x)
,
>
.
covered in the following
is
:
Theorem
8.
If f(x)
is
of the
form
=g(x) +h(x) of finite grade L and where
f(x)
where g(x) form
is
a fimction
h(x)
,
= hi/x
(
h.2 /x'~ [
h 3 /x*

h(x)
is
of the
,
then the summable equivalent of the function,
u(x)
=F(z)
>f(x)
,
is a differential operator with constant and has the form
in which F(z) exists
where
W
3
/3!
+
,
coefficients,
(5.3)
231
GRADES DEFINED BY OPERATORS
L <
where Q is the radius of convergence of F(z) ; and m exist such that Q(t) F( t) < t ) is of limited variation for 0
provided, (a)
Q,
(b) positive values k, A,
Aemt
\
\
>
,
in the interval
^t ^ k. 00
Proof:
It will
be clear that the series
b L 2 ^^
i
l
forms a majorant
for the series
U(x) by hypothesis L is the grade of g (x) and is smaller than the radius of convergence of the series expansion of F(z). Since U(x) is thus uniformly convergent we may form the derivative since
(5.4)
.
From theorem
1,
we know
section 3
that z n > g(x)
is
of grade
GO
L and
thus that the series
L n 2 &** forms
a majorant for
U
(n)
(x).
i^i
thus appears that U(x) is at most of degree L. It will be observed that this argument is merely a special case of the one employed in the proof of theorem 6, the annulus in this instance being merely the circle about the origin. We now consider the equation It
= F(z)
V(x)
Applying the explicit expansion of the resolvent
V(x)
= h {b /x + h {b,/x
+ 2 Ib /x* 2 Ibjx* + 3 \b /x*
b^x*
2
2
Slbjx*
+
4 !& 3 /ff 5
z
2
to
In general this series will be divergent, but
h(x)
e~ s s n ds
=n
+
it is
}
usually sum
!
J.
from which we obtain V(x)
V(x)
= (l/x)
in the
form
/too
[e^h, */
+h
2
(s/x)
get
}
mable by the method of Borel. This makes use of the identity 00
we
+ h.(s/x)
2
/2l
THE THEORY OF LINEAR OPERATORS
232
Making
the substitution s
V(x)
=
=
we
tx,
obtain
f" e*'Q(t)F(t)dt
,
where Q(t) is defined by (5.3). The convergence of this result is easily established under hypotheses (b) and (c) stated in the theorem for we shall then have /*OO
\V(x)\<M/x + A J
e<
I
m
Q
forR(x)
The case where Q(t)
>m
.
t) has a pole of unit order in the inof as follows easily disposed Consider the function
terval (0,
F(
is
oo )
:
G(t)c=Q(t)F(t)R/(a where
R
is
that G(t)
the residue of
Q (t) F(
regular at
a so
is

t
The second
m=
its
clear
[e~<*R/(at)\dt.
\
(5.5)
is
divergent, but
is
seen to correspond for
lto the solution of the equation, d/dx)
(a
which has for
is
*^o
integral
mally for the case
at the point t a. It consider the function
/'oo
eG(t)dt+
Jo
,
=
t)
we may
/oo
V(x)i=\
t)
m
>W(x)'=R/x
,
particular integral the function
More generally, if on the positive real axis a,, a 2 , a m are poles of unit order of the function Q(t)/F( t), then the solution of the ,
original equation will be
m
/oo
V(x)<=\/0 where the
R
t
f*x
SRt
e^'GWdt
(e'^/t}dt
are the residues of Q(t)
G(t)=Q(t) F(
t)
,
(5.6)
Jin
i=i
F( "j^R
t)
t
/(a
at the points a,
i
t)
and
.
i=l
The case by writing
of poles of multiplicity in the form,
G()
G(t)=Q(t)
m
is
m
F(t)
^
l
treated in similar fashion
GRADES DEFINED BY OPERATORS
A m are the Laurent coefficients Q(t)/F(t). One then adds to the integral,
where the
233 the expansion
in
of
/* 00
e
tx
G(t)dt
,
JQ
m functions obtained from W (x) m and replacing R by A m 3,
by
letting
m
assume the values
1, 2,
.
,
We
shall consider a
Example
few simple examples
:
Let us assume the functions,
1.
and h(x)
We
then compute,
F(z) > h(x)
= l/x
2 l/x ~
Q()
= l/x
er
f ,
and hence obtain,
+ 5/x
2/x*
3
16/s
4
+ 65 A
5
/*00
Jo
Example 2. In the preceding example following totally divergent series, h (x)
We
= l/x
then compute, Q(t)
F(z) >h(x) *=l/x
l/x
2
+ 2 l/x*
let
us replace h(x) by the
3 l/x 4
.

= I/ (1+0, and hence get, 4!/ 2!/x + 3!/r 3
2
=
=
Let us assume, F(z) 1, 1), Q(t) l/(cso we t unit at a that has note 0, pole t) F( I/a?. compute the residue at this point and hence construct the function,
Example
S.
=
We
=
Now
O+l/t
e*/(l G(t) =I/(e* l)+l/* employing formula (5.6) for the case
m
=
1,
we
readily
obtain,
F(z) >h(x)*={l/z
= + log x + c
1/2
+ B&/21
(l/x)
{
i/2
B
+
2
(
z*/4l
1) >B f /2px"}
Jfcrl
= fV^eVd Jo
c')
+ 1/t} dt+
(*dt/t, Jfc
,
R(x)
THE THEORY OF LINEAR OPERATORS
234
F(z) *h(x)
= + log
+I/t}dt
*')
,
where J5 n are the Bernoulli numbers. The theorem which we have just proved may be generalized in a significant manner by introducing the function f(x) g(x)h(x) This theorem may be stated as instead of f(x) g(x) f h(x) .
follows
:
Theorem 9. Under the following assumptions: (a) f(x) g(x)h(x), where g(or) is a function of finite grade g and h(x) is
=
of the functions Q(t) (b) f2 and F(z) a^ 2 ^ are a, a,z h, h*t /2! of finite grades Q and F respectively, then the operation F(z) > f(x) defines a series, in general divergent, which is summable by the
the form, h(x)
^
+
= h^/x
W
+
method of Borel
h*/x*
+
to the
F(z) +f(x)
+
h 2 /x 2
=
;
+
+
form,
=
where the real part of

e'
x,
t
Q(t)g^(x)F^(t)/nl
dt
#(.r), satisfies the inequality,
Q\F
< R^R(x) ^R' <
a)
.
Proof: The proof of this theorem is similar to that used in the preceding discussion. Employing the results of theorem 8, we first define the summable equivalent of F (M) (z) * h(x), where F (n) (z) is the nth derivative of F(z) to be, J
V n (x) =F<>(2)
= p Jo = g(x)
+h(x) (x)
We next
e*(t) dt
operate upon f(x) h(x) with F(z) and for the of our result the interpretation employ generalized formula of Leibformula of in we write u which nitz, (2.2) chapter 4, h(x), v == This the g(x). yields equation,
=
>f(x)=g(x)
F(z)
(F^(z) >h(x)}/n\
,
(5.7)
.
n~o
In order to discuss the convergence of this result 1 and hypothesis (b) that the function F (n) (
theorem F.
Hence
F
(n) (
t) is
we t)
notice is
bounded by the following inequality:
from
of grade
GRADES DEFINED BY OPERATORS n
where S n
is
= F and 6
235 
'
a positive quantity independent of

'
such that lim (S n ) 3/n w=oo
t
is an arbitrarily small positive number independent of n. There exists a similar majorant for Q(t) where T(t) is substituted for S n (t), T for S, and Q for F.
We
are thus able to attain the inequalities,
V n (x)
< Sn T
p
{
er" dt
Jo
+
P
J
5
>
dt}
t
< Sn T{(ler^)/p + c'*^/(p2d)}^ S n where we have p
=#
F
Q
> Q
R(x)
,
F
\
2d
\
.
Referring now to equation (5.7), we employ the inequality just < written down and hypothesis (a), from which we have g (n) (x) n 1/n R n g where lim (B) 1. We may then write, ,
=
W = CXD
^%R gS,,T(l+c*)/n\p
^f(x)
\F(Z)

H
> Q + F + 2<5
R(x)
,
.
Hence the series for F(z) * f(x) converges uniformly for values of x in some region Q f F < R < R(x) ^ R' < oo Therefore, since each integral representation of V n (x) exists in the specified region, we may interchange integration and summation signs and thus attain the statement of the theorem. .
The Summability
of Operators of Laplace Type. In section 5 considered the summability problem for differential operators with constant coefficients. It will be useful in another place to extend these ideas so as to include operators of the type considered 6.
we have
in
theorem
We
7.
theorem
shall thus prove the following
:
Theorem 10. Under the following assumptions: g(x)h(x) is a function of finite grade g and h(x) w Ij ft/
/
\ \JL> )
+
/y
/T //y. z^z ivi/ v
exists
I
r
and
7)
//V2 *h
fwn/
I
7)
/'V'J
~p~ 'v\/
it

~p"
/ Vv \ )
f~\ (
\^
*'*>
4~
\
/I
.
^vv^/

f(x)
(a)
of the /}
/
^1 ~~ '^2^

~~
^
form
/2 /O '^3^ / " 7j
f
defines a function
P(s)
f^
Q(t) dt
^S
throughout the interval
^ s ^ the following inequalities hold :
(c)
oo,
Y(t)
Cn n
,
w
e mf
\J
,
^
*S
is
/ 6
^
<
*
^
a function such that
/ t/o
>
^
oo
,
THE THEORY OF LINEAR OPERATORS
236
where Urn
(C,
1/n t
)
= C, C
w+1
> Cn m > ,
0,
thenF(x,

z)
/(#)
woo
we
write Cz
f>6 x
?\ rj? (\x, z)
I I
xt t> t/
V (+} ^t'/ J.
a series, in general divergent, which od of Borel to the form,
defines
/7/ U't'
is
Tfi 1 ^
^u./
,
summable by
the meth
F(x,z) Jf"e*'Q(t){Yg<>(x)A rTl
n
(x,t)/nlg(0)P(t)Y(t)}dt
m < R * R (x)
for x in the region, abbreviation, A n (x,t)
D=
=
^ R'
<
oo,
where we employ the x) > Y(z)] z = 
(x) n }/(D +
n
\_{D
,
t
,
d/dz.
Proof By an argument which coincides with that of the previous theorem since it is unaffected by the parameter x in F(x, z), we express the operation F(x,z)  f(x) in the form, :
F(.r, z)
=flr
+/(.)
(
"'(a:)T
n_o
where we abbreviate TF n
(a:)
=
e
xt

Q(t)F s <>(x,
t)

dt
,
JQ in
which we write
Since is
I(x, z)
we have Fs (n) (x, defined by (3.4),
z)
we
= e^(d/dz
a:)
from
and
get
(3.6)
(n)

its
/(^,
2;),
where
consequences,
e*'Q(t)A n (x, t)dt
where
A n (x,
t)
is
defined above.
Substituting this value in F(x, z) * f(x) and noting that ff(0)
=
00 flf
n^o
we
get
<">(*)(
)"/'.,
GRADES DEFINED BY OPERATORS
F(X,Z)
237

(6.2) (z,
t)dt
.
Considerng the second integral we write,
The second equality sign is seen to be justified by hypotheses (b) and (c) of the theorem which permit the interchange of the order of integration by Dirichlet's formula and insure the existence of the infinite integral.
tain
Referring finally to the from hypothesis (a), i.
and from hypothesis
/
*R
n
g
that
(n)
(x)
0
R
where x
is
n
g
JQ
C
n
n
are able to atlim (R n ) l/n 1,
=
Hl
00
Q(t)Y<">(t)\
i
r
\dt\
\x\
r=o
\x\)} (1
^
\
^ Rr>g
(c), the inequality
e*t
n
we
of (6.2)
first integral e.
i(l
+ R')
n
(1
+ e^'^
+ e^^}/\x
limited to the region
m < R
^
R(x)
m\ ^
,
R'
<
oo.
Hence within the stated region the integrals In all
exist
and the
(x)= f" JQ
series
converges uniformly. Therefore we can interchange summation and integral signs and thus attain the statement of the theorem.
CHAPTER
VI
DIFFERENTIAL EQUATIONS OF INFINITE ORDER WITH CONSTANT COEFFICIENTS 1. Introduction. In the first chapter we have presented the history of methods devised to solve the differential equation of infinite
order Oou(x)
+w'(x) +a u(x) 2
+..
=
/(o;)
(1.1)
,
coefficients are assumed to be constants. It is the purof this chapter to explain some of these methods in more detail, pose attention special being paid to conditions imposed upon the function f(x). It is clear that equation (1.1) is of rather general application since its theory is closely associated with the theory of functional
in
which the
equations of the following types
(a)
+
u(x)
f
^e^
^x
x t
+
u(x)
>u(t)dtc=f(x)
s,
,
>
,
^\ *
(b)
:
K(t)u(x
+ ct)dt = f(x)
,
(c)
(d)
A
u(x)
+A
n
u(x{n)f(x)
.
This relationship is formally exhibited by expanding u(T) in a Taylor's series about x,
u(T)
^n(x)
+
(Tx)u'(x)
+ (Tx)
2
u"(x)/2l\
If we replace T by t and substitute in (a) equation in which the coefficients are, n
Oo*=l+ H Similarly, if
we
l/s m
a,
,
=
we
.
get a differential
n
l/s m
t+1 ,
>
i
.
set T*=^x\ct, equation (b) is seen to reduce with the coefficients
to a differential equation of type (1.1)
a
= l+ J
K(t)dt
,
a,^=
f \K(t) *J a
o,
238
(ct) */i!] dt
,
i
>0
.
EQUATIONS WITH CONSTANT COEFFICIENTS Letting
T
=x
f 6,
239
equation (c) reduces to a differential equa
tion with coefficients 0,0
=1+
1
,
,
= &'/!
i>0
,
.
As
for the general difference equation with constant coefficients given by (d) we note from equation (12.1) chapter 2, the equivalence ny of u(x n) and e u(x) and hence we may write (d) in the form, ,
,
+
[A
+ Atf* + A
2
ew
+ A,e
3~
Hh
4
n
e*] *u(x)
= f(x)
.
2. Expansion of the Resolvent Generatrix. In order to attain the formal solution of equation (1.1), we first write it in the form
F(z) *u(x)=f(x)
,
where
F (z) is
=^ a
+ a,z + a z 2

2

,
the generatrix function.
In order to determine the resolvent generatrix G(z), as defined 4, we set Bourlet's operational product equal
in section 10, chapter to 1,
G(z) *F<*) Since F(z)
is
= [G.F](z)=l
independent of
x, this
.
immediately yields the re
sult
G(z)=l/F(z)
.
Simple as this result appears, it invokes certain questions which can best be raised by means of an elementary example. If the generatrix is F(s) z), then the resolvent is G(z) t=. (1 z) (2 the has three which expansions, z) (2 z), 1/(1
= V + 32/4 +7^/8 + 15^/16 + G,(z) = (1/2 + 2/4 + V8 + G(z)
G
2
(2)
... ,
\z\
<
1
;
= 1A + 3/s + 7/2* + 15/z 2
3
A
question of primary concern, of course, is to determine which of these three forms is a valid operator. Applying each to the function f(x) and taking account of the identity,
=
(xt)f(t)dt/nl Jf* a
THE THEORY OP LINEAR OPERATORS
240
we
obtain the results,
*
f Ja
For
[e
illustrative purposes
2 <* (
e"
>
f(t)dt
we may now
.
specialize, f(x)
<=
1,
and
thus obtain,
GOO
We now
> 1
= 1/2,
note that
all
1
Gi(z)
e*e?
1/2
,
three functions are solutions of the equa
tion,
and that the difference of any two of them geneous equation,
FOO
*u(x)
This fact is very fundamental methods to the equation considered
form
in the following
=0
a solution of the homo
is
.
to the application of operational and we may state
in this chapter
theorem
it
in general
in
Theorem 1. If G(z) designates a Laurent expansion of G(z) an annulus formed by two concentric circles about the origin and G(z) is any other expansion about the origin, then the function
if
U(x)
= {Gi(^)
:
G(z)}  f(x)
,
where f(x) is arbitrary to within the limits of the existence of the right hand member, is a solution of the homogeneous equation F(z) >u(x) t=0 Proof: In order to prove F(z) is of the form
FOO where r t=
a.
this
.
theorem
let
= (za)/
us assume
first
that
,
has no singularity within or upon the circle of radius The inverse operator, G(z)
(z)
=
expansions,
^ x
(z)
{1+
={l/z
+ a/z* + a /z + 2
3
.
}
(z)
,
EQUATIONS WITH CONSTANT COEFFICIENTS
241
the first expansion being valid within the circle of radius a and the second in the region exterior to it. Without essentially impairing the generality of the discussion we may write X
( f(t)dt
.
^o
It will
be convenient to replace
Q n (x,z)
= {1
n
\/:.
by
its
equivalent generatrix,
+ xz + x*z*/2l +
[1
which we have discussed in section 6 of chapter 2. n in the expansion of Letting ff(z) 1, and replacing l/z
=
by
generatrix function
its
we
get:
be clear from this explicit expansion that
It will
CO (jr i
(0)
G/(0)
7j n=l
ct
n
=
x n / yi Id f
aw
'
l
===
*

(c
^Y[(
ax
1)
= _{^'(a.r G/' (0)
and
{r (a
a
a
2
+ 2)
2a.T
2}/a^
]
,
in general,
HH nl]
If
we
and then
nl}/a
n+l .
replace these values in the expansion
tu collect the coefficient of e ,
we
see that the operator reduces
to the expression,
Gj (z)
c'
We now which
may
lx
e
x~
(I/a
+ z/a + z /a* z
2
j
)
* y ~> ^ n , note the operational equivalence, z n e*~ == be established from Bourlet's formula [See (3.1), chap
THE THEORY OF LINEAR OPERATORS
242 ter 4]
by setting
= {z _ 2
sin
X
3.32*1+1 __
e*
s
F
and
x z* m + 2 /2
=z
 }
2
1

n
For example,
.
sin x =
(
1)
x cos #) t= 0. Hence we see that z n er* s > f(x) Taking account of this fact, we get
m
z 2 e x
~

sin x
(sin x cos x
=f
(n)
(0)
.
:
a2 Similarly for
(2)
=
,?,
we
obtain
and more generally, for
Assuming that
+
>
we then
= Ce
U (x)
Since
ax
+
obviously a solution of the equation
is
F(z) the truth of the theorem
is
>u(x)=0
,
demonstrated for the special case assumed
above.
Let us next assume that F(z)
F(z)
= (2
i)
is
of the
(202)
form
(zdn)/
.
a n are points within an annulus formed by two conand R, (the latter having the larger radius), about and where (z) has no singularity within or upon R.
where a l9 a 2
,
,
centric circles r
the origin,
Then the generatrix may be written G(z)
=^(
+ l/[P
/
where
w!e
(a,
l
)(^a)]}
,
employ the abbreviation (2
02)
(2
aw )
.
(2.1)
EQUATIONS WITH CONSTANT COEFFICIENTS
243
Let us designate the expansion of the resolvent within the circle r by G(z) and the expansion in the region exterior to R by Gi(z). We then have from the result of the case of one pole, the expansion
G(z) >/(.r)
(2.2)
,
+ G(z) Hence the difference
U(x)={G (s)G(z)}*f(x) 1
(2.3)
,
a solution of the homogeneous equation. We should make specific note of one exceptional case, i. e. where e xx , the value of A coinciding with one of the zeros a h let us f(x) say for simplicity of notation the value a. We shall then have is
=
.)
(a,
a)}
+ lim[c/{F'(a) (a
i)
where JS"r means that a has been omitted from the sum. Taking the limit we thus obtain
(2.4)
i=i
Example Let us :
evaluate,
F() = (2 2) (2 We have F'(5) = 3,
where
* e
The case of multiple resolvent
5)
=
F"(5)
z2
=
7z 2,
= eVS +
+ 10.
F'(2)
=
(e"/3)
poles is treated
(r
Hence we get
1/3)
.
by a simple device
is
G(z)
3.
=
,
.
If the
THE THEORY OF LINEAR OPERATORS
244
we may
write
it
in the
G(z)
=
form
O'Vto'
1
)
(rl)
I
.
Hence we have G(z) >/(&)
[I/
(rl)
!]
rz/a
(a
+r (r+2)sY
.
3!
+
}
/(a?)
.
The difference between the left hand member and the second term of the right hand side is again seen to be a solution of the homogenous equation. The general proof is then easily constructed from this fact. It remains for us to discuss the values of the solution and its deobtain the following theorem rivatives at the point x == 0.
We
Theorem
2.
If
G
}
:
denotes the Laurent expansion of the func
(z)
tion
G(z) =
<s
nomial of degree x c= for r t= 0, 1,
2,
,
n)
,
i, cu, a,,, and if
in the region exterior to the poles
m < n,
(za
a,)
,
Proof: Writing G(z) in the form (2.1) we have from the results of the last theorem,
{/(0)/a
+
/'
(0) /a,*
+ ...} i=:i
f /(')(a;)/o,
s
+ ...}
.
Recalling the algebraic identity
we
see that
t
EQUATIONS WITH CONSTANT COEFFICIENTS
(r)
(0)
=
{
+ ^a,
h

^a,"} m/ {/(O) /a,
1=1 <
r
245
;0
^"(0) /a,
2
+ ..}//"
(a,,)
,
2) /'>)+...}
>
+'(
2) (/.,/""">
(0)+.}
,
;~o
D/(0)
=
If r f y
m we have r
1^7?
2,
m
^ n
1,
=
then ?^ r) (0)
which
is
Since j does not exceed
0.
the statement of the theorem.
Corollary. If f(x) is a function which vanishes together with its derivatives at x c= 0, then n (n (0) n f q m. for r
first q
t
In order to illustrate the preceding results to several
we
shall apply
them
examples:
Example 1. The fundamental equation satisfied by the polynomials representing the sums of powers of the natural numbers, S n (x) ln 2" x\ is obviously
= +
+
Sn(x)S (xl) n
which
may
The
=a:
,
be written symbolically as
solution follows immediately
Sn(x)=I/(I
C
s
)
+X"
:
,
= (l/z + 1/2 f n,z/2
1
B.2 z V4
!
+ B^/6
1
 }
THE THEORY OF LINEAR OPERATORS
246
= x ^J n
S n (x)
( ft
f 1) f x*/2
+ nB^x^/2
!
_ n (n_l) (n f tt(tt_l)
(>^2)
3)
(wr
(w4)
(^5)
w 5 3
/6!
where the B n are the Bernoulli numbers. The formula ognized as the classical solution of Bernoulli.
is

,
at once rec
Example 2. The following problem is taken from Bromwich [See Bibliography, Bromwich (1), p. 413], and will be used in the illustration of other methods.
We
shall solve the equation, 2
{z
+ 2vz +
(v
2
+n
2
)
= Fe~
}>u(x)
v*
cos (nx f
co)
,
subject to the condition that u(x) shall vanish to as high an order as possible at the origin. It is clear from the boundary condition that this problem is a special case of theorem 2. The generatrix is written in the form
F(z)
=2
2
f 2vz\ (v
=
2
+n
+
2
(z
)
a 2 )
Oi) (z
,
ni a z v v ni where we abbreviate, a t By theorem 2 we must take the expansion of the resolvent generatrix in its outer annulus, which leads us to write, .
,
a,)
l/(z
a,)]
vx Designating by f(x) the function Fe~ cos (nx
('[l
Jo ('{.l Jo
+
co),
we
get
+ a (xt)^a^(x 1
+ a*(xt) ~\aS(x
r
[c^<**>
t)V2
!
+
]
f(t)dt]
e<"<**>
Jo
By
simple integration this leads to the solution,
v 2 ^.(#) ^= [Fe V(2n )] [nx sin(nx
which
is
seen to vanish together with
Example
3.
We
} co)
its first
sin
nx
sin co]
,
derivative at x
shall illustrate the efficacy of
formula
=
0.
(2.3)
by
applying it to the following elementary Sturmian problem, in which we seek to solve the differential equation,
EQUATIONS WITH CONSTANT COEFFICIENTS
=
u"(x) +tfu(x)
247
,
subject to the boundary conditions,
u(Q)
The resolvent generatrix l/F(z)
=w(l) is
from which we obtain by means of
_/<3>(
The boundary /(O)
)
/A'
+.}]
,
(2.3) the solution,
= (I/A
4 (0)/A 
(4)
l/(z+Ai)]
ti)
)}*f(x)
+/
.
obviously,
(I/2ti) [1/0?
~
=0
2
)
[coste {/(O)
isinAx{/'(0)/A
}
.
=0,
condition, u(Q)
leads to the equation,
/" (2.5)
gives us the condition,
Similarly the assumption that u(l) sin A {/'(O) /A If
we now
/>(0)/A'
+ /<*>(0)/A'  = }
define characteristic
sinA
.
numbers by the equation,
=
,
then a solution of the original system will be obtained by operating upon any function which satisfied (2.5). We might choose, for example, f(x) e= x and thus obtain,
u(x) ^={G l (z)
G(z)}*x 2
A /z*
= (I/A)
+ A /s 4
 )
I/A
2
+ z*/V
X
C
$inl(xt)tdt
:
sin Ax/A 3
.
Jo
Let us
which
now
consider a special case of the homogeneous equation We shall con
be of importance to us in another section. sider the solution of the equation will
F(z) tt(ar)
where u(x)
is
,
(2.6)
subject to the boundary conditions:
(2.7)
THE THEORY OF LINEAR OPERATORS
248
and F(z)
is
a polynomial of the nth degree,
F (z)
=z
Az n 
n
1
+ Bz
= (zaj (za
n
Cz n  H
2
(za
(za,)
2 )
we
Referring to theorem 1 and expansion (2.2)
.
n )
see that the func
tion
.
w =l^^
a solution of equation (2.6). Let us now determine the function p(a ) in such a way that u(x) shall satisfy the boundary conditions (2.7). For this purpose is
t
we
write p(t)
=pJ + p. t*\hp 2
fl
<"
.
Hence making use of the abbreviation
we can
write (2.8) in the form
=
u(x)
Moreover we have u (r >(x)
=
Letting a; 0, we obtain the following set of equations for the determination of the coefficients p t
:
n
t
,
r
= 0,l,,..
,
n
l
.
(2.9)
The function I(p}
will be seen to
I(p)
=0
I(p)=l I(P) =(7,
=
,
,
,
have the following properties:
nl p^nl Q^p <
,
(2.10)
,
p
=
7i ?
_ 1
.
Jl'a ^ = ^o. a a h> etc., where where we abbreviate a 2'a,, a, the the summations range independently over integers from 1 to n. Since the a are symmetric functions they can be expressed in terms of the elementary symmetric functions and hence in terms of l
r
t
(
y
,
l
j
EQUATIONS WITH CONSTANT COEFFICIENTS
249
the coefficients of F(z). The calculations are somewhat involved, although entirjp elementary, and we shall give only a few values as follows:* (T!
=A
Vn
^A*
,
B
,
(2.11)
^ ^ A* =A
+C ZA^B + 2AC + B* 2AB
4
(Tl
,
D
,
Returning to (2.9) and making use of (2.10) we obtain the following system of equations for the determination of the coefficients
+ VlPn Pn2 + ViPni + <*& Pn1
Employing the
Ilt
,
explicit values given in (2.11),
lowing solutions for the cases, n
=4; Pt=
(
 + (7^

2(7^0
4
^cr 1
(
cal
()
3
+
2o 1 cr 2
C?i
cr d
3o 3
f 2(7io3
i
2, 3,
f
Bu
o 3
2
+
)?<
2 o 2
+
(^i
2

Au>
o 4
(err f

^4
we have
the fol
:
2 o 2
)^!+
^2)^1
a^a +
) II Q
a

U f W 4 l
,
,
The reader may readily verify that these coefficients are identiwith the coefficients of the polynomial part of the product v(t)F(t)t *See M. Bocher: Introduction
18.
4 and 5
to
,
Higher Algebra.
New York
(1921), chap.
THE THEORY OF LINEAR OPERATORS
250
where we
define,
V (t)
The
= Mo/t + U^/t + U /t + 2
3

2
U n .Jt*
.
significance of this relationship
As an example,
let
is discussed in the next section. us consider the special case,
=
F(z) >ii(x)
where we
specialize,
F(z) = (zI) (z
We
,
2) (z
3)
4)
(
=
10z 3
z*
+ 35z then have,
*>(!)
=6
and employingu(x)
=
u (r)
F'(2) =2
we
(2.8)
,
F'(3)
=2
,
50z
F'(4)
+ 24
=6
+
9?< 2
?O <"/6
+
=
u,
,
(127(
easily verify that this solution satisfies the (0)
.
attain the solution
26u,
(24tto
One may conditions,
,
2
,
r
=
0, 1, 2,
3
boundary
.
Let us make the final observation that the solution (2.8) valent to the development,
u(x)
= lp(z)/F(z)lI
,
is
equi
(2.12)
is first expanded in its outer Laurent annulus. This observation then leads to the following rule, the significant character of which will appear in section 5 of the next chapter:
where the operator
The solution of equation (2.7),
may
be attained by
(2.6), subject to the
boundary values
first solving the algebraic equation,
F(z)
U^p(z)
,
(2.13)
The
solution, u(x), is then explicitly found by interpreting the operation [p(z)/F(z)~\ * 1 as equivalent to (2.8).
of CauchyBromivich. We have already indicated the in the first chapter important part which T. J. TA. Bromwich has played in laying a rationelle for the Heaviside calculus by an adaptation of a method of circuit integrals originally due to Cauchy.* We shall discuss this method for the case where the resolvent generatrix has the following form: 3.
The Method
*This method has much in common with what is called the method of Appell The reader will find it profitable at this point to consult section 13,
polynomials. chapter 8.
EQUATIONS WITH CONSTANT COEFFICIENTS
[/Vfz
a,)
+ F /(za z
2
251
\F n /(za n )]
)\
.
(3.1)
assumed that
i(x)
= (I/2ni)
f c"v(t)dt Jc
(3.2)
.
In another chapter we shall apply this Laplace transformation to discover the solution of a more general equation in which the coefficient of z n is a polynomial. Obviously here we are dealing with a degenerate case of this method since (3.2) substituted in (1.1) leads to the equation
:
f e**F(t)v(t)dt Jo
=
In order to interpret this result we see that v(t) must be a funcwhich erases the singularities of F(t) within the contour C, but That is to say, we may have simple poles at the points a lf 2
tion it
may
,
,
,*
write,
(3.3)
Choosing C as a closed circuit containing are led to the solution
= v(a,)A^ + v(a,)A,c s (l
u(x)
the points a lf
all
\p(a, n )A n &*
we
.
\
It will be seen that this furnishes a solution of equation (1.1) A n In order to specify with the n arbitrary constants A lf A.>, .
,
these
we
note the equations,
A
l
+ ^(n
2
)A 2
=w

\
\
,
= .......... = A l a 1 {'^(a,)A 2 a 2 \^(a n )A n a n
(aO Aitt^"
1
+
(a 2 ) Aid*"

1
~\
where we employ the abbreviation, u (p)
Bromwich
1
(0)
(a n )
ul
A
=u
finally observes that if (3.3)
p
n
an
n~
,
l
(3.4)
u n ^
.
.
be expanded in the Lau
rent annulus exterior to all the values of the a if then the coefficients are precisely the boundary conditions which form the right hand member of the system (3.4), that is,
THE THEORY OF LINEAR OPERATORS
252 v(t)
where
V (t)
= = 0(*
w 1
(3.5)
)
Since, moreover, in the present case the function,
p(z)=F(z)v(z) is
a polynomial, the value of v(z)
,
readily computed by first evalu
is
p(z) through the direct multiplication of (3.5) with F(z) and the subsequent rejection of negative powers of z. ating
Example: Let us solve, [z + 2vz to the boundary conditions u (0)
+
2
ject
=
=u
Since v(z) 0>
2
f
n 2 ), we
nomial
we r
v(z)
?i/c
2
n^z
p(#)
get,
?f
+ + +
()
,
+ ^ )3 2
+
z2 and F(z) Employing
2r^
0,
sub
?t,.
=
,
?*i
=
?t'(0)
 n (x) <=
2r,~
+
this poly
.
then obtain,
= (n + n, + 2m)/[(za (}
z
where we abbreviate, B  niu )/2ni
?ii
+
/z
2
(>'
a
v
{
+
(r?/
,
(}
f
l
)
ni,
(za,
2
a>
)]
==
^
ni,
2m)
?rMf,>)/(
?*i
At=
(y?/
[
.
Integral (3.2) obviously yields the solution
u(x)
=Ae' ^ 1
vf
=: (c
We of
+ Be ^ a
/n)
,
(VU Q
[
\
?/0 sin
nx
f
w?/
cos ^.T]
.
consider next the nonhomogeneous case and for simplicity shall assume that f(x) has the expansion,
argument we
We
first
f(x)= /o
+ ftf + f x*/2 + f^/3 +
note that
we may
!
2


.
write formally
(1/2*0
f(x)

!
f Jc
e rt
V(t)dt
,
where
y(t)=/
/i
+ /i/* + /2/*' + 2

(3.6)
,
and the path of integration is a circle about the origin. If we assume that the solution may be written as the integral (3.2) we must have after substitution in (1.1) /2,70
and hence F(t)v(t)
f e"F(t)v(t)dt Jc
= V(t),
that
is to
= f(x)
say,
v(t)=V(t)/F(t)
.
,
(3.7)
EQUATIONS WITH CONSTANT COEFFICIENTS
Bromwich
limits himself to the case
253
where the impressed force
is
^=Aemt
f(x)
from which he
,
obtains,
V(t)
t
The path of
integration, it should be noted, can be in this case about alone, or about this value and any number of zeros of F(t). Perhaps the most significant fact in the situation can be sum
tz=m
marized in the following theorem
Theorem
:
If in the integral
3.
u(x)
= (l/2ni)
( Jc
e xt
\V(t)/F(t)\dt
(3.8)
the function V(t) is defined by (3.6) and the path of integration is a circle about the origin sufficiently large to include the zeros of F(t), then u(x) is formally equivalent to the operational solution (2.2),
The proof of this theorem can be constructed by the reader from a consideration of the following special case. Let us wirite l/F(t) so that
= l/(ta) = l/t + a/t* + a A + 3
2
,
we have
We then obtain fx
from
+
(3.8) the
(A
+ a/
)
expansion
x*/2
!
+
+ Aa
( A>
a x /2 + (A/a') (A/a + A/a + A/a + (I/a) (/ + hx + A.T /2 + (I/a (A + hx + A* /2 + (I/a (A + A* + f& /2 + z
2
(e**
*=
2
e
!
ax
1)
+
3
)
2
!
)
3
ff
e" (/
/a
+ A/a + A/a + 2
d/a)/(x) e (A/a
!
)
!
)
2
)
=
)
2
2
3
2


)
(l/a )/'(s)
+ A/a + A/a + 2

3
3 (l/a )/"(a?) 
)+ [l/F (z)
}
~> / (x)
.

THE THEORY OF LINEAR OPERATORS
254
Example. As an illustration we shall apply this method problem solved in example 2 of the preceding section.
From
the fact that
f(x)
we
to the
= Fe~
vx
% F(e
+ CD)
cos (nx
a **+ Mi
a *> 1 ) f e
,
get at once,
^ where we abbreviate A = V(t)
We
=
7?
,
where we write a this
=
we
= l/2m
&
.
derive the expansion
V(t)/F(t) =Aa/(ta)*\ (aB
which when introduced u(x)
+ Ab) [a/(ta) + b/(t a)] +
into integral (3.8) yields the solution
= Aaxe?* + (o?B + Aab) e^ + x
c= [Fe va? / (2n 2 )
The
i/
have
also
From
Fe wi
i/2
solution
is
] [?io;
sin
(tio;
+
(abB
+ A6
co)
sin
2
nx
)
6^
+ bBxe?*
sin
w]
.
thus seen to be identical with the one previously
obtained.
PROBLEMS Discuss the solution of
1.
(1
when
(a) /(*)
Show
2.
is
=
;
fxsins) tt(o?)
(b) f(x)
';
=f(x)
(c) /(ar)
,
= I/a.
that
a solution of the equation 2 u(x)
Reduce the solution
3.
R(x)
>
where k
=
to the
f(x)
Given u(a;4l) + w(ar) c, has the form
is
a constant and p(x)
/;
form
~2 is
/(,r),
show that
bounded, then
if
f(x) in the half plane,
EQUATIONS WITH CONSTANT COEFFICIENTS (l/'2 7rn l
u(x)
f(t)g(x
I
t)dt
R(x)
,
255
>a>c
,
where we abbreviate
g
(s)
=
[2e*V(l + H)] dt
I
= *[% (sf 1)]
'
(Hilb).
Prove that
4.
31
e~
Hence show that the
is
I/a

11
1
+V ^
.
(
^
2wtfi
J
solution of the equation
given by ,n
/(t)cft
Jr
+ 22 Nl
pZ c
J
where n(.r)
is an arbitrary periodic function of unit period and the lower limits of the integrals are arbitrary constants the imaginary parts of which are all equal. (Wedderburn).
Prove the following theorem [see Bibliography: Sheffer
5.
(3), p. 280]:
00
If u(x)
2
u n x n /nl
is
a function of grade q
<
R, which satisfies the
WO
equation
where F(z) in the
^
an
zn
converges for
<
z 
<

<
R, then / (x)
may
be expanded
form 00
where
P n (x)
Show
are Appell polynomials generated by
also that the converse is true.
(For the definition and discussion of Appell polynomials the reader ferred to section 6, chapter 1 and section 13, chapter 8). 4.
The Method of Carson.
We
is
re
have commented in the first chapmay be regarded from one
ter to the effect that the Heaviside calculus
point of view as a special case of the generatrix calculus. This connection is essentially the contribution of J. R. Carson, whose ideas we shall now reconstruct.
THE THEORY OF LINEAR OPERATORS
256 Let us
first
consider the equation
F(z) *v(x)
From
we then
(2.2)
=
v(x)
1
(4.1)
.
obtain
e 1=1
We now
note the following formal equivalence
:
(* Jo
But
this
expansion
is
identically equal to
l/[^(~)]
so that
we
attain the result, (4.2)
(
Jo
In this manner the solution of equation (4.1) is reduced to the inversion of this Laplace integral equation. For practical applications rather extensive tables have been prepared to facilitate the inversion and this mechanization of the method has given it a merited brief table is appended at the end of this section. popularity.
A
Let us
now
see
how
to proceed in solving the
more general equa
tion,
F(z)
We
shall
show ,
,
+u(x)=f(x)
.
that in terms of v(x) the solution
is
given by
d
(4.3)
v'(xt)dt In order to prove this statement Referring to the latter solution
(2.2).
we we
.
shall identify (4.3) with see that we can write,
EQUATIONS WITH CONSTANT COEFFICIENTS
u(x)
=
{e""/
(;  1)
(0)/a
l
257
/""(a?) /a,'/" (a,)}
'/?"(a l )
*i
>=i
+ /(x)/F(0)+
f(x)/a,,F'(a
t
)
,
i=i
=
f ^0
I
(
t=t
This last expression
is
seen to be the equivalent of T
f(x)v(0)+ ( f(t)v'(xt)dt Jo which establishes the validity of It will
Example.
,
(4.3).
be instructive to apply this method to the soproblem given in example 2 of section 2.
lution of the illustrative
In this case w'e begin with the integral equation,
= l/{2 + 2vz +
/*GO
v
() e''ds
Jo
+
(v
j>*
2
) ] }
,
where we abbreviate
Noting the integral /oo c?"do:
== I/a
,
Jo
we
at once derive,
= A f ?
v (s)
Since v(0) <== A \evaluation of the integral
(
flj
"
.
{
0,
V(xt)f(t)dt
only with the
.
Jo
This leads immediately to the solution,
u(x)
eral
vx
[Fe~ / (2n
z
) ]
[nx sin(nx
{co)
sin
nx
sin
CD]
.
It should be pointed out here that the Carson theory is less genthan that of Bromwich from the fact that it is designed to solve
THE THEORY OF LINEAR OPERATORS
258
a single type of boundary value problem, namely, the one most interesting to electrical engineers where the solution vanishes to as high an order as possible at the origin. Within the scope of this problem,
however, the Carson theory is unusually effective. For the sake of applications, a brief table is given below for the inversion of the integral,
In applying the theory of this section to more complicated operators, the reader will find very useful the great work of D. Bierens de Haan Nouvellcs Tables d'Integralcs Dcfinies, published in (18221895) :
1867.
TABLE OF LAPLACE TRANSFORMS F(z) 4
(1)
(z
(2)
sVw!
\)/z
X)
(4)
+ \)/z (^ 4 X2)/X~
(5)
(S2
(6)
[(25
(3)
(z
4
X2)/~2
4
sin \s
X2)
cos \s
X2)
tl^ sin Xs
0)2 4
X~']/Xz '
ls
(7)
COS
1
X.S
4
(8)
(32 4 X')/X*
(9)
(z*
+
\*)/(z
(10)
(s 4 a)
(11)
(z
+
a)2
(12)
(3
+
a)
(13)
(z
+
o)
1
cos
1
2 sin Xs
3
e*jVl (s)]/a 1 4 as 4a2 S 2/2!
[1 />(
,
_3 ( 8 )
4 (14)
(z
4
X.s
a) (s
+
l/rt&
b)
4
ai.s"V
(!)!
(ezV6
(16)
(z 4 a) (z
(
4
a)
+b)/z
(z 4

a) (z
4
ftx
6)]
a)(z4 6)]
(
be
&*
e
"s
)/(rt
a^'"
1
)
/
a) 6)
(
ft
a)
EQUATIONS WITH CONSTANT COEFFICIENTS
1/D^OO]
F(z)
(17)
259
z(z+a)(z 6)
(18)
(z+a)(z+b)
X
(19)
+
(z
(z
a) (2
X (20)
X (22
(22)
(2
(23)
(2
+
X
c)
+
(2
2
(Z
+
+
X X X
C)]
(a
o)]
(6
c)]
(6
c)]
+e e*V[c(<
X
X
(a^c)(6
[cosh as
i
c)]
(e/a) sinh as
c)/2
+
a2)
+
X2)*/2
a2 )
(c/a) sinew
(2
J
(Xs)
eA<*W
for s
^
X (24)
(a/2)
(22
J n (08)
+
*] 025)
e^)"Y2
(26)
<>'"
7[(a/4s)] where
1
)
Pe^d Jo
(27)
^
(28)
2
(29)
2
(30)
e/^
n
(e
/~)/2 a
(31)
l/[2(2+X)4]
r Jo
(32)
(33)
c)
6)
C)/S
a*)
(22
(21)
+
(2
+ X)l/2 r' Jo
X
THE THEORY OF LINEAR OPERATORS
260
Inversion as a Problem of the Fourier Transform. Unquestionably the most important single method for the inversion of the operators which appear in the problems of mathematical physics is that founded upon the theorems of Fourier series and the Fourier in5.
The method when it is warily applied exhibits extraordinary the existence of a large literature of both theoretical and and power formal results makes its application a comparatively simple one. We shall begin by stating without proof the fundamental theorem on tegral.
Fourier series: If f(x) is a single valued function which has at most a finite number of infinite discontinuities and is of limited variation in the interval a x ;g a ivhen such discontinuities httve been excluded, and
^
for other values of x
and
if,
is
defined by the equation,
moreover, the integral
EQUATIONS WITH CONSTANT COEFFICIENTS exists,
261
then f(x) can be represented by means of the Fourier series,
f(x)
i/o
A
+ A! cos(7i/a) + A
2

cos(2.ra;/a)
)
(5.1)
+ where the
coefficients are
J. a
tt
sin (2.
Z> 2
determined from the integrals
ra
i
A M ^:
+
sin (7i/a)
Z?!
f(t) cos(nnt/a)dt
ra
B n = i
,
f(t)
a J_ a
sm(nnt/a)ds
.
Fourier series gives the value, lim
1/2
[/<*
+ *)
imtY existing except where the function
is infinite.
In order to apply this theorem to operational analysis,
let
us
write,
cos
px
^
i/o
[e
pxi
~
e~ ;m ]
,
sin
px
=
and note the operational identity: F(z) > Hence we get,
i/oi
e
(/
'
O
e^
^^ F(a)c
l
]
,
ar .
F(z) F(z) with propobtained
If these operations are substituted in equation (5.1).
er specialization of p, the following formal expansion
is
:
F(z)
e
w/o
F(
mni/a,)
e
mr * i/a ]
mi
(5.2) If F(z) is an even function, that is to say, then this expansion takes the simpler form,
if
F(z)
= F(
F(z) mi
(5.3) 00
B mF(mjii/a)
s
mi is an odd function, that then (5.2) becomes
Similarly, if F(z)
F(
z)
9
is
to say, if
F(z)
2
THE THEORY OF LINEAR OPERATORS
262
F(z) /(*)
=
oo 1/2
AoF(O)
+i i
J] m=i
A m F(mjii/a)
sin(mnx/a)
B m F(mni/a)
cos(mnx/a)
.
(5.4)
All these expansions, (5.2), (5.3) tations of the operator F(z) * f(x)
uniformly in the interval
and
(5.4), are valid represenprovided the series converge
a,a).
(
which we have stated above for expansibility in a Fourier series do not hold, then equation (5.1) must be replaced by the Fourier integral. A sufficient condition for this representation is found in the following theorem: If the conditions
// f(x) is a single valued function which has at most a finite of infinite discontinuities and is of limited variation in any interval finite from which such discontinuities have been excluded,
number and
if the integral,
exists,
then f(x)
may
f(x)^^ *n
be represented by the integral, a
P
ds
( f(t)coss(xt)dt.
J_oo
The integral gives the lim
(5.5)
J.QO
value, i
this limit existing except ivhere the function is infinite. (5.5) is written in exponential form and a in one of the terms, it will be found that made of variable change the integral can be written in the following form If the cosine in
:
1
f(x)^. 2^
ds J00
Hence the operation F(z) means of the integral, by F(z)
*/(&)= ^
/
foo
oo
e
8(x ~ t)i
f(t)dt
.
JOO

f(x) can be represented formally
("ds ( *F(8i) <&<** f (t)dt J 00 J 00
,
(5.6)
provided the integral exists. Choosing the plus sign in this representation for convenience,
now
resolve F(si) into its real
and imaginary parts,
we
EQUATIONS WITH CONSTANT COEFFICIENTS and thus obtain the expansion
=~ 2n
:
r oo
/*oo
1
F(z) >f(x)
263
ds J_ ro
\_G(s)coss(xt) J.^ \
H(s) sins (*)]
f(t) dt (5.7)
/*
ds
[H (s)
\
J_ ^
cos s
(xt)
f
+ G(s)
*)]/(*) dt
sin *(;r
.
Since in general F(z)  f(x) is real, the second integral will vanish provided the path of integration does not pass over singularities of the integrand. In this case equation (5.7) becomes
r
r
i
cr2jl
ds
H(s)
cos s(x
J
JK
sin s(#
*)]/(*) dt
t)
.
(5.8)
A
problem of peculiar interest that meets us on the threshold of the theory is the determination of the form of the Fourier integral when f(x) is the unit function of the Heaviside calculus, that is,
f(x)=l f(x)
=0
,
,
x^O x <0
,
.
It is obvious that the criterion of the theorem quoted above is not met since the integral of f(x) over the infinite range does not converge. Several devices have been formulated to overcome this difficulty. N. Wiener [see Bibliography: Weiner (4)] has employed a method for separating a function into Mghandlow frequency ranges, the purpose of which is to segregate "those difficulties which come from the singularity of f(x) at the origin from those which arrive from its behaviour at infinity/' T. C. Fry [see Bibliography: Fry (2)] has used a "fortuitously chosen set of definitions" of a certain Stieltjes integral which, together with the Cesaro limit of a class of divergent integrals, is employed to interpret Fourier integrals that "in any ordinary sense, are without meaning/' G. Giorgi [see Bibliography: Giorgi (4)] makes use of the idea of improper functions to the class of which the so called impulsive functions belong. He
says in part: "By the symbol
Fu(t)
and by the name of impulsive unitary function I mean a function of for all values of t, except in an infinites!t which is everywhere
=
THE THEORY OF LINEAR OPERATORS
264
=
mal interval containing the point t 0, in which interval the funcbecomes infinitely great with such values that
tion
This Fu(t) may be considered as a limiting form of a rectangle or of a function of a kind A e \ or of others, according to the degree of continuity or of vicinity required. Following these ideas, we may, ll
~
l
by differentiating Fu(t), define impulsive functions of the second, third order and so on.
"At first sight it may appear that the use of these impulsive functions is strange and illegitimate, as it involves the consideration of actual infinitesimals and infinities. But in fact it is very useful because it simplifies the formulae very greatly and removes the exceptions; in fact not one of the most rigorous writers on dynamics has refrained from introducing the impulsions. As regards the theit is to be remarked that actual infinitesimals and be introduced with perfect rigour as a class of non
oretipal standpoint, infinities
may
archimedean numbers, involving special postulates or, what amounts to the same thing, we may say that all formulae containing improper functions are formulae wherein the sign of Urn, is understood, so that Fu(t) and similar symbols may be regarded as a kind of short;
hand notation/' In order to include the unit function of the Heaviside calculus shall adopt the device offered by the concept of impulsive functions and thus introduce a function, Q(a,t), which has the following properties
mentioned above, we
:
I.
limQ(a,0
1
t^O
,
,
fcO
II.
III.
^0 limQ(a,t) =0 Q(a,t)
t
,
a
,
<0
,
>
,
/^a>
the order of vanishing being sufficient to assure the convergence of the integrals,
/0 I
Jo
cos(#
t)
and
Q(a,t)dt
sms(
It is sufficient to select for this function,
this is
t)
Q(a,t)dt
=
at
eQ(a,t) then substituted in equation (5.8), there results, 1
:=; "2jl
/ oo
/
d8
J co
.
,
t
^
0.
oo
[G(S) COSS(Z
t)
Jo
H (s)
sin s (x
t) e~
at
dt
If
EQUATIONS WITH CONSTANT COEFFICIENTS
^
C
1
J ^^
71
.1
+ Taking the described above
\irnF a (z)
f J_
:rji
a
limit, as
>
F (z)
tion of the operation
,
rssin#s
.
+
G(s)\r
acos.rs
,
j 6
(I
I
(I
(
)
[
we
0,
a2
S
j
ashlars!
Hrj, s vTscosffs iy)
265
a2
 s
[
,
ds
6'
.
J
obtain the following representawhere f(x) is the unit function
> f(x),
:
+f(x)=F(z) *f(x) cosxs
7
_
(5.0)
we
If
_
1 2ji
then
made regarding
note from the assumptions

tion over the negative range that
C
^ ^
/
Js>
we can
x
G(s)
sin xs
r
ds
s
* f(
F(z)
+,1
f
2:i
TT
^
x) ,
^
cos
n
is
to say,
T6>

J_^

00
1
that
H(s)
write (5.9) as follows:
, x ., x F(z) *f(x) *=
the unit func
0,
f x,/ .sinxs ds G(s) s J n
= n1
f J
OT
^

cosxs
H(s)
ds
s
(5.10)
In case G(s) t= can be written:
 2
F(z) +f(x)
71
G(
(^
J
s)
and
G(s)
T(s) ^^
H(
s), this equation
M
S ^l Js^ S (
H(s)
J(
Jl
S
(5.11)
A
few examples
Example
Jf
.
will illustrate the application of these formulas.
Let us
first
consider the operator F(z)
= z/ (z

A)
.
Since
= 6/(P + ^) + s//U + =G( *),ff(s) = //(), we have 2 f sinars 2  f^^r/x ds = G(s) J J
andG(s)
TT
is
2
A
F(si)
6'
s2 )
,
.
71
A" f S
The connection between this method and the method of Carson immediately ascertained by referring to the first formula in the
table of section 4.
THE THEORY OF LINEAR OPERATORS
266
Example
2.
from which we
As a second
derive,
+ i?
P) 2
and H(s)
=
G(s)= 2s*t*l/ [ (s
Since G(s)
from
illustration consider the operator,
= G(s)
2
+ 4A>
2
]
,
H(s), we
then obtain
(5.11)
C
=2
sin
J
This answer will be found to coincide with that obtained by the Carson method as recorded in (6) of the table in section 4.
Example 3. As a somewhat more complicated example, let us consider the fractional operator,
=z
F(z)
derive, G (s) Since both G(s) sin xs/s
from which we
a
cos
<
a
,
i/>
1
a n sa
,
,
sin
H(s)
l/>
a n sa
.
and H(s) cos xs/s have branch points at the origin, the path of integration in (5.7) must be deformed so as to pass below the origin. One then obtains, 1
2n
f
,
sin xs
.
G(s)
J_^
.
ds s
sin
1/2
a n cos
1/2
anP(a)
(1
+ e~*
ai )
a / (2 n X )
s
= sin 1
2n
2
1/2
s
rr/
J^
(
cos xs
f J
= COS 2n
aTiT(a)
,
H(s)

2
l/o
aTir(a)
(1
e* ai )
/ (2
7i
xa )
,
ds
s
= sin
1/2
a
7i
cos
i/o
a
7i
T(a)
Substituting these values in the formula,
(1
TI
,
EQUATIONS WITH CONSTANT COEFFICIENTS
r
(*)=4 271
we
ds
J
4
S
ds
H(s) 2ji
obtain,
By means
*f(x)
= T(a)
of the identity,
J
2n
s
J_ 00 Za
267
F(a)
G(s) _ a)
Sin a n/ (n x a )
= 77/[sin a
,
.
F(l
JT
ds s
a)], this
may
be simplified to
which agrees with the result stated
in (27) of the table given in sec
tion 4.
One
of the most useful features of the Fourier theory of opera
tional process is found in the ease with which inverses may be constructed. This inversion is obtained through the use of what is called
the Fourier transform, which
_
fa
2/7i
_ 2A _
f2/n
write as follows
cosxsg(s)ds
2/n
_
we
:
,
*^o
Too
cossxf(x)dx
(5.12a)
;
Jo /00
sin.r *^o
f
sinsxf(x)dx
(5.12b)
.
^o
These inversions are valid under the general conditions stated in the Fourier integral theorem previously quoted. They may be used, in particular, to reconstruct operators from functions derived from
their application to the unit function of the Heaviside calculus. Thus, if we designate by J (x) the function F(z) > f(x),
we
have from (5.11),
shall
J (x)
From
= 2
this
we
r
G(s) ^
/
.
.
sin xs ds
= 2
f*>
H(s) 
.
cos xs ds
.
obtain by means of (5.12a) and (5.12b) the inver
sions, 00
G(s)=s
f Jo
The operator F(z)
J(t) sinsf dt
,
H(s)=s
f
J(t) cos st dt
^o is
then constructed from the equation,
F(z)=G(iz)+iH(iz)
.
.
THE THEORY OF LINEAR OPERATORS
268
For example,
if
and the corresponding operator F(z)
^=z /(a 2
2
we
e~ at ,
J(t)
z
2
obtain
is,
i(iz)a/(a
)
2
z2 )
=
As an aid to application the following table has been appended which gives a few of the Fourier transforms frequently encountered. An exhaustive table to which reference has been made in the first chapter was prepared in 1931 by G. A. Campbell and R. M. Foster.* Numerous complicated examples are also to be found in the work of D. Bierens de Haan: Nouvelles Tables d'Integrales Definies, referred to in the preceding section.
The Solution of Fractional Equations. In preceding sections applied the method of operators to differential equations in which z appeared to integral powers. Many of the most interesting 6.
we have
applications of the operational calculus, however, will be found associated with problems in which z enters with a fractional index. It is desirable, therefore, to consider a rather general formulization of the
operational solution of equations of the following forms:
L (u)
= A u + A^u
(l/n)
A
f
M (2/n H>
2
+ A uw=f(x) M (u) = B u + B&W + B u + + B u^^^=g(x) n 1
,
(6.1)
,
(6.2)
(2/n >
2
n.I
where
A A 0t
lf
,
A n^
and
J5
B
,

l9
,
B n i
are constants.
Let us designate the generatrix functions of these equations by F(z) and G(z) respectively, that is to say,
F (z) G (z)
+ A^ + A
A =B
1 "*
+
Bjz
1 /
+ +
2 z*'
+B
2

z~ 2/n
An^z**'"
,
Bn^z"*
.
us consider F(z) for which we shall first find a rationalizing factor. In order to accomplish this we multiply F(z) l/n z z/n successively by z etc., thus obtaining the system of equations,
To begin with
let
,
,
= A + A&' + A z Hh An^v* z 'F (z) = zA Ao2 + A,z ^ H An z"' z^F (z) = zA + zAn^z F (z)
2 /"
2
l
n _i ~
n .2
*
,
2
l/n
f
2
,
1 '"
Fourier Integrals for Practical Applications. Bell Telephone System. Monograph B584. 177 p.
nical Publications (1931),
EQUATIONS WITH CONSTANT COEFFICIENTS
Joo o
sin
269
xtg(t) dt /(*)
(la)
t^i
(2a)
1/t
sin
/2 '7r
,
,
1
/2 '7r
(3a) (4a) (5a)
,
'
t/Kt* + 0,2)2 + t(& + o 2 / .
n
_. r,
i
(8a)
ev
(9a)
(e
a*
+
\
* x sin
62]
n
!
cos
T~^
l/2x
1)
a
'
(c
(lla)
1/sinhot
(12a)
t 2m +*
(13a)
J n (at)/t
1)
a;*
/*
x*
jtt
^/(2asinh cosh x'Tr/a)
1
(2asinh x^r/a)
2x
('TT
(lOa)
<
+i 2 )
sin at/ (a 2
)
>
>
x x x
1
"
sinh
;
2a cosh x7T/2a
eM /i 1
sin{?i
arc sin (x/a)
an n{x
}
^a ,x g a
x
,
sin y2nvr
4 (x 2
,
> _i
/^( 7^ )
.
sin{n arc sin (x/a) (14a)
X ^
(0,2
or
oo
x
,
X2

(15a)
J
(af)
+
a2 )i{ X
x>a (x
12
,
2
=a
oo
x x
,
,
,
,
a
> x >
(w)
a?)*
x  a
,
X2
(
}
2)i
a n cos (
,
a^)*}^
,
=a
,
<
,
a.
2)l/2}n
2
a
,
.
7T
(16a) (17a) (18a)
(7T/4X)
arc cot (t/a)
a /2
e*
*2X 2
=
+
sin x)
e^)
sin (p arc tan x/a)
(19a)
(20a)
(cos x
(7T/2x) (1
+
e
QdX
e* t
+
*ff
2 cos a
~__

gar
_______
.
^
Q,
^
<77*
(a*
=
( t^n(^) See A. Adamoff
:
Ann. de
St. Pttersbourg, vol. 5 (1906), pp. 127143.
THE THEORY OF LINEAB OPERATORS
270
JQ
cosxtg(t) dt
3
/(*)
g(t) cos
db) a/(a2
(2b) (3b)
+
2)
+
cos xt/ (a2
2)
(4b) (/*
X sin
H
cos px*
/
2
<5b)
(2
+
tt
(Xoospx*
2)2 f 62
(6b) (7b)
I/ cosh at (9b)
e*)
(1
(10b) (lib) (12b)
1
l/(x 2
wJ
(jcX)
>
x
,
.
arc sin (o;/a)
^
x
a
}
,
COS
x g! a
,
>0
R(n)
.
cos{n arc sin (x/a)
}
(13b)
x
<
oo or
,
a
,
a
x
,
a sin /l
>
x
a
(x
(14b)
R(x)
,
2
a 2 )* cc
(15b)
>
1
<
.
,
x
a
,
,
x^a
,
,
x
>
,
a
4a
(16b) (17b)
(77/4.r)
log (1
+
!/ 2
(cos x
(TT/X) (1
(18b)
>
f
(19b)
+ *
2
,
(
arc tan x/a)
c*
e* 4 2 cos
+ 6 )i fa D w n (^)/0 2
sin x) e<w)
a 2/i
2
=
(a*
(x), where
+
fc2)i_ a 2
(ic)
=
e**
.
1
,0 n
(
==cfn
(a)/d
EQUATIONS WITH CONSTANT COEFFICIENTS n
Eliminating z^' equations we obtain
z'2/n ,
,
F(z)
,
z
i
~'l/n
from the right hand
AI
,
,
A
2
,
>
A ^i
>
A sio
271 side of these
= >y ,
From
this
it
A
***
^il2
...
/J
^^^^
>
.
>
>
follows that
where we abbreviate, 1
,
z^ /n
A
,
i
A n_
,
,
2
^2/n
(6.3)
and
A
,
A (2) zA. n i f
is the same determinant with the zA n 2, "' zAi.
Since A(z)
n
first
column replaced by
>
is
a polynomial in z of degree less than or equal to A (z) is a rationalizing factor and equation (6.1)
1 it is clear that
can be formally replaced by the ordinary linear equation
A(z) >u(x)=A(z) +f(x)
To
this equation there corresponds
u(x)
=
(6.4)
.
an integral of the form
f* W(x,t) \_A(d/dt) Jc
/()]
dt
,
(6.5)
where W(x,t) is the Cauchy function associated with (6.4) [See section 2, chapter 11], provided W(x,t) [A(d/dt)  f(t)~\ does not become infinite to a higher order than u < 1 at t c.
=
It will be noticed, however, that equation (6.5) is not a complete solution of the original equation since we must adjoin to it such solutions as exist for the homogeneous equation,
F(z) +u(x)
=0
(6.7)
.
Let us assume that a solution exists of the form,
u(x)
where v(x) must be further
=A(z) *v(x) specified.
,
Substituting this in (6.7) and and A (z), we readily
noting the commutability of the operators F(z) obtain,
(6.8)
THE THEORY OF LINEAR OPERATORS
272
F(z) *u(x)
Hence
if
v(x)
=F(z)
> [A(z) >
= A(z)
*v(x)
=
.
any solution of the ordinary
is
tion
=
A(Z) ~*V(X)
differential equa
,
then (6.8) is a formal solution of (6.7). Since, moreover, there will be in general n 1 such functions v(x), we see that in general equation (6.7) will have n 1 formal solutions. Some of these solutions, however, may be extraneous. It is of some interest to record the explicit solutions for the three cases, namely, where n 2, 3 and 4.
=
Case
We
1.
have for the equation,
shall
+ A^
(Ao the solution,
is
is to
say, v (x)
Case
2.
2
,
*v(x)
,
Ary'(aO
=0
where k = A
,
2
Q
,
/A^
.
For the equation,
+ A^ + A^
>u(x)=Q
2
/')
,
the solutions,
u(x)
= [Ao
2
AoA^'/^ (AS AA l
where the v
l
Case
3.
AtAJz*'* 2
z
+A
2
2
^ 4A: ] > v, (x)
,
(x) are solutions of the differential equation,
AoM#)
+
SAoAiAa)^'^) +A?v"(x) t=
(AS
.
Considering the equation,
(A
we
=
>u(x)
A^)
v(x)
^C e^
(Ao
we have
)
a solution of the differential equation,
A that
1/2
= (A
u(x)
where v(x)
first
+ A^ + A,z^ + A,z^) 1 /*
>u(x)=Q
1
,
find the solutions,
u(x)
= [Ao* A A z + + (A + 2A A A 2
1
/4
(^.Ai
1

<
A A
2
2
A A
)z
2
2
3)
z1
+
2
A A )^ (ArA + 2A A A l
1
A A
2
2
2
/4
3
3
2
3
2
+
)s
2 /*
(A X
2
A
2
A.A^z'^
2A A A 3 1
EQUATIONS WITH CONSTANT COEFFICIENTS
where the functions Vi(x) are solutions of the
(4A
AA 2
273
differential equation
4A 1 A 2 A 3 2
2
3
In the application of these formulas use the following derivative:
it is
frequently necessary to
If
gx,s
r I
e
<*
Jc
where
continuous in the interval c ^ s ^ x and g(x,s) in the and where < a < 1, we then have
ti(s) is
triangle c ^ s ^ x ^ x
:
S)
(6.9)
This formula can be established by the method of "partie finie" due simultaneously to R. d'Adhemar and J. Hadamard in 1904.* A generalization of their
from which
method
is
Theorem
IfF(x)
4.
is
defined by the integral
J*A(x) B(X)
where
contained in the following theorem
(6.9) is derived as a special case:
f[u(x,s)]h(x,s)ds
,
A (x)
and B(x) are continuous functions of x with continuous derivatives such that either Urn f[u(x,s)] or Urn f[u(x,s)] or both are infinite, although f(x,s) is finite and continuous elsewhere
first
in the interval (A,B)
functions having variables,
and
if
T(e,x)
first
T(e,x)
and where u(x,s) and h(x s) are continuous and second derivatives with respect to both y
defined
is
=[7 +
T
(x,A
by the following expression: s) ]
f(x,
Ae)h(x, Ae)
*R.
THE THEORY OF LINEAR OPERATORS
274
*(x,s) t=
where
aJb
/
r
n=
_
du du
[D x h
I
dx
,
then the derivative of F(x) will be
T
Dh
hD
8
s i~\f
ds
4
lim jT(,#) ,(6.10)
JB
provided lim T(s,x) and the integral both
exist.
Proof Since both / and h are regular in s in the open interval s < A, we may apply the ordinary rule for differentiation under :
B <
the sign of integration to the integral
F
^ e
AB
(x)<=
f\u(x,s)~\h(x,s)ds
,
JB+B
where
e is
.
Since
an
df
dx
arbitrarily chosen positive quantity,
df dfdu = dfdu 1= and ds duds dudx
df
dx
= i(x,s) /
x
3/
,
where
ds
....
,
we shall have .
, i(x,s) t=
du ,du / dx' ds
Substituting above and integrating by parts
= f*'* ^f dx JB
[D f h
iDJi
and thus obtain
AD.i]
we
.
shall
fds+T (e,x)
have ,
^E
where T(e,x)
is
the function already defined.
and T(e,x) approach
If the integral
zero, then r
ax
finite limits
will be the limit of the right
as
e
approaches
hand member of
equation.
Equation (6.9) follows as a special case by letting
f(u)= ua
,
We then have t (x,s) T(e,x)
u(x,s) z= (x
=
1
s)
,
h(x,s)
and
= (1
which when substituted in (6.10) yield the desired derivative.
this
EQUATIONS WITH CONSTANT COEFFICIENTS Let us
now
Example
1.
consider two simple examples.
We
shall solve,
+ <>D/u(x)
lu(x) Rationalizing
we
the solution of which
this
=f(r)
r~n
= oD;f(x)
is
we must add
the solution of the homogeneous equation,
=0
v(x) 4oAiM.r)
which we readily
We
find to be,
+
!
(JT.T)
x*/l 3
!
+ 2e
+
) }
+ Example
2.
1,
where the
=
f(x)
r
:
(ar/.i)
+1
(2*)
VI
(1
ar/3
e*
2c f
+
3
+ * /5 2
!
(ar/.i)
)
2
!
(1
(2a)Vl35
+
or/3
..
+
oJD,
1/J
u(x)
+DV*U(X) =0
.
obtain at once from case 2 above the solutions, Ut (x) v, (x)
= (I
z 1 '3
z
+ z"'*)
*v,(x)
,
are the solutions of the differential equation,
v(x) is
=
Let us consider the equation,
u(x)
We
1
= C"
C'{e*
that
,
Let us specialize these results by setting then obtain,
u(x)
.
obtain the ordinary equation
du/dx
To
275
to say, Vi (x)
eT
2v'(x) \v"(x)
,v
2
(x)
= xe
r .
=0
,
1.
THE THEORY OF LINEAR OPERATORS
276
If we introduce the first of these functions into the solution, reach the conclusion that
we
3r(2/3) a solution of the original equation. This is obviously not correct. 4/3 2 1/3 further examination of the expression (1 z ) 1/3 shows that it may be factored into the product (1 z) (1 z ) Eliminating the extraneous factor (1 z), we conclude that the original equation has only one solution, is
A
+
'
.
u(x)
where v(x)
is
=
(1
is to
)
*v(x)
,
a solution of the equation,
v(x) that
1/r>
v'(x)
=
,
say
Returning now to the second solution, u^(x) obtained from v z (x), we find by direct computation that u 2 (x) is identical, except for a constant factor, with the value of u(x) just written down. From this example we learn that extraneous solutions may appear in the formal solution of fractional differential equations just as they appear in the analogous solution of algebraic equations with 9
fractional index.
Integral Equations of Fractional Index
We now proceed to a discussion of integral index, that
is to say,
G(z)
where G(z)
As
is
equations of fractional
the integral equation,
*u(x)=g(x)
(6.11)
9
the generatrix of the operator defined by (6.2).
in the preceding case of differential equations of fractional
index
we
ment
essentially the
first
obtain a rationalizing factor, A'(z), which by an argusame as that used above we find to be, 1
,
" (6.12)
EQUATIONS WITH CONSTANT COEFFICIENTS
Hence we may replace equation ized equation,
B(z) +u(x)
by the rational
(6.11) formally
= A'(z)
*g(x)
277
(6.13)
,
where B(z) is the determinant A'(z) in which the elements of the first column have been replaced by B 0f z~ B n . l9 z~ B H . 29 z~ l B lt l
l
,
obvious that the solution of (6.13) will be unique since the only continuous solution of the integral equation, It is
=
B(z) *u(x)
,
=
in general u(x) 0. The only exception to be noted here case where the integral z~ l has an infinite limit. is
is
the
As an example let us consider the inversion of the generalized integral equation of Abel, a special case of which first appeared in the tautochrone problem. This equation may be written in the form, g(x)
=
f"
(xt) m
*
a
u(t)dt
1<
,
a
<
1
,
m = 0, 1,2,(6.14)
In terms of fractional operators this
g(x)
= T (m+a+l)
c
Dx
~
may
^
(m
Operating on both sides with the symbol
n(x)
=
D f m " g(x)/r(m+ar\l)
l)
be written,
u(x)
c Dj.
.
m+tl+1
we have
1
c
(x
,
~tya
J (6.15)
Recalling elementary properties of the gamma function and of (6.9) we can write (6.15) in the form use ing
u(x)
mak
(xc)' g'(c) (
'
r(a)
(x
" ta c)"
""
>"
r(a)(.r
c)
(6.16)
is
But it is clear that, if the value of u(x) given by equation (6.16) substituted in (6.14) the integral will not be convergent unless
Under
these conditions u(x) defined by (6.16) will be the desummarized as follows
sired solution. These results can be
:
THE THEORY OF LINEAR OPERATORS
278
first
Theorem 5. // in equation mj2 derivatives in the
=Q
wi
g'(c) c=...c=:0< )(c) in the open interval c
(6.14), g(x) exists together with its interval and if g(c)^=^ , then the solution of equation (6.14) exists
f
<
^
x
c^x^x
XQ
The solution
.
is
given by formula
(6.16).
The efficacy of the methods which we have stated ther illustrated by means of the following examples: Example
1.
Dxu + Ait = f(x)
whose rational equivalent
T~
is
if (x)
solution of this equation
= Ce*/* + ^ 1
e
j
f
[
Jc
(6.17)
,
the equation,
u
u
ax
u(x)
be fur
Let us consider the equation c
The
may
c
D
x *f
(x)
.
(6.18)
is
r/'(0
^
A
cDt*
i
(6.19)
C
In order to determine i
r
^/n J
"(o
we
ty
(x
.
It is clear that for
(6.19)
consider equation (6.17) in the form
x
=
c
we have
iu(c) <= f(c).
Then from
get
u(c)^Cc<^ i/(c) and (6.19) can be written
in the
form
(6.20)
As a
specialization
Hence replacing )*, we have from
c
we may
write
by
by
0,
A
and noting that Dr*
1
(6.20) the solution 1
J* o
er t
==
V^
dt
> 1
=
EQUATIONS WITH CONSTANT COEFFICIENTS This solution can be expanded in two ways i
r
,
,
(2^)
,
279
:
2
...1
,
13
\/JlX
J
and 2T
\/nX
The second, although everywhere divergent, pansion of the Example,
is
the asymptotic ex
first.
Let us consider the equation,
2.
(6.21)
which can be written in the form,
Obviously
we have 1
B(z)t=
r(2/3)
2^(2/3)
1
c
i
r(2/3)
1
and similarly
Hence equation
(6.21)
is
equivalent to the rationalized equation, *
M(ar)
in
+r
j
(2/3)
(x
f
t)u(t) dt
= F(x)
,
JQ
which we abbreviate J(*
The
(x
{
)
solution of this equation
u(x)
=F(x)
+;.
1/
is
r sinl(tx)f(t)dt
,
Jo
where
Wie abbreviate A
=r
3/2
(2/3). This function
be the unique solution of (6.21)
also.
may be shown
to
THE THEORY OF LINEAR OPERATORS
280
Example u(x)
3.
Let us consider,
= t*x**/r(2
a)
_{A/T(1 a)} f'{u(t)/(xt)}dt
,
(6.22)
which we may write
in the form,
Bsr*)>u(x)=g(x)
(l
(6.23)
,
where we use the abbreviations,
Bc=
l
g(x)
,
=u t
Since the numerical value of b
not specified
is
we cannot now
employ the method of rationalization, but instead can invert the equation formally as follows ti(a?)
:
Bz)
=!/(!
*g(x)
= (14B^ + B &
2
^
2&
,
+ ^ + .^ Z?
3
6
)
*g(x)
.
(6.24)
Noting the formula, zr**
*x b
= J) *x* = r(l + &)a
we immediately
6 < l+M
x
+ b + nb)
>/T(l
obtain the following expansion for (6.24)
(1+6) .
}
,
:
+Aa V 2
+
2
(/i/A)
(6.25)
.
The function which we have obtained as a solution has more than a passing interest on its own account, since it is intimately connected with a generalization of the exponential function which was first studied by G. MittagLeffler* and later by E. Lindeloff and E.
W. Barnes.t
A
resume of some of the results
first define
will be stated here.
We
shall
as the MittagLeffler function,
E
a
(x)
x/F(l
+ an)
,
a
^
,
*Un generalisation de I'lntegrale de LaplaceAbel. Comptes Rendus, vol. 136 (1903), pp. 537539. Sur la nouvelle fonction E a (x). Comptes Rendus, vol. 137 (1903), pp. 554558. fSur la determination de la croissance des fonctions entieres definies par un developpement de Taylor. Bulletin des Sciences Mathcmatiqucs, vol. 27 (2nd series ) (1903), pp. 213226; in particular pp. 224225. JThe Asymptotic Expansion of Integral Functions Defined by Taylor's Series. Trans, of London Phil. Soc., vol. 206 (A), (1906), pp. 249297; in particular, pp. 285289. See also: On Functions Defined by Simple Types of Hypergeometric Series. Cambridge Phil. Transactions, vol. 20 (1907), pp. 253279.
EQUATIONS WITH CONSTANT COEFFICIENTS in terms of
which the solution (6.25) may then be written:
=
u(x)
For a Ei(x) *=
=
=
e?
,
E
becomes
E
2
{}
)
(6.26)
.
1/(1
(x)
= cosh V~#
(x)
b
,
for a
x), for a
=
.5
,
t== 1,
#. 5
(a;)
r* t2
n.
(l
^2
f or a
,
(u/l)E b (lx
(/i/A)
0, this function
2
2
e*
281
e~ dt), etc.
I
It is
connected with the Laplace inte
gral through the equation,
The most interesting properties of this function and those which led to its investigation by MittagLeffler are associated with its asymptotic development in various sectors of the complex plane, these properties being summarized as follows For the case, < a < 2, we have the following representations: :
(a)
E
a
provided yaji (b)
E
_

(x)
< amp
<
x
yan
2n
^
1 a
exp(o;
(x)
T/a )
oo ni
provided
(C)
where
/*
amp
y^an ^
For the
case, a
tf a
^
2,
x ^ y^an we have
.
(s)co!j
takes
all
integral values (positive, negative, or zero)
such
that 2jjti
4 cimp x ^ y^aTi
,
+
n and n inclusive. x having any value between These results, first announced by MittagLeffler, were proved also by Barnes, who availed himself of the following representations
amp
:
(a)
M
Ea(x)=^ 2ni
ds n/1 [ Jf r(l+as)sms.i .
,
,
the path of integration being taken along the positive half of the real axis and enclosing the points s 0, 1, 2, oo, but no other poles of the integrand. ,
/,
v
(b)
n E
/
a
x
(x)
1 a
/
exp(a;
i/ x 1/a )
1
=r
2m
7
fJ J \
(
as)sin7i(l
sm
Tis
a)s
.
x s ds
,
THE THEORY OF LINEAR OPERATORS
282
the contour being taken along the positive half of the real axis and oo as) and the points s 0, 1, 2, enclosing the poles of F(
=
E
(c)
a
p

(x)
^
1
exp (x l/a C
as ) sin(2p
{^(
J
L^Tll
.
,

a) s #*/sin ns} ds
1
,
where the path of integration is the positive half of the real axis and The oo encloses the poles of F( as) and the points s 0, 1, 2, integer p is chosen so as to satisfy either of the inequalities, .
,
<
a/2
2p
~\
1
<
a
or
,
a
<
1
2p f
<
8a/2
.
PROBLEMS 1.
Show
that the equation
u(x) is
4
I
Jo
t(xt)*u(t)dt
= f(x)
equivalent to
u(x)Vt*
I
(t(x+t)}u(t)
dt
= F(x)
Jo
where we abbreviate
F(x)=f(x) 2.
Prove that the equation
f*
3
Jo is
equivalent to
H
f*tt2(4Ax*
5 243
+ 40x + 75 3t
2
t2
+ 4 Qxt3 + 44 1*)
where we abbreviate
4 T2(2/3) [x*
D r 2
/3
16
y
L(it)
and
if
p
is
x
t( X >
M)
H
:
9
=
gJ0
ar
if
=
(x
a) (x
so chosen that (
D X*/ 3 f(x)
352
X
Prove that
3.
2x*
/(x)
(a..^)
p(p
b) u" (x) 4 (c\hx) u' (x) 41)
4 pft 4
&
=
0,
then
u(t) dt
=
EQUATIONS WITH CONSTANT COEFFICIENTS
283
Hence employing the abbreviation y(x)
=
reduce the solution of the equation L(u)
=
to
an equation of
first order.
(Let
nikoff). 4.
Discuss Legendre's equation (1
by the method given
#2) in
u
2xu'(x)
(x)
problem
+ n(?Hl)
u(x)
^0
3.
Special Applications of the Fractional Calculus. In the next we shall exhibit the efficacy of the fractional calculus in the solution of several types of problems which arise in the application 7.
chapter
of partial differential equations. It will be sufficient for our present purpose to discuss four problems from essentially different fields in
which fractional equations appear.
Example 1. The first problem to be solved by integral equations was due to N. H. Abel, who by a curious coincidence also employed halfderivatives in attaining his inversion.*
Abel considered the
fol
lowing question: Suppose that a heavy bead is constrained to move under gravity on a curved wire situated in a vertical plane. If the time of descent from a height h to the lowest point of the curve is supposed to be a function T(h) of the height, determine the equation of the curve, the initial velocity
being zero.
FIGURE
1
=
x (y) Let us suppose that the equation of the curve is x the element of arc will be ds u(y)dy where we abbreviate,
=
u(y)
= [1+
.
Then
,
(7.1)
*Auflosung einer mechanischen Aufgabe. Journal jur Mathematik, vol. 1 (1826), pp, 153157. Also Oeuvres. Christiania (1881), vol. 1, pp. 97101.
THE THEORY OF LINEAR OPERATORS
284
From
the energy formula 1/2
we
m(ds/dt)
2
= mg (h
y)
are led to the equation,
(7.2)
Employing fractional derivatives
may
this
be written,
from which we obtain by inversion,
V*] d/dh
(hy)*T(y) dy
,
h,
When
the function thus obtained is substituted in (7.1), the desired equation follows by a single integration,
As a specific example let us consider the tautochrone curve (the c and h^ t= 0. curve of equal descent) where we suppose that T(h) which when substituted in (7.3) We then get, u(h) (c/n) (2g/h),
yields,
where we employ the abbreviation,
Making
x=^ a (2t
2
2
/ji
:
f sin 2t) 2t
~
y
gc
y = 2a sin
the transformation,
Employing the substitution the form:
a
= a(l
=
ft,
sn cosi?)
.
2
t,
we
easily find
.
the values of x and y take
,
,
which are recognized as the standard parametric representation of a cycloid generated by a circle of radius a.
EQUATIONS WITH CONSTANT COEFFICIENTS
285
Example 2. The problem of determining the shape of a weir notch so that the flow of water through it shall be a given function of the height has been solved by W. C. Brenke.* In the figure the shaded area represents the cross section of a weir notch, which is symmetrical with respect to the Taxis. The quantity of water which flows through the notch per unit of time is
given by the equation
^C F (htyf(t)dt
(7.4)
,
VQ
where C is a physical constant and the form of the notch mined by y f(t), t ^ O.f
=
is
deter
yf(t)
FIGURE 2 If
we
replace
Q/C by g(h)
it is
clear that
we can
write (7.4)
in the form,
from which we obtain by
inversion,
J
As a khm
.
We
special application
we
(h
t)*g"(t)dt]
.
consider the case where y(h) ^=
then obtain,
*An Application of Abel's Integral Equation. American Math. Monthly, vol. 29 (1922), pp. 5860. fFor the dynamical considerations involved here the reader is referred to any standard work on hydraulics.
THE THEORY OF LINEAR OPERATORS
286
> If
we
=
set
m=
where n
n,
is
an integer,
this
1/2
formula reduces to
m= +
and when n 3) ]/i"(2n (k/n) [2 tt!/l 3 5 */2, it becomes f(h) (2?z4l)/2" (ul) I]/*" From these fc[l 3 5 results we have the interesting specializations that when the flow is directly proportional to the depth of the stream the form of the connotch is y 3/2 the notch is rectangular, y *, when
f(h)
;/J
,
=
1
.
m= = 5/2 the notch stant, and when m
is
a parabola.
Example 3. The following problem, taken from J. Liouville,* is introduced as an example of the use of series in attaining the solution of a fractional equation. The problem does not strictly belong in this place since the coefficients are not constants, but the method of inversion employed is instructive. Let AB and CD be two straight lines perpendicular to one another, the first terminating at the point A and extending to infinity on the side of B, the second infinite in both directions. We extend the line AB to the point P, where it cuts the other straight line. Midis placed which is attracted way between A and P a small mass the of and force elements AB a CD represented by a function by by u(r) of the distance. We then seek to find the function u(r) such that the attraction of CD shall be twice the attraction of AB, whatever the distance PA 2y.
M
=
ds
FIGURE 3 once clear that the force be written,
It is at
may
F
l
exerted by the line
AB
on
M
*Memoire sur quelques Questions de Geometric et de Mecanique, et sur un nouveau genre de Calcul pour resoudre ces Questions. Journal de I'Ecole Polytechnique, cahier 21 (1832), pp. 1G9.
EQUATIONS WITH CONSTANT COEFFICIENTS
Making the transformation Fl
=
=
s2
:
t,
y
= 2y where we abbreviate, r = F
2
write
.
CD may
be written,
,
I
/
s2
f
the transformation
y :
2 .
s \ y~
=
t,
y
2
=
z,
this
becomes
dt
r Now
we may
u(z*)/z* line
\u(r)/r]ds
2
2
F
z,
/*00
where we employ the abbreviation, v(z) Similarly the force F 2 exerted by the
Making
=
2
287
introducing the condition of the problem, namely that
= 2Fi we reach the equation,
(7.5) ^D^v(z) =i(*z)* m D s*v(z). From physical consideration we see that v(z) approaches
as z indefinitely increases. Hence it an expansion for v (z) of the form,
v(z)
When
^2 A
n
/z
zero
not unreasonable to assume
is
n .
(7.5) and the coefficients of terms obtain the following equation for we equated, corresponding v: the determination of n
this series is substituted in
+
This equation has the single solution n j v 3/2, from which obtain v(z) ^ A/z* /2 and hence the desired law of
we immediately force,
u(y)
=
4. The following example, taken from the field of bibeen furnished the author by Dr. Kenneth S. Cole.* has ology, A living nerve can be stimulated by passing a direct current through a short portion of it between two electrodes, provided the potential difference exceeds a certain critical value known as the rheobase. As the duration of the potential applied across the elec
Example
*For a more extensive description of the background of this problem the is referred to the following papers by Dr. Cole: Electric Conductance of Biological Systems, and Electric Excitation in Nerves, Cold Spring Harbor Symposia on Quantitative Biology, vol. 1 (1933). reader
THE THEORY OF LINEAR OPERATORS
288
trodes decreases, it is found that the intensity necessary for stimulation increases rapidly in a hyperbolic manner. The following analysis is designed to explain this phenomenon.
An
idealized nerve fiber consists of a cylindrical core of electrowith a thin sheath or membrane. It is assumed that a covered lyte local threshold change of the normal potential difference across the membrane will stimulate the fiber and cause an impulse to be propagated. The problem is then to express analytically the strength of stimulus which, when applied to the nerve bundle as a whole, will change the potential difference across the membrane of an individual fiber by a threshold amount in a given time.
1 FIGURE 4
To begin with, experimental evidence points to the conclusion that the electrical behaviour of the nerve fiber may be simulated by the type of circuit illustrated in the figure, where r and R are constant resistances and the element P, called the polarization element, has an impedance* defined by the following equation :
0
(7.6)
The function IP (t) is the current in the element P, e p (t) the poand p the operator d/dt. The positive constant K is
tential across it
determined experimentally. No combination of electrical circuits with ordinary resistances and capacities is known to lead to an impedance of the form postulated, but experimental evidence appears to indicate that such an impedance is essential to the description of the curious electrical behavor of biological materials in general and of nerve fibers in particular. *For a
definition of
impedance see section
2,
chapter
7.
EQUATIONS WITH CONSTANT COEFFICIENTS Referring
now
tween Ip (t) and
we compute the when a constant e. m. f., E,
to the figure,
e p (t),
the electrodes. This
is
289
relationship beapplied across
is
found to be
easily
L
where I r (t) and Hence we get,
are the instantaneous currents in r and P.
I P (t)
7?4r
PI
(7.7)
Marking use of (7.6) to eliminate I p (t), we obtain
or JR
+r
a
(l\jfrKp
+e p (t)
)
This equation
may
(E/R)
be more simply written
&*) +e P (t)=g(t)
(1+ where
=Kp>
,
wte abbreviate,
r
g(t)
,
^KE
t
a
/\_R
F (I
+ a)]
This equation has already been inverted in example and the solution found to be
_ KE MO =^4' .
[
r( i
+
EV
V
K+ = ER S^pr ?

I
[
72
,

+r
at(
_U
where ^0(0:)
a)^
>
_
~t~ 7 7
+r
section 6
,1
+ 2<0 +"J
.
ER 7J
It*
r(1
q)
3,
.
/
'
is the MittagLeffler function discussed in the example referred to above. From the asymptotic representation of E a (x) over the negative axis of reals and from the fact that A is a positive constant, we know that
lime P (t)
=,
THE THEORY OF LINEAR OPERATORS
290
=
and decreases Moreover, since E a ( x) equals unity for x zero e x it to as is clear that builds P (t) increases, uniformly up unifrom to its value limiting formly Er/ (R f r) .
Now
tial
E
equal the rhcobase, that is to say, the critical potenacross the electrodes for which stimulation of the nerve will just let
take place. In terms of this potential, the E n r/ (R f r) clearly e P ( co )
=
maximum
value of e v (t)
is
.
But if a higher potential, E > E n is put across the electrodes, then the critical value of c P (t) necessary to stimulate the nerve will be attained in a finite time determined from the equation, ,
Er
r
R+r that
is to say,
(7.8)
In the study of the electrical stimulation of nerves a characteristhe chronaxie, has been widely used. By the chronaxie is meant the duration of time, t 7, which is necessary to build
tic time, called
Y R
FIGURE 5 Gp(t) up to stimulus strength, when the initial potential, E, 2E From (7.8) we then obtain, twice the rheobase, E
=
is
just
{} .
or (7.9)
where we abbreviate, x
The tion of a
a
Ay
.
inversion of this series is ,
x
= x (a)
,
we may
now assumed and
write
since
,
= [logx(a)
log A] /a
.
is
a func
EQUATIONS WITH CONSTANT COEFFICIENTS
291
PROBLEMS
=
Problem 1. (Liouville). We require the equation, y u(oc), of a curve which has the following property: Let us select any ordinate MP of the curve and with P as a vertex draw a parabola PQR with its axis along OX and its directrix midway between P and the origin O. We then construct a third curve PNV the ordinates of which are the product of the corresponding ordinates of the first two, that is, NS = BS X QS. This proposed, we finally require that the area XPNV under the new curve shall be a given function f (jc) of the ab
AMB
scissa of the point P.
Show
(See Figure 5).
that the function u(x)
n(x)
For the case f(x)
=
a2
,
is
given by
(V2/7T)
this yields the solution
u(x)
(V2t/T)
a'2 /x 2 .
Problem 2. (Liouville). Suppose that a uniform distribution of masses symmetric with respect to the x axis is taken along a straight line y of infinite situated extent, and suppose that these masses exert an attraction upon a mass on the x axis at a distance x from the line y. The total attraction f (x) in the direction of x is known, but the law of attraction is unknown except that it defrom the given masses. The problem is to find pends upon the distance r of this law of attraction, u(r). (See Figure 6).
M
M
Y
O
M
* FIGURE 6
Show
that u(r)
is
determined from the equation,
_1
d
TOO /(#)
dt '
~~vlTz
For the case f(x)
=
/*/#, this
t*
I
leads to
i<
(r)
(**>*
=
Equations Involving the Logarithmic Operator. Similar in kind although not so extensively found in applied problems as are 8.
equations involving fractional operators, equations in which the operator log z appears have attracted some attention. Their introduction and the theory of their inverse are due to V. Volterra who employed
THE THEORY OF LINEAR OPERATORS
292
them in his calculus of functions of composition which we have introduced in chapter 4. We shall give a brief account of the application of the logarithmic operator by first considering the integral equation, f(x)
= Jf\log(xt) + A] u(t)dt
/<0)
,
=0
,
(8.1)
o
which Volterra
first discussed by other means. Referring to the definitions of section 11, chapter we can write (8.1) symbolically in the form,
(zr*logz
A
where we abbreviate a
C
+ az*)*u(x)
2,
we
see that
(8.2)
,
(Euler's constant).
In order to attain the inversion we first note from the corollary 1 1 z" log z f o^r ) may be of section 3, chapter 4, that the operator ( inverted by means of the Bourlet operational product. Hence, since x does not explicitly appear, we have the formal solution of (8.2) in the form,
In order to interpret this symbol we now appeal to the theory developed in section 11, chapter 2, in particular formula (11.14),
where we
find r*OO
zp l d(n)dn */
=z
v
/'CO
e<
2^1
r
e
<
J in
which we write
$([*)
lo
^o
o
.)/ (a;_ t)dt
,
(8.3)
= ^ and /GO
{ew/r(/i
I(s)t=
+ l)K<<
.
^0
Since
we have e*>*
we immediately v
=
1,
that
>f(x)=f(t)
,
obtain the inversion of (8.2) out of (8.3) by setting
is,
f(t)I(xt)dt
.
It is of some interest in connection with this solution to note that the function I(s) is asymptotic to e Es in the infinite interval, a To see this we apply the where we use the abbreviation E e Es 1 Maclaurin integral test* for convergence to the series e Es
=
+
2
(Es) /2l *T. J.
+(
r
s)
Bromwich:
3
/3!+
,
Infinite Series,
.
= +
and thus obtain the inequality 2nd
ed.
(1926), p. 33.
EQUATIONS WITH CONSTANT COEFFICIENTS
Dividing by
e
ES ,
293
we have
which, for large positive values of s, establishes the desired property. It should be pointed out further that the function I(s) is also intimately related to the MittagLeffler function, E a (x) derived in the solution of problem 3, section 6. If we make the transformation, H~at, x^= e aa s a , then r J. is seen to be the continuous counterpart of E a (x), in which the sign of integration replaces the sign of summation. We shall next consider the integral equation,
f(x)
=
*[lotf(xt)+Alog(xt)
+ B]u(t)dt
,
(8.4)
an equation also due to Volterra. This equation can be written symbolically in the form f(x)
=z
where we abbreviate,
The function v'(#) is the psi function, which, together with its first four derivatives, has been extensively tabulated.* Numerically we 1 C (Euler's constant) have y(l) .5772157, y/(l) 2 2 1.6449341. 1/2 4 1/3 H
The
solution then appears in the
=
= +
=
=
form
A 2 ), and where /U and A 2 are roots of l/(/U 9^2
where
+
+ =
= =
= 3"^
f'R(xt)f(t)dt
,
JQ
where we abbreviate
BOO
=
00 I*
{*"/*
^o
*See H. T. Davis: (1933); vol. 2 (1935).
Tables of the Higher Mathematical Functions, vol. 1
THE THEORY OF LINEAR OPERATORS
294
PROBLEMS Show
1.
lowing
that the operator inverse to X(e)
1
0,
xz, is
e
the fol
:
= =
Y(0)
l
eeoli(e)
i
ee~'
where we abbreviate
li(eo)
=
C*>
J 0 and
C
is
Euler's constant.
(1
u(x)
,
Obtain the solution of the equation
2.
in the
(e*/t)dt
(?)
tt(a)=/(a:)
form
= Y(e)
*f(x)=Ax +
x
1
\ogx~ V*"/ (ll) (0)/[(w
(u
1)
1)!]
Convergent and Asymptotic Series. In the preceding sections set up the formal machinery for the solution of differential equations of infinite order with constant coefficients. We shall now consider the nature of the convergence of the formal inverses that have been obtained. Leaving to section 10 the problem of the homogeneous equation, we shall consider the convergence of the formal P.
we have
operator
v
'
F(z)
The following theorem
Theorem
6.
is
pertinent:
If the operator l/F(z) has the expansion 1
T(z)
=
^
mZ
'
u
which is convergent in the annnlus R' function of finite grade q such that R'
<
^
< R, and R, then
z 
q
<
if
f(x)
is
a
(9.1)
a function of finite grade at most equal to q. If, moreover, F(z) is expansible as a power series within the circle \z\ < R, then u(x) is a solution of the equation
is
F(z)*u(x)*=f(x)
.
(9.2)
EQUATIONS WITH CONSTANT COEFFICIENTS
295
Proof: Since the conditions of theorem 6, chapter 5 are fulfilled, it follows that [l/F(z)~\ f(x) exists and defines a function of to q. Since u(x) as given by (9.1) is thus a funcgrade at most equal tion of grade at most equal to q, it follows by the theorem just cited and the conditions of the theorem that F(z)  n(x) exists.
Hence, since
/<*>/<*>. *>]from the properties of the operator,
follows that (9.1)
it
is
a solution
of (9.2).
In order to establish the uniformity of the convergence of the function F(z) ^ u(x) to /(#), we write
F(z)
= ^a,
a
zm
(9.3)
.
in o
Representing the theorem 5, chapter 5,
first
we
n terms
of this series by
f(x)
F n (z)
>
n
(z)
and noting
00
00
I
F
obtain
u(x)
^M V


2
a m (q+e)
^ MM' 2
\
(Q/R)
m
2 /v
=M 1
I
\
bk
\
MQ*
7
y/A
k_j,
M
where and M' are properly chosen constants independent of n and k and Q is subject to the inequality ft' < q < Q < /?. Hence, since Q < R, and the series in the right hand member F n (z)+ ii(x)\ converges uniconverges, it follows that f(x) formly to zero as n > oo .
The theorem which we have to include
many important
just established is not broad enough applications, as, for example, the inver
sion of the equation
where the righthand member js a function of infinite grade. By extending the domain of solutions to include divergent series which are summable by the method of Borel, it is possible to enlarge the field of application of operators.
The following theory
indicates the nature of this extension
:
THE THEORY OF LINEAR OPERATORS
296
Theorem
If in equation (9.2) the function f(x) has the ex
7.
pansion f(x)
=hi/x + h2 /x 2 + h
3
/x
fc n /a n
3
\
+ .,
R<
x
,
\
then a solution of (9.2) exists of the form
cxtdt
(9 4) '
'
/GO where we write
provided positive values of (a)
(b) t
^

m exist* such that F(t) < A emt for < k ^ t^
A, and
k,
Q(t)/F(t) and
\Q(t)/F(
is
t)
oo
\
of limited variation in the interval
,
^
k.
The function u(x),
in general, represents a solution of (9.2) the sense in asymptotically of Poincare (sec section 4, chapter 5).
Proof: The conditions imposed by the theorem are those of theorem 8, chapter 5, and hence u(x) determined formally is summable by the method of Borel. It is thus represented by the integral (9.4).
In general u(x)
is
a function of
=
f
t
n
infinite grade, since
[Q(t)/F(t)']
we have
dt
Joo
and hence
^
t
n
A
e mt dt

Jo
= A n\/\ xm l^
1 ,
R(x)
>m
.
In order to establish the asymptotic character of the solution we in the series (9.3). Representing the first n
assume F(z) expansible terms by F n (z), we find
Q(t)
J ^
f 00
=A From
e~ xt
2
A
2
(n+l)
this inequality
e 2tnt t n + l
!/
x
we
lim/(aO
co
2 a m *dt\ ^'frrfT L) r V WW41
dt
2m
n+2 \
,
R(x)
> 2m
see that for a fixed n,
.
we have
F n (z) >u(x)\t=0
x=an
which establishes the desired asymptotic convergence.
*We
should note that both k and
m
may assume
the value
0.
EQUATIONS WITH CONSTANT COEFFICIENTS
297
10. The Homogeneous Case. In the foregoing theory we have touched rather lightly the problem of the homogeneous equation, namely, the problem of solving the equation
F(z) *u(x) If i n is a zero of
(10.1)
.
w
F(z) of multiplicity
?{ ,
then
it is
clear that the
function
where we write Pn (X)
= Po 00 (10.2)
will
be a particular solution of (10.1) questions of interest present themselves
Two
conditions will the
:
First,
under what
sum 00
u n (x)
tt(a;)=
(10.3)
n=i
converge? second, under what conditions will u(x) furnish a solution of the equation (10.1) ? In order to answer the first of these questions we first find a majorant for the polynomial (10.2). If Cn represents the absolute
value of the greatest of the coefficients and if the values of x in a circle of radius R > 1, then we shall have \P n (x)
< mn C

R" * 1
lie
with
1
n
(10.4)
In order to obtain a lower bound for \P n (x) ing theorem due to H. Cartan
we note
the follow
:
Let PI, p 2 p n be any set of n distinct points in the plane and let be an arbitrary positive number. Then the points X of the plane for which one has the inequality 
,
,
H
Xp,
XXP*XXP*XX Xp
n
s ff
can be enclosed within the interior of circumferences in number at most equal to n, the sum of whose radii is equal to 2eH, where e is the
Naperian base.*
Now
pt=mn
let 1,
R
xp be a number such that all the zeros, x l9 x*>, He within the circle x t= 2R, and let x' be some ,
,
of P n (%)
*Sur les systemes de fonctions holomorphes. Annales de I' E cole Normale Superieure, vol. 45 (3rd ser.) (1928), pp. 255346; in particular pp. 272 et seq. This theorem is a generalization of one originally given by A. Bloch: Annales de I'Ecole Normale Superieure, vol. 43 (3rd ser.) 1926), pp. 309362; in particular, p. 321.
THE THEORY OF LINEAR OPERATORS
298
point on the circumference of the circle of radius R. The point x' obviously be chosen so that P n (x') is at least as great as C. shall then have 
\
r ^
7^
may
We
/
1
Q
i
>
(2R)~ P II
I
x
x
\
But by Cartan's theorem the
last product is greater than to set of p circles the sum of whose exterior a (R n /2e) , radii is at most equal to R n Hence we get the inequality p
when x
is
.
\P H (x) If
we now assume
>C*(R
ft
m **
/4eR)
(10.5)
.
that n /a*)
(10.6)
,
it is clear that inequalities (10.4) and (10.5) limit the study of the convergence of (10.3) to that of the series
then
U (x)
=2C
c x '
rt
* .
Ml
The problem
of the convergence of (10.3) has thus been reduced and we shall postpone further discussion
to that of a Dirichlet series
of this question to the next section. We turn next to a consideration of the second question proposed above, namely, the validity of the solution. In this connection we shall prove the following theorem due to J. F. Eitt (see Bibliog
raphy)
:
Theorem U2 (x)
^i (x)
,
8.
,
,
If F(z) is an entire function of genus zero un (x) are particular solutions of
and
if
,
F(z) *u(x)
(10.7)
,
then the function
u(x)
=%u
n
(10.8)
(x)
Wl
is
a solution of (10.7) in any region within ivhich the righthand mem
ber converges.
Proof: Let us consider the difference
\F(z) +u(x)
F(z) *U m (x)
=
00
\F(z)
OJ
(*)!
,
(10.9)
where
Um (x)
is
the
*This conclusion Bibliography}.
sum
of the first
m
terms of (10.8).
and the foregoing analysis
is
due to G. Valiron
(see
EQUATIONS WITH CONSTANT COEFFICIENTS
299
But by Cauchy's inequality [see (2.1), chapter 5], we have for every function
the following inequality
r,
z p >
where
M
the
is
a
,
j

<
<
x
a \

f r
,
of y(x) on the circumference of the
circle.
Hence we
shall
have
<M2^ = MK
\F(Z) >
sum
,
*
member converges
in the righthand
to the value K.
But
and
this implies (see section 2, chapter 5) that (pi hence that F(z) is an entire function of genus zero.
ap
J//7
)
>0
Returning now to (10.9) we have from the uniform convergence of the series and
from (10.10) the equivalence
F(z) >
un
2,
(x)
=
n~tn+\
v
[F(z)
*/(#)]
.
HJ//+I
Since U in (x) and all the U H (X) are solutions of (10.7), it follows that F(z) > u (x) converges uniformly to zero, which establishes the
theorem.
PROBLEMS 1.
Solve the equation
2.
Since the function
"" is
V
IT
,~
cos
=
(2??
(2^1)2
a solution of the equation of problem lim u tn (x)
is
m
=4
1,
explain ty'xx
ty^r
why
<
,
x
not a solution. 3.
Show
that
Jo reduces to the linear differential equation (2
Hence 4.
3
+
/3
5

/5 H
)
/(a?)
=
.
solve the equation.
Solve the equation u' (x)
=. au(x)
+
6 sin
u(x)
.
<
2
,
THE THEORY OF LINEAR OPERATORS
300
Prove the following theorem:
5.
(Polya)
Given two series of numbers {a n } and (b n ] such that sup lim &J 1/n 1/n 0, construct the new set of numbers (finite) and lim &J
=
Cn
where
n
Cr
are zero,
is
we
=K
the shall
Hh
bn
+ n^i
+
b ni
i
n^2
binomial coefficient.
2
+
'
'
'
+
6Q }
<*
,
If not all the values of the set {b n }
have a
sup lim [cj 1 /" 6.
6 n~2
Employing the g( X )
definitions of
=
1Z+J1Z2
(b o
where / (x) has one singular point, x o following development about x
=
f(x)
problem
= a /x + Q
Prove, in particular, that g(x) has x
=
a,
on
5,
.
discuss the function
...)+ its circle
f( X )
,
of convergence and has the
:
Oj/aca
+ ajx*
\
= a as a singular
.
point.
(Polya).
M. Kalecki [Econometrica, vol. 3 (1935), pp. 327344] has reduced his 7. macrodynamic theory of business cycles to the solution of the following mixed difference and differential equation u'(t)
=au(t)
bu(tO)
where
9 is a constant. Discuss the solution of this equation in general. Using Kalecki's values a 0.6 discuss the solution of the equation. .i279, .158, b This problem under the stimulus of Kalecki's application has been examined by R. Frisch and H. Holme [Kconometrica, vol. 3 (1935, pp. 225239]. Prior to this the equation in more general form was studied in a series of papers by F.
=
Schiirer (see Bibliography).
11. DirichleVs Series. We have seen from the preceding section that the existence of solutions of (10.1) are fundamentally associated with the convergence properties of the Dirichlet's series
tt(s)=ZX
x "*
(11.1)
and {c n } are sets of complex numbers. We may in parassume that these numbers are real and that the set {A n } is monotonically increasing with oo as its limiting value. For this case the convergence of the series (11.1) is a half plane bounded on the left by a line of convercngce. The point (
{A w }
ticular
:
= lim sup log *

UO
EQUATIONS WITH CONSTANT COEFFICIENTS
t= lim sup log
c n /i w
I
H = GO
The
and the second
is
be used
.
w _ /rt
when
the abscissa of convergence w'hen the abscissa of convergence limit, a*,
first limit, a, is to
is positive
301
negative,
The
of these limits is generally attributed to E. Cahen: Sur la foncde Riemann et sur des fonctions analogues, Annales de I'Ecole Normale Sup6rieure, vol. 11 (series 3) ( 1894), pp. 75164. The second was given by S. Pincherle: Alcune spigolature nel campo delle funzioni determinant i, Atti del IV Congresso del Matematici, vol. 2 (1908), pp. 4448 and W. Schnee: Uber die Koeffizientendarstellungsformel in der Theorie der Dirichletschen Reihen. Gottinger Nachrichten (1910), pp. 142. Formulas which yield the abscissa of convergence for either sign have been designed by K. Knopp: Dber die Abszisse der Grenzgeraden einer Dirichletschen Reihe, Sitzungsberichte der Berliner Math. Gesellschaft Jahrg. 10 (1910), pp. 17 and T. Kojima: On the ConvergenceAbscissa of General Dirichlet Series, Tohoku Math. Journal, vol. 6 (1914), pp.
tion
f
first
(s)
t
134139.
An
excellent
summary
of the theory of Dirichlet series will be found in a
monograph by G. H. Hardy and M. Riesz: The General Theory of Dirichlet'x Series, Cambridge (1915), 78 p. For recent developments the reader is referred to E. Hille: Note on Dirichlet's Series with Complex Exponents, Annals of Mathematics, vol. 25 (2nd series) (192324), pp. 261278, G. Polya (see Bibliography} and G. Valiron (see Bibliography).
The half plane within which the series (11.1) is absolutely convergent is similarly defined by the abscissa of convergence computed from the following formulas:
>
a* c= lim sup log xiuo
y LA
!/!/; \^n\/ATH
,
n=m
where as before the first limit is to be used when the abscissa is positive and the second when the abscissa is negative. If we now remove the restriction that the values of the set form a monotonically increasing series of positive numbers with oo as a limit and assume that they may be any set of complex numbers, we no longer find the region of convergence to be in general a halfplane. For convenience it will be assumed that series (11.1) con00
verges at the origin, that
is
to say, the series
cn
converges.*
no
It will first be shown that R, the region of absolute convergence of (11.1), is convex, that is to say, if the series converges for two points XL and x 2 in R, then it converges for all values on the linear
segment connecting X L and x 2 *This discussion
is
.
taken mainly from Hille:
loc. cit.
THE THEORY OF LINEAR OPERATORS
302
To prove this, suppose that the series is absolutely convergent for the two points x t and x 2 Then any point on the linear segment (XiX 2 ) can be represented by .
= tx +
x
We
t)x 2
(I
1
Qztzl
,
.
then have
Vj
CO
CD
00
V /_j 1
]/
C n
pXnX C/
V
^ 
>\ntlt p Xnf>(l~t) t'/i On ^
!/
1
\
j
I
no
(> (y tt

A I^H
i
\
# n UO &
>
wo
wO
where we use the abbreviations A /I n
1/JAnJil V I
But the
RJ
,
/3Xn^ C/
/t
I
.


series GO
Lc t/
A
M /l n
t
(1 Tl >n
 f)
n^O
converges since (1) A* Bn ^)B W < (l ^ tA n 2'c n A n and Z\c n \B n are convergent by hypothesis.
+
(1 ~
For complex
Dirichlet's series Hille defines a
convergence as follows: Let us write
and
let
^
Lo ment.
L n ~ \og\c n
A
rt
+B
n,
and
(2)
maximal region of
\
{L} be the set of the limit points of L n Now take any point of the set (L) and join it to the origin by a straight line segThrough L draw a line perpendicular to (L 0), thus divid.
ing the plane into two parts. Denote that half plane in which
R(x/L) ^1
R
,
denotes "the real part of", When this construction has been repeated for all the limit points, a region will be defined common to all the half planes, J J,,, and this may be denoted by J. Ji, If any of the limit points is zero then the construction fails, but the division of the plane may be accomplished usually by a limiting process. Thus a set of half planes is constructed for some set having the limit point zero and the limit of this set of half planes is adjoined to
where
by J
.
,
the set
zl/.
not difficult to show that any point exterior a point of divergence for series (11.1).
It is
A
is
It
may
also be
proved that lim  j:
then the Dirichlet's series
region
if
UJ/ log^c=
is
to the
co
,
absolutely convergent in the region A',
where A' denotes the open region A minus
its
boundary
set.
EQUATIONS WITH CONSTANT COEFFICIENTS
303
minimal region of convergence, assuming a point of absolute convergence for the series (11.1). This definition furnishes sufficient conditions, independent of the magnitude of the set {A,,}, for the absolute convergence of (11.1). Under the condition that the series ~c>, converges, it is clear that Hille also defines a
only that the origin
is
converges to zero.
Now
consider the sets of points {p n }, where
pn
= log R /An
define

n
This set has a set of limit points which
we
we may
designate by {P n }*
Employing this set of limit points exactly as we did the limit points , A, , {L,,} above, we can construct a new set of half planes,
D
similar to the set of half planes assume that J9 k
,
l<>,
Ji,
K(X/P n ) <1 common to the set {D n },
,
J,,,,
except
we now
.
The region D,
A and It
is
included in the region
series (11.1) converges absolutely within
has long been
known
it.
that the convergence of a Dirichlet's
series also implies the convergence of factorial and generalized binomial (Newton) series. In order to establish this connection con
sider the following series
:

cn



,
(1L2)
Wl 00
(l) n c (xa,)(x
w(x) <=
fl 2
n

)
(xa
tl
)
n=i
(11.3)
Let us
now employ
the following abbreviations
:
n
^n^El/am, m~i
E
n
(x)
=c
n e*
T ,
H
n
(x)
=/} (lx/a^e"** m=i

Using these abbreviations we can now write (11.1), (11.2) and (11.3) as follows:
THE THEORY OF LINEAR OPERATORS
304
We now
invoke AbeFs lemma for complex series which states the series ZA n is convergent, and if the series B n+i ) (B n B then the series 2A n is convergent.* H absolutely convergent, It will now be shown that if
that is
if
converges, then the convergence of IE n (x) implies the convergence of v(x) and w(x). In order to prove this, consider the difference
H ^(x)\^=\ n (Ix/a m )
\H n (x)
n
*=\H n (x)\K n where K n Moreover
is
H
x\*/\a n+l \*
uniformly bounded in every n
converges by
,
finite
also similarly bounded (x) hypothesis, the series is
(lx/dn^e"**^
1
region of the #plane.
and hence, since Jl/a m
2 
converges absolutely.
Combining this with Abel's lemma we see that the convergence similar conclusion may of u(x) implies the convergence of w(ce). in the essential be drawn without argument for the converchange
A
gence of
*See
I.
Bromwich: The Theory
of Infinite Series.
(2nd edition), London
(1926), pp. 242243.
fThe argument here follows that given by Hille whose proof is based upon Sulle serie di fattoriali generalizthat given by S. Pincherle for u(x) and v(x) zate. Rendiconti del Circolo Matematico di Palenmo, vol. 37 (1914), pp. 379390. :
CHAPTER
VII.
LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS OF INFINITE ORDER
WITH CONSTANT COEFFICIENTS 1.
chapter
THE HEAVISIDE CALCULUS.
Some General Dynamical Considerations. In the preceding we have discussed a number of methods for the solution of a
differential equation of infinite order
with constant coefficients. These methods have a general validity beyond the applications which were made there and need only slight modification in order to be applied systems of linear differential equations with constant coefficients. is doubtless well known to the reader such systems are fundamental in many problems in dynamics. It is obviously beyond the scope of this work to enter fully into these dynamical considerations except through some special application. For this application we shall choose the theory of electrical conduction because it involves most of the elements to be found in dynamical problems of the closed cycle type and thus may serve as a prototype of oscillatory phenomena in general. It would seem to be useful to set forth a few elementary concepts. Let us assume that y is the displacement of some dynamic variable from a position of equilibrium and that this displacement is a
to
As
function of time,
y
= y(t)

Examples of such displacements are very numerous. Perhaps the simplest is the displacement of the bob of a pendulum from its position of equilibrium. Also y might represent the displacement of a bead on an elastic wire fastened at each end to rigid supports it might represent the moving charge in a portion of an electric circuit or the displacement of a price index from its line of trend. If y(t) represents a simple harmonic motion, we can then write it in the form, ;
y
(t )
= A cos (2nt/T) + B sin (2nt/T) = (A f sin (2nt/T + a) 2
Z?
2
>
)
,
where a arc tan A/B. In this representation T is called the period, B 2 ) the amplitude and a the phase angle. The reciprocal of T (A 2 is called the frequency of the harmonic motion.
+
J
More generally a motion simple harmonic terms ]/(*)
=^o + I A
n
may
cos(2nt/T n )
sum
be expressed as the
+
B n sm(2jit/T n
)
.
of several
(1.1)
w=i
n=i
If T n t== 1/w, and the summation extends to infinity, of a Fourier series.
305
we have
the case
THE THEORY OF LINEAR OPERATORS
306 It
should be noted that the total energy (E) of such a system
is
given by,
where the C n are the weighting factors determined from the physical conditions of the problem. All dynamical systems, when not sustained by impressed forces, tend toward positions of equilibrium. Free energy is dissipated and disappears into the lowest energy frequencies of the system, that is to say, into molecular frequencies. Under these circumstances we say that a damping factor has been present. In many dynamical systems this
damping factor r[ cr where
function,
,
is
accurately represented by the exponential and depends upon the physical
r is positive
properties of the system. It is obviously possible, therefore, to improve the description of the actual motion of a dynamical system by
means of the damped harmonic
series,
+7? H
sin (2
Tke Problem of Electrical Networks. In the problem of the 2. flow of current in an electrical network we are concerned with certain physical quantities called resistance, inductance and capacity which are represented customarily by the letters 72, L and C. Resistance plays a role similar to that of friction, inductance to inertia, and capacity
to the spring potential of ordinary material systems. In addition to these quantities, the description of an electrical system includes impressed forces designated as c. m. f. s. (electromotive
forces)
.
The laws which govern the flow of electricity in a network be stated as follows:
may
The algebraic sum of the currents entering a branch point
(a)
of the network
is
The
always zero.
impressed force around any complete circuit in equal to the potential drop due to resistance, inductive and reaction capacity reactance in the circuit. (6)
the network
(c)
ductance
where Q
total
is
The
potential drop in a given branch with resistance R, in
L and is
equation /
capacity
C
is
equal to
L
d 2 Q/dt*
the moving charge. The current dQ/dt
=
.
is
+ R dQ/dt

Q/C
,
computed from the
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS These laws
may
be illustrated by means of the following two 2 by Z (p) the operator L p ~\ R, p From (a) we have, d/dt
typical circuits. Let us designate i
,
where p
=
l
l
.
= dQ /dt 2
where the subscripts refer ure
307
j
to the three
dQ./dt
(2.1)
,
branches of the circuit in
fig
1.
By means
of laws (b) and (c)
we
Eliminating Qj by means of (2.1)
>Q
Z
a
Z
l
2
These equations
we
Q
(p)
obtain s
*=E
i
(p)
FIGURE '
arrive at the equations
may
1
be written more specifically,
(1/C2
(Rt
+ 1/Cs)
]
In the second circuit diagramed in figure 2 we designate the muby M. The equations of the two branches are written down by inspection as follows
tual inductance
:
or
more
specifically,
(L,
p2
+ Rt p
THE THEORY OF LINEAR OPERATORS
308
Other more complicated circuits are similarly reduced to operators.
In terms of the simple circuit,
l/C) the function,
is
referred to as the impedance function.
When an
alternating e.m.f.
is
imposed upon the
circuit,
then the
function, z(coi)
=R+ [Leo
I/ (Ceo)]
i
,
the impedance of the alternating current and the real and imaginary parts are called the resistance and reactance respectively. is called
Figure
The on the
solution of the equation
circuit at
t
~
0,
2.
when
unit e.m.f. has been impressed
namely,
z(p)
>/()=!,
t
>
,
defined to be the indicial admittance. This solution nated by A(t). is
is
usually desig
PROBLEMS If a condenser of capacity C 3 discharges into two circuits with imped1. ances equal respectively to Z.L (p)/p and Z 2 (p)/p, show that the charge Q on the conductor is governed by the equation
2.
If in problem 1 the first circuit contains no capacity
and the second no
inductance, show that the equation reduces to
3.
A
circuit is
this parallel circuit
composed of an inductance and a capacity in parallel and in series with a resistance R^ and an e.m.f., E. If the im
is
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS
309
pedances of the inductance and capacity branches are designated respectively by
and
z 2 (p)
z 3 (p)
and the currents by
/2
*a
+
If /j is the current in the
and
J3
,
derive the equations
^ + Val

E

R^ how may
branch with the resistance
it
be
computed? electric filter is meant a system of circuits connected in tandem as The impedances z l (p) and z.2 (p) for the circuit in figure 1 we shall call the series impedances and the impedance z 3 (p) the shunt impedance. Let us consider a filter of n circuits in which the shunt impedances are all equal to z 2 (p) and the series impedances are all equal to z l (p) except in the first and last circuits where they are each equal to I/z^ (p). If the first circuit l contains an e.m.f., E, show that the circuits are governed by the following equa
By an
4.
in figure 1.
9
tions:
V2 z, (p)

7
3 X (/>)>/,
+z
2
(p) > (7 X
+S,(p) 
(/,
72 )
=E
/, +1 )
,
MP>
/r
*(/,!
^
r
)=0
,
1, n.
5. Discuss the electrical filter of which the following problem is the mechanical analogue: Consider the motion of an elastic string on which are fastened n beads, each of mass ra, whose distances apart are equal to the constant length a. If y lt y 2 yn represent the displacements of the respective beads from the position of ,
.
,
equilibrium of the string, if these displacements are supposed to be at right angles to the string and in the same plane, and if S is the tension of the string,
show that the following system of differential equations govern the motion of the beads:
+
mi//'
(S/tt) (y,
+y
1/ 2 )
+2/,
3/3)
=
,
"
w?/ 2
" >>J n
f
+
(S/a) O/o
(S/a) (y n
?/,
y n _i
+ yn
0)
.
=
.
[This problem was originally treated by J. Lagrange: Mecanique analytiSee also: A. G. Webster: The Dynamics of 1, Paris (1788), p. 390. and of Rigid, Elastic, and Fluid Bodies, Leipzig (1904), pp. 164173].
quc, vol. Particles
3.
Fundamental Theorems.
We
proceed next to a discussion of
the solution of systems of equations of the type derived in the last section. It will be convenient to limit the exposition to the case of a
system of two equations, but no essential impairment of generality We shall thus consider the system
thus introduced.
*Q*(t)=E,(t)
C(p)Q (t)+D(p)Q (t)=E l
2
2
(t)
,
,
(3.1)
is
THE THEORY OF LINEAR OPERATORS
310
where the operators are linear and of
finite order with constant are concerned here particularly with the Heaviside problem, we shall assume that the two functions, E l (t) and E 2 (t), are zero prior to t 0. In the last chapter we showed that this assumption is equivalent to the determination of the functions Qi(t) and Q?(t) so that they shall vanish to as high an order as possible at t Q. In the actual circuit problem the operators are polynomials of second degree, but there is no reason in the ensuing analysis to make
coefficients.
Since
we
=
this limiting assumption.
As a simplification which does not impair generality we will note that the solutions of the system can be obtained from the addition of the solutions of the two problems :
A(p)
Vi (3.2a)
C(p)
A(p)
V\ + D(P) V,=
;
*
(3.2b)
=
+W
F 2 (0 That is to say, Q, (t) c=V*(t) +W^(t) and Q 2 () 2 (t). Hence no loss of generality is suffered if wfe consider a problem of the form (3.2a). A further simplification of the problem is possible by replacing Ei(t) with a unit e. m. f. By what is known as the superposition theorem, which we shall discuss later, it is possible to derive the solution of the general system (3.1) by means of a single quadrature of the solution of this system in which E^(t) and E 2 (t) have been replaced by 1. Let us therefore consider the following theorem: Theorem
1.
If
^(t) and h2 (t) are solutions of the system
A(p)
>M*
C(p)
>M*
(3.3)
where A (z) is a polynomial of degree a, B of degree b, C of degree c, and D of degree d, then solutions h (t) and ho(t) exist which vanish 1 together with their derivatives of orders up to and including a c .* 1 respectively provided a f d > & and b L
+
Proof: If tional symbols
*No than b
replace fci(t) by H^ in (3.3) and drop the operaobtain the generatrix equations
+
>
d b 4 c rather essential restriction is imposed by assuming that a a c d, because this assumption merely fixes the subscripts of
4
h and h
we we
>
+
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS
311
which have for solutions
= D(p)/[A(p)D(p) C(p)B(p)1 H, = C(p)/\_A(p)D(p)C(p)B(p)] H
1
.
Let us write A(p) =A(p)D(p) C(p)B(p) as a polynomial of a ,p" f a,p n ~ \ a n where it is clear that degree n, i.e., A (p) n a ~\ d
=
=
l
(
,
.
Since D(p)
#i
=
(do2>*
of degree d
is
+
'
rfip"
= (1/P")
(vl.
wa may
write HI in the form
H h
+
Giving to 1/p"  /(f) the customary interpretation
=
(t
s) 1
*Jo
we
obtain as the value of h^(t) the function
+ .]
.
(3.4)
vanishes together with its ?i d 1 a 1, the part of the d 1 0. Since n be true. A similar argument aptheorem relating to hi(t) is seen to c 1 derivaplies also to Jiv(t) except now the solution and its n c tives vanish at t 1 > b 0. From the inequality n 1, we establish that part of the theorem which applies to h 2 (t). It should be noticed that these results might have been stated directly as corollaries of theorem 2, chapter 6, since the operational expansions which were employed were those of the outer Laurent annulus. We have previously discussed the method of Carson as it applies to single differential equations. (See section 4, chapter 6). This method immediately generalizes for systems of equations and we may state it as it applies to system (3.3). It is clear that this function
derivatives at the point
t
=
=
THE THEORY OF LINEAR OPERATORS
312
Theorem 2. The problem of solving system (3.3) is formally equivalent to finding the solutions of the integral equations:
(3.5)
r
Jo '
Proof: The proof of the theorem is easily effected by showing the formal equivalence of (3.4) and (3.5). Noting the integral POD
pPl c
I I
fin tt
rff ^^ a6 i
v) rc
T
P
//rittn
'/
y
Jo
we have
directly
,()d<
=
We are now in a position which may be stated
Theorem
3.
If
to derive the solution of
problem (3.2a)
in the following superposition theorem:
h
and h 2 (t) are solutions of system
(t)
(3.3)
vanishing together with their derivatives of orders a 1 and b 1 respectively at t 0, then solutions of (3.2a) are obtained from the
~
formula t
V>(t)=p*
l
E
l
(s)h
l
(ts)ds
i*=l,2,
,
(3.6)
Jo
provided a\d>b\c. Moreover Vi(t) and V2 (t) will vanish together with their derivatives up to and including orders a 1 and b
1 respectively.
Proof: In section lution of the equation
4,
chapter 6
we have
identified the
Carson
so
F(z) *u(x) with the solution described generatrix
is
of the
in
theorem
tively, its
6.
If the resolvent
form
l/F(z)=P m (z)/Q where
chapter
2,
n
(z)
,
P m (z) and Q n (z) are polynomials of degrees m m < n, it will be recalled that the solution u(x)
derivatives
up
to
and including order n
m
and n respectogether with
1 vanishes at
x =0,
provided that the expansion of the generatrix in the region exterior to its poles is employed. Since the Carson solution is identical with
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS
must vanish together with
this one, it
its first
(n
m
313
1) derivatives
at the origin.
Hence we see that V (t) are given formally by (3.6). Also from the explicit forms of t and 2 it is clear that V^(t) together with its first (n d 1) derivatives and V (t) together with its first (n c 1) derivatives will vanish at t t= 0. Since we have n d 1 L
#
H
f2
ncl
=a
1 and > 61, it follows that V,(t) and V*(t) will vanish together with their derivatives up to and including orders a 1 and 6 1 respectively.
The
solution of system (3.2b) and 3 and
plication of theorems 2
E where the functions
fc,
(*),
2
(s)
k
i
=
1, 2,
t
(t
obtained
is is
now by a simple
ap
found to be s) ds
i=l,2
,
,
are inversions of the equations,
B(p)/[pA(p)]
W^(t) will vanish up to and including order and including order d 1. z (t) up From this it follows that Q (t) of the complete system (3.1) will vanish up to and including order p 1, where p is the smaller of the numbers a and c and that Q 2 (t) will vanish up to and including order q 1 where q is the smaller of the numbers 6 and d. It is also clear that
c
1
and
W
to
{
Example: The following example
will illustrate the application
of these theorems.
Let us
find the solution of the
system
which vanishes to as high an order as possible at the We first consider the reduced system
hS 4 2h,
Then
hi(t)
f
1^" f 3/h'
and h 2
(t) will
P(P
D (P+D
P(P
D
3
+ 3/i = 2
origin.
.
be solutions of the equations
THE THEORY OF LINEAR OPERATORS
314 Since [p
we have
+ 3p + 3]/[p(p = 3/P +
2
1)
3
(P+D]
+
U/8) /(p+1) (ll/4)/(p
= 2/p we can
1)
(l/8)/(p+l)
+
/(p1)
(7/4)
l
n
/nl) e
xt
d
+
2
(15/8) 2
1)
(7/2) /(p
I)
3 ,
/(p1) (3/2) /(p1)
8 ,
h2 (t) from the formula
derive the values of h t (t) and e* (t
(23/8) /(p
=
o
We
then get (1/8)
^2
(1/8)
e*+ (23/8) (15/8) ^
e'
+
f
(11/4) *6*
e<
(7/4) te
f
+
(7/4)
(3/4)
a
t e*
and we observe that
Similarly for the system
1=
we
,
get
(141/8)e
'+ where
MO)
= */(0)
A/'(0)
fc 2
+
(64/4)e
*
(33/4)
(0)
= A,V(0)=
Making use of theorem 3 we obtain
(21/4) te*
f
e^
(85/8)
=
(
e
+
(9/4)
a
t e*
,
0.
the complete solution of the
system in the form
We also note that Q, (0)
,
= Q', (0) = Q
2
(0)
= Q'
2
(0)
=
.
LINEAR SYSTEMS
THE HEAVISIDE CALCULUS
315
The Heaviside Expansion Theorem. Central to many appli4. cations of the Heaviside calculus is the so called expansion theorem. Numerous proofs of this theorem have been given and Heaviside himself gave two as has been pointed out by M. S. Vallarta (See Bibliography). The expansion theorem is easily written down from the results which we have previously obtained, being generalized, as a matter of fact, by formula (2.2) of chapter 6.
To put the theorem in the form in which it appears in the theory of electric circuits, we consider the operational symbol G(p) where
Pm (p)
= Pm(p)/Q
Q n (p)
and
n
(p)
m
,
,
are polynomials of degrees
m
and n respec
tively.
Referring to formula (2.2) of chapter 6 we see that we may write the expansion of the outer Laurent annulus, that is to say, the Heaviside expansion, in the form 
G(p)
J=l
+ E"(0)/a,* + ...} + {G(0)
+G'(0)p
+ G"(0)p .
whereof^,
This
a n are zeros of Q n (p) be formulated in the following theorem:
0*2,
.
,
may
Theorem
5.
G(p)
If
is
the operational symbol,
G(p)^=P m (p)/Q n (p) then G(p) upon E(t)
(4.1)
,
m
,

E(t) as defined by (4.1) is equivalent to the operation of the expansion of G(p) in its outer annulus of conver
gence.
Two a unit
e.
special applications are to be noted, the first
m.
/.,
=
E (t)
=
1,
and the second where
E (t)
is
where E(t) is an alternating
xt
e* e.m.f.,E(t) In the first case the expansion theorem reduces to .
G(p)>l = and in the second
G(p)*c'
.
to
= ;{J (4.2)
This formula must be modified when li equals some one of the values of a,, let us say a. This modification is made by means of
THE THEORY OF LINEAR OPERATORS
316
formula (2.4) of chapter
Referring to this formula and noting that
6.
= Q'
F'(a)
= [P
F" (a)
we modify
Q" n
m (a)
(a)/P m (a)
n
(a)
(a)
2P',,,
,
(a)
Q',,
]
/P m
(a)
,
(4.2) to read
G(p) *e'"
=
{PM (a,)e ;i
+ {
Example
a has been omitted from the sum.
ti
As an example
:
n '(a)}{t
us solve the following equation,
let
which is the equation of the charge in a simple been imposed a unit c.m.f.
Making
4L/C > R~
the assumption that
of the equation,
Lp
Rp f 1/C =
2

upon which has
circuit
we obtain
as the roots
the values &i
f coi,
,
= & ~ a R/2L abbreviations,
,
aj where we employ the 2 2 =^ that 1/C Q'(p) Lp + Rp (1/CL Q(p) ^V4L )^. Noting == 2Lp f /? we obtain Q'(a,) ^=^ 2Lo> and Q'(&2 ) t^ 2Lo> Employing these values in the expansion theorem we at once
&
cu
u>i
,
+
,
,
.
,
obtain,
G(p)
1=
(
== (e
By means
atwt
[c
of or
" (,
/rt
>V(2L?wa T
v
/2L)
f
&2
r [
e" .
~
,
1
)
co
2
+
e (a
^ V( i)
daji

.,
2Licwtt 2 )] .
e"
o>
2
4
ttcoi
_

2
]
T + C
l/CL, this expansion reduces
In order to obtain the current, I(t),
+C
we compute
f
_,
to,
Q'(t) and thus
find,
2 assumption 4L/C > R' we have 4L/C the formulas for Q(t) and Q'(t) are found to be
If instead of the
Q(t)*=C where we abbreviate, o/
,
C
=
e
(li
2
(cosh o/t
(K /4L
2
+ a sinh o//o/)
1/CL) 1/2
.
,
< R
2 ,
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS
is
317
It is interesting to note that considerable algebraic simplification attained if we compute the current directly. Thus we should have
I(t)=pG(p) >1
=
at
(l/Lco)e
&m
.
PROBLEMS Solve the equation
1.
(Lp2
+ Rp f
1/C)
 Q (t)
E cos mt
,
and obtain the solution
E cos (mt
=
a) f
F(t)
where a2 f b 2 R 2 m 2 _j_ [(l/C) tnsLp tan a #m/[(l/C) m2L] and F(t) is a function which damps to zero. This problem furnishes a simple example of forced vibration, since the system is forced to assume the same period as the impressed force. It will be further observed that if (1/C) m2L and if R is small, the amplitude of the vibration will be large. This is the phenomenon of resonance, which is found in the heavy rolling of ships, the vibration of bridges under marching troops, etc. ,
,
=
Solve the system
2.
^! (P)
^Qi+L
2
*Q 2 = Ei cos mt
(p)
,
and discuss the condition for resonance.
Employing the methods of the Heaviside
3.
system
:
30" y"
A particle
2y'hs/
+
%'
Zy
+ 6s" + 22"
3s' 2s'
calculus, solve the following
+ 4z = e** + z e2*
,
.
under the influence of a central force moving directed toward the origin and proportional to the distance of the particle from the origin. Show that the equations of motion are 4.
in the .r?/plane
is
x"= and hence deduce that the path 5.
We
is
an
a2 y
,
ellipse.
are given a set of data forming a time series
Vv
v
and from
o.2.r,
this set
we
derive the A?/
:
#2> y**
first
Ay 19
9
yn
and second difference
A// 2 A7/ 3 , ...
,
AT/^
series
,
If r12 is the correlation coefficient of the second derived series with the first, r 33 the coefficient of the second derived series with the original series, and r23 the
THE THEORY OF LINEAR OPERATORS
318
coefficient of the first derived series with the original, show that the equation connecting the three series is the following:
where
A 2 2/
,
c^,
^o
v2 and
<^ standard deviations respectively of ky, 3 are the anc^ ^o are ^ ne respective averages of the series. ,
1) * y and Replacing Ay by (eP for i/. Obtain the particular solution y 5
AB + C
X
in
i/&
log
(
which we designate the
by
AS?/
=
u
e
,
2

#, solve
y,
and
the equation
where
arctan
)
I)
(eP
&y and
(4AC
2AB
coefficients of the difference equation
by A, B, C
res
pectively. 6.
to 1914,
The Dow Jones when subjected
market averages by months from 1897 computations described in problem 5, yield the
industrial stock to the
following numerical values: o1 cr
(foTtoy)=
2
=
.1129,
r 13
=
3
(for?y)
r 23
.552,
,
2.358
,
= 15.150
,
(for Ay)
r J2
.512
.234.
Under the assumption that the averages of all the series are zero, show that the difference equation for the original series is the following :
1.847A2y
+
.OOOlAy
+ M4y =
.
Hence show that the period of y is 46 months. (Data furnished by the Cowles Commission for Research in Economics).
=
=
e? 1 and since & 2 Since the operator A (eP p2 + p f9 replace the difference equation of problem 6 by a differential equation of second order on the assumption that derivatives of order higher than 2 can be neglected. Determine the period of y from this equation. What criterion is suggested by this problem for the justification of the neglect of higher differ7.
%
.
.
.
,
I)2r=wp2+.,.
ences in a statistical problem of this type? If the initial values of x, x' and y, y' are respectively designated by # 8. and y Q y lt show that the value of x determined by the following system:
,
x 19
,
(p
2
4p)

 x (P+6) is
given by
4 2/i)
+
x
(1/4) e3
(1/12) e* (6a? (
2x,
f
3^
x
(p
+ (P*P) y
a^
2/o
1)
+
2/!>

y
> y
+ y,)
,
=
,
(l/3)e
2
'(
3x
+ 2^
?^

[E. J. Routh's Rigid Dynamics, vol. 2 (1892), art. 367;
Bromwich:
(1), p.
407]. 9. Using the notation of problem from the following system:
8,
show that the value of x determined
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS (p
(2p
~
is
M
2p) >
> x
1)
M
P >
y=
,
2/
,
2
f
%M
+ N e*
e< (M f t i 4given by # 2 t*) are determined by the equations and
where the values of
f
N
2
2M 2 = 2Af
[Routh: 10.
,
+
.r
+ A/
2
a
=
f
1
f
?/
+
x
//,
2ar 1
+
i/
Using the notation of problem
+
(^2
+ M,
2A/
,
+ + N=x xn
A/
,
a?
1
M M
x,
,
.
373; Bromwich: (1), pp. 407408].
Zoc. czf., art.
ing system:
a +
l)
(p is
2
319
+
2
8,
show that the
p) >
i/
2p)
(/;*
+P
a?
2
~>
2/
solution of the follow
= =
,
,
given by
aj=
provided 11.
we assume
+ or/3)
(ar
that
o.*
=
JT
(1/3) ^e"
f 2y/ t
I
,
[Bromwich:
.
(1), pp. 409410].
Prove that the system (pa
+
1)
>
+
u.
(;^
+
p+x +
(p
>
f 1)
y>
+
z/
*y
1)
=r
=
t
,
et
,
has the unique solution (without arbitrary constants) x
=
1
3ef
 t
,
y
,
13.
1
.
p. 145].
Solve the following difference system:
u(t+2)
~3 u(t+l)
M(f + l)
2v(f + 2)
f
2u(t) v(t
f
+3
+ l)
u(t
+ l)
2v(t)
=*
^
a
+1 *
,
2
.
Find the Heaviside solution of the equation (P
14.
2e'
London (1927),
[E. L. Ince: Ordinary Differential Equations, 12.
=:
7p
!
+ 6) >M() =1
.
Find the Heaviside solution of the equation (p
4
1)
> u(t)
5. Applications to Certain Partial Differential Equations of Mathematical Physics. One of the most striking applications of the Heaviside calculus is to be found in the solution of certain partial dif
ferential equations of the general type, A
WU
.
d
2
U
d
2
U
where A, B, and C are constants and the solution u(x, y) is subject boundary conditions suggested by physical or geometrical consider
to
ations.
THE THEORY OF LINEAR OPERATORS
320
The
method of procedure can best be
actual
illustrated
by cer
tain physical examples. a.
The Fourier Heat Problem
We shall begin by considering Fourier's problem of determining the steady state of temperature in a thin, rectangular plate of breadth L and of infinite length with faces impervious to heat. We shall suppose that the two long faces of the plate, AB and A'B', are kept at zero temperature and that the distribution of temperature along the short face, A A', is a given function of x, f(x). The temperature is also assumed to approach zero as we proceed indefinitely far from the base. A special case of this problem has already been solved by other means in section 8 of chapter 3.
Y B
uo
A'
L FIGURE 3
The problem,
then,
to determine a solution of the equation
is
^+ where u(x, y) u(0,y)
z=Q
is
,
lL
=
o
(5.2)
,
subject to the boundary conditions:
u(L,y)
=0
,
u(x,Q)
=f(x)
,
now regard
To begin with, let us able y, replacing d'2 /dx 2 by ential system,
2
z'

.
We
\imu(x,y)0
.
this as a problem in the varithen are led to consider the differ
+ z~u z=
,
= /(a:),n'(0) =0
.
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS
=
321
2 2 and referring to forEmploying* the abbreviation, d' /dyq' mula (2.13) of chapter 6, we see that this statement of the problem is formally equivalent to solving the algebraic equation, ,
+ z*)U = q f(x) n
(q*
(5.3)
from which the desired
solution, u(x,y), is to be attained by interpreting the operation U(q,x)  1 by means of equation (2.8) of chapter 6. We now come to the magical aspect of formula (5.3) which is to be regarded as a differential equation in x by replacing z'2 by its 2 Hence we turn to the solution of the difequivalent symbol 9Vd# .
ferential equation
(^ + tf)U = q*f(x) where
U (x)
is
(5.4)
,
subject to the boundary conditions: [7(0)
=U(L) =0
(5.5)
.
In order to solve this problem, we must have recourse to the theory of Green's functions of a single variable, a theory which is extensively developed in section 5 of chapter 11. We first observe that a set of fundamental solutions of the homogeneous equation,
=
is furnished by Ui(x) cos qx, U 2 (x) sin qx. From a linear combination of these solutions we must construct a continuous function which satisfies the boundary conditions (5.5). This is the desired Green's function. With reference to equation (5.4) of chapter 11, we then compute the Green's function to be,
G(x,s)
crrr
sin qx sin
q(L
G(x,s) t= sin qs sin q(L
s) / (q sin
qL)
x) / (q sin qL)
x^s x^s
,
,
,
.
we
designate the first of these functions by GI(X,S) and the second by G 2 (.r,s) the following limit is easily verified If
:
,
lim
[A G! (x,s)
In terms of G(x,s)
we
~^G
2
(x,s)
]
*= 1
express the solution of the system (5.4),
(5.5) as the integral
U(x)
=
G(x,s) qf(s)ds
,
THE THEORY OF LINEAR OPERATORS
322
L
=
G
{* Jo
U(x)z=
2
g
[q sin qs sin
I
f Gi(x,s)q J
(x,s)q f(s)ds
2
L)/sin qL]f(e)ds
q(x
^0 [q sin qx sin q(s
I
f
Jx
L)/sin qL~\f(s)ds
.
Finally, in order to attain the desired solution, we invoke the U(q,x) is to be regarded as an principle enunciated above. U(x) in the which and solution, u(x,y), of the original f)/3i/, operator q
=
=
problem
is
supplied from the operation,
= U(q,x)
u(x,y)
> 1
.
This expression
To
is interpreted as the expansion (2.8) of chapter arrive at the explicit development we first abbreviate,
F F
l
(q)
= sin qL/\_q sin qs sin q(x
L)]
,
2
(q)
=sin qL/[q
L)]
,
sin qx siuq(s
and note that the zeros, a n of each function are given by a n t= Hence we compute, ,
Fi'(a)
= F '(a 2
t=L /[nn 2
n)
We
sin(nns/L) sin(nnx/L)']
6.
nji/L.
.
next take account of the fourth boundary condition of our 0, which immediately excludes all problem, namely, lim u(x,y)
=
V='T)
terms of the form a > 0, from our summation. In order to avoid this obvious difficulty and to obtain a convergent series we replace y by y for all positive values of n and hence reach the following solution e ay
,
:
cx
f
(M*! (*)
J
where
1 ._
I
f^)ds\~
I
Jx
6 is positive or negative as n is positive or negative. obvious symmetry, this reduces to the familiar solution,
From
u(x,y) =
en
(2/L) 6.
n7r?//L J] e~
rr
2Vie
rL sin(unx/L) J sin(nji5/L)/(s)cfe ~ I
.
Problem of the Elastic String
It is illuminating to begin our discussion with a derivation of the equations which govern the vibrations of an elastic string. We shall assume that the position of equilibrium of the string is along the axis of and that the ends of the string are attached at the
points x
=
X
and x
= L.
Also
let
T be
the initial tension in the string,
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS
323
that is to say, T is the force necessary to hold a point P in position if the string were severed at that point. We shall assume further that the weight of the string is and that its modulus of elasticity is E that is to (Young's modulus), say, E measures the force necessary to increase the length of the string from L to L f E d L. This law asserting the linear increase in the tension with the extension of the string was enunciated by Robert Hooke (16351703) in 1676 and is assumed to hold within the elastic limits of the material.
W
FIGURE 4
We now let the string be deformed in such P with coordinates (# ,0,0) jroes into a point
point
a manner that the P' with coordinates
x ~\ , r], C, in which C, are functions of the variables x and t, t with their derivatives denoting time. We shall regard >/, f together D f D t >], D as small variations of the first order the higher powers of which can be neglected in comparison with lower powers. Let T p be the force necessary to accomplish the described deformation, that is to say, T will be the tension acting in each direction from the p ,
*]
,
,
,
,
point
P'.
Let us
Z
If
1,
now
//,
v
X SL p ,
T pt
into its
components
Xp Yp ,
,
,
>
are the direction cosines of the tangent to the curve at
we
the point P',
resolve the tension
have, t
= _ uT
*J Z p
J
,
To x we now give an increment dx and obtain the point Q. In the deformation just described the point Q is displaced to Q' with coD f dx, tj f D f dx, ; f ArC dx. ordinates x ~\ dx ff~ j]
Denoting the tension at Q' by into the components,
T Qf we ,
see that
it
can be resolved
THE THEORY OF LINEAR OPERATORS
324
Hence the force acting upon the elementary segment ds of the elastic
curve will be,
+Y Zp + Z
F,,*=Y p
F By Newton's
8
,
,
Q
,*=D,Y p ,dx *=
(r
Df Z
pt
dx
(5.6)
,
.
second law of motion these forces must be equal
;*...
to the
components of acceleration,
where A
M is the mass of ds, A M
tion of gravity.
(5.7)
W ds/gL, g denoting the accelera
=
We now observe that ds D x )dx, provided differentials (1 of a higher order than the first are neglected, and that the direction D x v\ v D T C cosines are respectively: A 1, // Hence by an application of Hooke's law and the neglect of terms involving differentials of higher order than the first, we get
=
Xp =
(T
,
+ED
f
)
+
=
Y p ,^
,
=
9
T DM
.
Z
pf
Substituting these values in equations (5.6),
=
TD t;. x
we
readily obtain, (5.8)
.
Since
we have
also,
stitute this value in
A
M
obtain by equating (5.7)
~~
'
~"^42
(W/gL)
(1
f
D
x
we may
)dx,
sub
and upon neglecting second order terms, and (5.8), the equations:
(5.7)
^
' ?L
<=
^49.
'
^^'.>.
~^41
O^2
'
\
i7
/
=
where we abbreviate: w~ TLg/W. ELg/W, v 2 The first of these equations defines the longitudinal vibration of the string and the other two the components of the transverse vibration. The longitudinal vibration can usually be neglected in comparison with the transverse motion and this assumption we shall make here.
In order to simplify the problem for the sake of the application shall limit our discussion to a deformation
which we contemplate, we
LINEAR SYSTEMS
Hence we
in a single plane.
THE HEAVISIDE CALCULUS
325
shall consider the solution of the single
equation,
= imposing upon the function u(x,t) the following boundary conditions
:
(1)
u(x,0)*=f(x)
(2)
(5.11)
,
(3) It will be readily observed that setting the constant v equal to unity imposes no essential restriction on the problem, since it may be absorbed in a transformation of the variable t. In order to solve this fundamental problem by the operational methods of the Heaviside calculus, we now proceed exactly as in the case of the flow of heat discussed in section (a).
Abbreviating d/dt by (3.10) and the tional equation
^ where
U must
P
2
U==
replace the differential equation (3), by the single opera
Wf(x)+pg(x)]
(5.12)
,
be determined so as to satisfy the conditions [7(0)
A
we
p,
boundary conditions (2) and
= U(L)
:
(5.13)
.
2 fundamental solutions of the equation, d'2 U/dx 2 p U *= *= sinh px. From these, cosh px, U 2 (x) 0, is given by Ui(x) with reference to equation (5.4) of chapter 11, we then compute the Green's function for the system (5.12) (5.13) to be,
set of
= sinh px sinh p(L G(x,s) = sinh ps sinh p(L
G(x,s)
In terms of G(x,$)
we
s) / (p shih
pL)
,
x) / (p sinh pL)
,
x^s x^s.
,
express the solution of the system as the
integral,
U(x) *=
G(x,s)[p*f(s) ^o
=
[sinh
I
px sinh p(L
#)/sinhpL] [pf(s)
+ 0(s)]ds
*^o
CL
+J
[sinh
I IE
px sinh p (L
x)/sinhpL][pf(s)}g(s)]ds.
THE THEORY OF LINEAR OPERATORS
326
example we now regard the function U(x) t= as an U(p,x) operator and obtain the desired function u(x,t) from an interpretation of the symbol,
As
in the previous
u(x,t) e= U(p,x)  1
.
This interpretation, as before, is made by means of the expansion (2.8) of chapter 6. To attain the explicit development we first abbreviate,
F F
l
2
= sinh pL/[p sinh ps sinh p(L (p) = sinh pL/ [p sinh px sinh p(L (p)
#)] x)~]
,
,
and note that the zeros, a n of each function are given by a n Hence we compute, ,
dp
= L/[sin(njis/L)sm(n7ix/L)~\
dp
The desired
solution
is
thus immediately expressed in the form

u(x,t)
g (s)
obvious symmetry this reduces to the explicit solution 00
= (2/L)
Y
l
,
Pfln
a*F'(an )
From
nni/L.
/
:
L /
sin(nnx/L) cos(nnt/L)
I
sin( n<ns/L)f(s)ds
*^o
n^ri
/*L
it/L)
si
I
^
PROBLEMS 1.
A
potential function, V(x,y), belonging to an electrostatic field satisfies
Laplace's equation .
So* 2
3
2 i/
Set up the operational equation for the potential of a plane, rectangular which is kept at zero potential on three sides and at a potential f(x) on
plate,
That
the fourth side.
2.
Show
V(x,y)
=
is,
assume
that the solution of problem 1 2
sinh{ra7r(&
is
given by
y)/a] sin(m'7rx/a)
C J
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS 3.
Solve the problem just proposed for f(x)
4.
Solve the above problem for f(x)
=
=
I.
x. rx .
5.
Solve the above problem
6.
Determine the value of V(x,y) when the plate
if
e
f(x)
in the direction of the yaxis, namely, if b 7.
If a string
boundary conditions
327
oo
is
of infinite dimensions
.
vibrating in a resisting medium, equation (5.10) and the are replaced by
is
(5.11)
do
u . t
2
3
t
3
u 2
u(Q,t)
D
n
= v*+ 2ku x oi
(x,Q)
~ u(L,t) :^0
=
,
,
.
Set up the operational equation for this problem. 8. If an elastic string of length L, fastened at each end, is displaced at its center through a distance k, show that its subsequent vibration is given by the
series GO
(8k/w 2 )
u(x,t)
1
V
sin
V2 n
9. Show that the actual motion of the elastic string of problem 8 consists of three straight pieces, the center piece always moving parallel to the axis of x. [For a diagram of the motion see Lord Rayleigh: Tlwory of Sound, London
(18941896), vol.
1, art.
146].
If the elastic string of problem 8 is displaced through a distance k at a 10. point (l/m)th the length of the string, show that the motion is given by the series 2
km 2
u(x,t) (ra
l)7r
V 2
^
1
n~
sin^TT/m) sin(n7r#/L) cos(nirt/L)
.
Applications in the Theory of Electrical Conduction. In the we discussed the classical problem of determining the vibrations of an elastic string which is subject to a given initial distortion. This problem is closely related to certain problems of electrical communication through a conducting cable which we shall now discuss. The situation with respect to conducting cables enjoys considerable simplification, however, due to the fact that the flow of electricity is set up by a known impressed voltage V(t) which is zero for values of the time prior to the time, conveniently chosen at t 0, 6.
last section
=
when
it is impressed upon the cable. This simplification of boundary conditions makes the Heaviside calculus an unusually effective tool for the development of this problem. We shall begin by considering the case of the flow of electricity in a noninductive cable with distributed resistance R and capacity
THE THEORY OF LINEAR OPERATORS
328
C x
per unit length subject to an impressed voltage
=
V
(t)
at the point
0.
The
differential equations of the cable are*
,
dx
~V^ 1
I
dx
dt
(6.1)
,
where x is the distance measured along the cable from a fixed point (x^= 0), /is the current at the point x, and V the corresponding potential.
Let us now replace 'b/'dt by p and through the elimination of / obtain the operational equation,
Then symbolically we have
V(x,t)=e+**V
=
where we abbreviate a
(p/vC)
l
5
+&>* V
(t)
2
and TM*) and
(t)
V
(6.2)
,
2
(t)
are arbitrary
functions.
=
Since I(x,t)
oX
V(x,t),
we
obtain from the differentia
tion of (6.2) the symbolic equation
I(x,
t)
~
=
Assuming that the wave is absent we may

cable set
V
is infinitely 2
^
&* >
> V,
[e
72 ]
.
long so that the reflected
and have as the symbolical
solu
tion of our problem r
= /"
where Vi(t)
is
Vi
(t) ==
V
(t)
from the boundary
to be determined
this asserts that
V(x
t)\ r ^
=V
(l
(t)
,
Hence the complete
.
in the expansion
f
it
is
conditions. Since
clear that
we may
solution of the problem
is
set
found
:
*For a discussion of the derivation and significance of these equations see H. Jeans: The Mathematical Theory of Electricity and Magnetism, Cambridge (1915) pp. 332335. See also H. Bateman: Partial Differential Equations of Mathematical Physics, Cambridge (1932), pp. 7376. J.
LINEAR SYSTEMS
THE HEAVISIDE CALCULUS
where for brevity we have written
x
=
If
V
0,
we
If
V
A
V/t C
Therefore
.
=
1, that is to say if a unit c.m.f. get the well known solution*
(t)
= sin<w*, that
(t)
pressed on the
circuit,
is to say, if
we have x
at x
C ^ d
C
=
is
329
we
get
impressed at
an alternating c.m.f
is
im
0,
r* sin
co
5
7
C* COSCOS
"
co
J
'
T^1^)"^
In establishing the convergence of the series representing 7(.r, t) in the general case for real values of t greater than 0, it will be sufficient to assume that (t) is analytic in the neighborhood of t t= 0.
V
Let us
first
consider I(x,
t)
as a function of
oc.
If
we
on
by a
aws
series
n
and make the transformation y
V
(s)
s),
we
replace
= s/(t
n=o
shall
have ^f
where
lV (8)/(t
is
8)^*8=
an analytic function without singularities at the origin.
Jeans: Loc.
cit.,
p. 334.
THE THEORY OF LINEAR OPERATORS
330
Hence for values of the inequality,
t
=
0,
and for n
sufficiently large,
k n nl
C
where &
a constant independent of n. proximation for nl, we obtain
A
.r
2
(6.3)
,
Making use
is
we have
of Stirling's ap
" 2
i
Similarly the nth root of the general term of the second series same limit and the convergence of I(x,t) regarded as a function of x is established. Regarding it as a function of t, we may use leads to the
on
Ax
2
" 2
?i!

feV(2n+2)
!
Wl
as a dominating series for the first series and on 
A
zn ~*
x
rc!
k n /(2n
1)
!
n^i
as a dominating series for the second. The case of a cable with distributed resistance R and capacity C with distributed leakage G per unit length leads similarly to the differential equations:
Ri^JL v dx
,
(C*+G)*V = Ll. If the cable is infinitely long
we
shall then
have
V ^=e 2!
4!
6!
=
where we have used the abbreviation Cp ) G. The problem is thus reduced to that of determining the meaning ?
to
be attached to the operations,
(Cp
+ G)* **V +
But from formula
Q
(t)
and (Cp
+ G)
n
>V
(t)
.
(12.9) of chapter 2, these expressions
interpreted as the following:
may be
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS
331
where
is either n\y% or n. In the former case the interpretation of // the operation is finally effected by means of the formulas of fractional
differentiation.
Hence employing the abbreviation
A
=
G/C, We obtain the ex
pansion,
(6.4) ^
t
P~ X * e
V
(<;}
~^d*.
f J
s
\/t
=
The solution for the case where unit e. m. f. is impressed at x can be obtained with considerable ease by this formula. We first recall that: ~
I
S)>
As A*
ds
=
___
(
rf^
ds
(
^VVAef^
,
,rfs
Substituting this value for the integral in (6.4) and taking the indicated derivatives, we get
V~
"2 ~ e X
+J
I
2
s)
(f
"
2
2 3
!
(
s)
t
(6.5) .
O
o
1
^' ^/CC/y
OOOt^t/'/ \v 4
Making
*
'
j
Li
*j
the transformation
H "i"
_ 
,,,(/,,t/
t
A/77 71 \f
.9
,
r~\i
J
t
jj.
~i
oQ^7/2 ^
.
*
J 2
we reduce
(6.5) to
(RC)
'
2!(45)
I.
45
3! (4s1 )
3
^

i
j
rrr
That this function, together with the value of I(x,t) derived from it by differentiation, forms a solution of the original differen
THE THEORY OF LINEAR OPERATORS
332
equations can be verified by direct substitution. condition is obviously satisfied.
tial
The boundary
A third problem of similar kind is that of determining the voltage at a cable terminal when an e.m.f., f(t), is impressed on a long cable of distributed resistance R and capacity C per unit length We have just seen that the cur* through a condenser of capacity C rent entering a cable, the terminal voltage of which is V, is given by (Cp/R) * V. The current flowing into the condenser will be V is the voltage across the condenCoP > [/(O V\ since f(t) .
ser.
Equating these two values and dividing by p lowing equation for the determination of V
we
obtain the
fol
:
The solution of this equation is easily found since it is identical with equation (6.17) of chapter 6. Properly specializing the solution of this equation we then obtain V(t)
= +
If f(t)
7(0
=1,
this reduces to
=
The general telegrapher's problem may be similarly treated, but the details will be left to the reader. If distributed inductance L is included, then the general equations become
(6.6)
of / from the system leads to the equation of E. T. Whittaker suggests might properly be called which telegraphy,
The elimination
Heaviside's equation:
.
(6.7)
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS If
we employ
the transformation I
= ie~
:
V=v
Ki/L ,
333
e~ Ri/L
,
both the system (6.6) and the equation (6.7) take simpler forms.
and
PROBLEMS If a conducting cable is of finite length L, then the second term of the light hand member of equation (6.2) cannot be neglected. Hence assuming that at x and at x 0, L, 0, show that 1.
V=E
V

sinh
V(x,t)=
E
If
is
a(L x) > aL
; sinh .
E
.
constant, derive the expansion
V(x,t)
= E(lx/L)
2. If for a conducting cable of minal conditions:
(1) attf
show that V(x,t)
is
0,
V=E
finite
length L,
(2) at x
;
=L
,
we have
the following ter
dV/dx
,
determined from the operational equation
.
cosh r/X If
7
is
a constant, derive the expansion
W< = 1
3.
If in problem 2 (1) at x =
show that
V (x,t)
is
we have ,
the conditions
V = E cos
co
/ ;
(2) at x
=L
,
dVAto
=
,
given by the expansion
cosh [V2 t
(#C)
m
3 si
^4 +
16
that the voltage, V, at a cable terminal when an e. m. a on long cable of distributive resistance R and capacity impressed length through a terminal resistance R n is given by the equation 4.
Show
,
f.,
C
/(),
is
per unit
THE THEORY OF LINEAR OPERATORS
334 Show
E
that the solution for f(t)
(a constant)
be put into either
may
of the following forms:
=E
V(t)
ER
= E [1
(]
(R *C/2Rt)
(C/RTTty>{l 2
exp (Rt/R
C)
]
Let an impedance 2^ be connected at the transmitting end, namely, with f., /(), of a cable of finite length L with distributive resistance R and capacity C, and let an impedance z 2 be connected at the receiving end. Employing the abbreviation a (RCp)*, show that the voltage in the cable is deter5.
an
m.
e.
=
mined from the following equation:
(R*+z^z2 &) sinhaL + ajRsinha(L
Show
6.
ditions of
that
problem
when 5
x) z2
j
=
(aR)
a2
0, z 2
~ E
and for f(t)
+
(z l
cosh
z.}
z2 )
a(L
jR
,
is
_
x) L, under the con
the current at # is
(a constant),
where the summation
cosh aL
given by
y2 sin2X n )
RL(\ n
over the roots of the equation
tanX
=
(R Q /RL)\
.
If the initial temperature of an infinite solid is given as a function, 1. /(#), of the distance from some fixed origin at an initial time t 0, and if the heat flows in one direction only, then the temperature at any subsequent time is
given by 3
T
32
T
where a is an experimentally determined constant and T(x, f(x). boundary condition: T(x, 0)
=
Show
t)
is
subject to the
that the solution is given by
T(x>
t)
=
'
f(x
I
+ 2a
Vt.s) e^ds
.
V T" J ^
Hint: Note that a formal solution (jr,
Now
is
given by
t
employ the integral
r oo e r2 r*n
dr
=V
Jo 8.
J<>
cools in air, the temperature, T (r, t), where r is the center and t is the time, is subject to the equation
If a sphere of radius
the distance from
r
2
R
32
(Tt)
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS
335
If the initial distribution of temperature is given as a function f(r) of the distance from the center of the sphere, and if the surface of the sphere is initially kept at a constant temperature T then the boundary conditions are: T(r,Q) ,
/(r), T(R,
Show
r o where R
t)
the radius of the sphere.
is
,
that
T(r,t)
=
r
on
2
2 ex P(
nta'mt/R*) sin(mrr/R)
2R V r
co
4
T
1 H
{
.
IT
(
s f (s)
sin(ntrs/R)ds
Jo
l)n
n
li
I
exp
n2
(
a*
^ t/R*)
sin (nirr/R)
}
.
9. If the sphere of the preceding problem is placed in air at a constant temperature of zero into which it may radiate, the boundary conditions become
r(r,0)=/(r)
a
d
where h
is
r
T(R,
,
+ hT(R
t)
t

,
a constant which depends upon the conductivity of the surface of the
sphere.
Derive the operational equivalent of this problem.
From the operator of the preceding problem, show that the solution 10. be expressed in the form T(r,
t)
=
(1/r)
V c n exp(
a*
V
t) sin \ n
r
may
,
n=l
where we abbreviate "
cn
^n"
=1 T>
/v
in
which
X n is
~"i
v "*"
*
!?> J_liJ?/M? A?/C (A?/v A Aj" /C* >
\
i
'
1\ 1)
I
I I
\ sf(x) sn /
^
a root of the equation A
R
cos A
R
f (ft/2
1) sin X fl
=
.
7. Infinite Systems of Eqiiations with Constant Coefficients. In the discussions which have preceded this section, attention was devoted mainly to the formal machinery of inverting systems of equations with constant coefficients. We shall now consider existence theorems for such systems in the general case where the differential
equations are of infinite order and the variables infinite in number. The principal paper on this subject is due to I. M. Sheffer [ See Bibliography: Sheffer (4)]. His results, however, are an almost immediate corollary of the CauchyBromwich method described in section 3, chapter 6 upon which one imposes properly specialized condi
from the existence theorem for from theorem 1, section 3, chapter 3. is, Let us assume a system of the form
infinite
tions derived
that
determinants,
00
" 00
" P
~* V
'
'
'
'
'
'
(7 .D
THE THEORY OF LINEAR OPERATORS
336
where we abbreviate GO
AH(P)= 2
o,/*)p*
(7.2)
.
fc^jo
We
make
shall
the following assumptions:
A
All the functions
(a)
R <
vergence:
\p
<
\
R'
tJ
(p) share a
common annulus
(b) The elements of the determinant J(p) the conditions of theorem 1, section 3, chapter 3 for the annulus of convergence.
A(t) has only a
(c)
The functions
(d)
where sup lim f
We

of con
.
finite

of zeros in
l}
(p)
satisfy
values of p in
R <
[
p
<
R'.
have the expansion
g, (t)
g im 1M rg r
number
A all
<
R'
.
= 00
consider
first
the homogeneous case of system (7.1), that
is,
the system
2 for which
Ai,(p)<;,(t)=0, 0'.=
oo,.,oo
)
,
(7.3)
we prove
Theorem
6.
If functions v>(t) are defined by the
Cauchy integral (7A)
where we
=
A
and Ai(t) is A(t) with the ith coll} (t) where Pi(t), P 2 (t), are arbi2 (0> replaced by Pi(t), <= and are so chosen that A (t) trary polynomials which vanish at t converges within the annulus of convergence, and if C is a path which lies within the annulus of convergence and includes the point t 1, where, A(l) *= 0, in its interior, then the functions v (t) furnish a define A(t)
\
P
umn
,
l
=
t
solution for (7.3)
Proof:
From
2
.
the operation
Aij (p) > Vt (t)
i=OQ
we
obtain formally
2 A i=oo
it
(
= _L Zjli
f \P,
J
(j
(t)
6*V<] dt
THE HEAVISIDE CALCULUS
LINEAR SYSTEMS since the right
hand member
is
337
zero from the fact that P,(t)/t has
no singularity on or within C.
The process of operation under the integral sign is justified by the corollary to theorem 1, section 3, chapter 3. Since the polynomials are bounded within the annulus of convergence, the expansion of A,(t) with respect to the ith column would converge uniformly within the
annulus and hence, in particular, upon the path C.
Considering prove
Theorem
where Ai(t)
7.
is
now
the nonhomogeneous equation (7.1),
we
shall
If the functions v,(t) are defined by
the determinant A(t) with the ith column replaced by
G,(*)= 2 ./"
,
))>
if C is a path in the annulus of convergence which Contains no zero of A(t) upon it and whose modulus exceeds r [see condition (d) above], then the functions v (t) furnish a solution for equation (7.1).
and
v
A
Proof: Operating upon (7.5) with respect to i, we get formally 00
1
2
Aij(p) ~> Vi(t) =2~~^
f
,
J<
tj
(p)
and summing with
^ A ()Ji() ^ tA(t) i;
dt
The justification for this formal summation under the sign of integration is derived immediately from the corollary to theorem 1, section 3, chapter 3. We first observe, from condition (d) above, that G(t)\ is dominated by the function 
M2r'VI t\***=M[l/(lr/\ WlO
f
I)],
t>r
.
Hence G(t) converges uniformly along C and thus by the corollary to theorem 1, section 3, chapter 3, the expansion of A(t) according to the ith column will converge uniformly along C. This is sufficient to justify the formal processes which we have employed. Corollary. less
than This
The
solutions obtained in theorem 7 will be of grade
R'.
may
be derived from theorem
6,
chapter
5.
CHAPTER
VIII
THE LAPLACE DIFFERENTIAL EQUATION OF Introduction.
1.
The object
INFINITE ORDER
of the present study
is
the Laplace
differential equation of infinite order uo
+a
2 (ano + ttni wO
nz x*
4
.....
+a
ntl
x*)u<
n
>
(x)
=f(x)
(1.1)
where p is a positive integer and not all the quantities a np are If we employ the abbreviations z *= d/dx and a n (x)
we can
=a
no f
+a
a tn x
\ a nii x p
2
n .,x'

zero.
(1.2)
,
write (1.1) in the abbreviated form, a.2
(x)z

2 \
(1.3)
.
The history and the present status of the theory of the solution of the Laplace differential equation of infinite order have already been summarized in section 6 of chapter 1. We shall attempt in this chapter to give the details of the formal solution of both the homogeneous and the nonhomogeneous cases of this equation and present in so far as possible existence theorems which apply to a broad class of functions. It will be observed immediately that the general theory of equation (1.1) formally unifies the theories of the following essentially different types of linear functional equations in which the functions
PL(X) are polynomials of degree not greater than p: r
u(x)
(a) in
+J
uo
QO
^p x
l
(x)q
l
(tx)n(t)dt^f(x)
>
1^1
which we assume that the
,
(x) approach infinity in such a
manner
Too
that
I
n (s)s ds exists for
all
values of n\
^0
(b)
u(x)+
(c)
p m (x)u(x+m) \p n (x)u(x)
~
(d)
The
=f(x)
;
Laplace differential equation of finite order.
The formal equivalence of these types with equation hibited by means of the Taylor's expansion,
u(T)<=u(x)
+
(Tx)u'(x)
+
338
(T
2
(1.1) is ex
THE LAPLACE DIFFERENTIAL EQUATION
we
If
T by
replace
form
of the
and substitute in
t
(ct)
we
339
obtain an equation
:
P*(x)u(x)
+ (1.4)
where the
coefficients are, /^O
Jo
>
n
we
Similarly for (b)
with the
let
=x
T
f ci
.
and obtain equation
(1.4)
coefficients,
P
(x)
=
s*b
m
22>.C) J
=i
r Equation
T by
ly
x
f r,
r
(c) is
=
transformed into the desired type by replacing TH. We thus obtain the coefficients,
0, 1, 2,
,
,
The Laplace equation of finite order is derived from (1.1) merefor all values of n greater than a fixed n\ by assuming that a nl
=
should be noted, incidentally, that the case for which p has already been extensively treated in chapter 6. It
Calculation of the Resolvent Generatrix for the Laplace Equashall first consider the formal aspects of the inversion of equation (1.1). It will be convenient to employ the abbreviation F(x,z), which we define as the operational series 2.
tion.
We
F(x,z)
=2(
Mo
+a
rtl
ff
+
a, J2 ff
2
n p H \a n y )z
.
,t
?(<)
It is clear from section 10 of chapter 4 that the formal solution of equation (1.1) depends upon the determination of a resolvent generatrix, X(x,z), which will satisfy the equation,
X(x,z) *F(x,z)
=1
,
or more explicitly,
F(x,z)X(x,z) H
1
(dF/dx) (dX/dz
n )
/n
!
+
.
=1
.
(2.1)
THE THEORY OF LINEAR OPERATORS
340
we make
If
the abbreviation, on
A
r
or
(#,)= n=0 2 ^a nm mlx m
'r
z n /(m
r) \r\
,
n> r
we can
A
write (2.1) in the form
+A
(x,z)X(x.z)
1
+A
(x,z)dX/?z
2
(x,z)d*X/?)z* l
In order to integrate (2.2) tion of the
form X(x,z)
with respect to d
n
z
X(x,z)/ox
we
r;
let
us assume that X(x,z) is a funcTaking successive derivatives
X(,?).
obtain,
=
n
(r
e
xs
(
\_X
"
nx
(z)
X
l)^Y^^( r)/2! l
When
(2.2)
.
(w  1)

(c)
(l)xX(z)'}
f
,
these values are substituted in (2.2) an interesting simand the generatrix equation reduces to
plification takes place
(A Q (z)X(z)
+ A>(z)X'(z) + A (z)X"(z) + .~ + A (z)X<(z)}e** = l 2
p
A
where the
We from
l
(z) are the functions,
shall
zero.
(2.3)
,
make
It will
the assumption at this point that a op is different be noted that this is a restrictive condition, which
be discussed in greater detail in section 15. The significance of found in the fact that the homogeneous equa
will
this restriction here is tion,
A
(z)X(z)
+A
1
+A (z)X(z) + + 4 (*)X<>>(*)=0
(*)X'(s)
2
...
p
will
have
exist
we
p
z
=
as a regular point provided a ()0
There
0.
=/
(2.4)
,
Will then
linearly independent solutions, regular at the origin,
shall designate
by X,
(z)
,
X
(z)
,
X. (z)
 
,
,
X
p
(z)
Similarly the adjoint of equation (2.4), which
which
.
we
shall call the
adjoint resolvent,
(2.5)
THE LAPLACE DIFFERENTIAL EQUATION have a
will
341
set of
p linearly independent solutions, Y^(z), Y^(z) regular at the origin. As is well known,* these solutions can be calculated in terms of X, (z) by means of the formula,
Y
3
(z),
,
t
Y p (z),
y,() where
W
is
= oiog WYax,'>)/A,,(z)
(2.6)
,
the Wronskian,
w x The
solution of equation (2.3) may be written in terms of the [Xi (z) }, {Yi (z) }, and we thus obtain as the formal value of the operator X(x,z) the function, sets,
f
X(x,z)=
\C p (x)X f (z)]
1
1
'
e xt W(z,t)dt
(2.7)
,
where we make use of the abbreviation,
)=Xi(z)Yi(t)+X*(z)Y*(t) The functions d(x) thus far enter the
H\Xp (z)Yp (t)
.
solution arbitrarily, but
we
shall see that they are uniquely defined. It is well
known
that
W
(z,t)
has the following properties :f
(2.8) 4=s
From
=( IVAMs)
these relations
it is
clear that
i
we
+ J^=P
shall have,
\
_
dz for r less than or equal to p
1.
If this last equation be expanded and the coefficients of x compared with the coefficients of x in equation (12.5) of chapter 4 we
obtain the following system of equations: "For this and other relations between the adjoint functions see G. Darboux: des surfaces. Paris (1889), part 2, book 4, pp. 99106. See also section 2, chapter 11 of this book. fSee Darboux: loc. cit., p. 103. See also section 2, chapter 11 of this book.
La
tJi&orie
THE THEORY OF LINEAR OPERATORS
342
the solution of (1.1) when /(#) *= a: 1 further specialize the functions X,(z) so as to satisfy the
where ^(#) If
we
is
.
conditions,
Xi<>(0)*=du where
d lj
=
d lj
1,
=
0, i
^
/,
(2.9)
,
the arbitrary functions are identified
with the special solutions u (x), namely C,(x) t= ^^(a;). may formulate these conclusions in the following theorem t
We
Theorem is
The resolvent generatrix X(x,z) of equation
1.
:
(1.1)
expansible in the form,
X(x,z)=
s
[%u^(x)X (z)+ Cer'WteMdf] *=i i
,
(2.10)
Jo
=
where the ui(x), i p 1, are solutions of equation (1.1) 0, 1, 2, when f(x) x% Xi(z) are solutions of equation (2.4) subject to the (} defining conditions, X '~^(Q) c== ^ t; where d l} ^= 1, a^ e= 0, i j, and W(z,t) is defined by (2.8). ,
t
We Ui(x).
^
,
proceed next to an explicit determination of the solutions result that we shall prove is contained in the following
The
theorem
:
Theorem ing functions
Let us designate by
2.
H
(i) }
(t)
and
Ik
(i)
(t)
the follow
:
(2.11)
t=
(
Jo
(2.12)
where the
Y
t
(t)
are the adjoints of Xi(t) as defined by (2.6) and
(2.9).
Then the solutions u from the formula,
t
(x) of theorem 1 are explicitly determined
(2.13)
343
THE LAPLACE DIFFERENTIAL EQUATION
where U(x) is a solution of the }Wmogeneous equation and I, is a path of integration between zero and any other point in the complex xt ,p, and /*<*> (t) e <= 0, i, i+1, plane for which J5T/> (t) e" t= 0, j
=
k
=
l, 2,
Proof: We first observe that our definition of the solutions X, (t) <= 0, condition (2.9) has uniquely determined the values of the <= z in equations adjoints Yi(t) at the same point. Thus setting t
at
t
,
we have
(2.8)
the bilinear relations,
X (z)Y (z) +X (2)Y,(H\X,(z)Y (z)<=Q X '(z)Y (z)+X '(z)Y (z) H(Xp'()rp (z)=0 1
Setting z
y,(0)
=
0, i
=
2
2
l
l
f
2
i
and noting the 1, 2,
,
y ((0)=0, 4
;
t
=
F,,(0)
we
= l,2
f
p
..,
F<>p. (0) tf
0,
i
=.
1,
2,
,
P2,
1,
q
=
,
values of X, (z) we obtain, 1/4 P (0). Similarly letting
get r/(0) <= and in general,
j <= 1 in equations (2.8)
0)
p1,
initial
,
(
1)VA,,(0)
.
(2.14)
From
these values and those obtained by successive differentiation of equations (2.8) we obtain the following numerical equations:
= (
l,t+l,,p ;H, *>(0)t=l
y^O,!,^.*
.
,
(2.15)
These equations can be used to compute the elements of the 1 2, we have Fmatrix, HIV' (0) For example, ifp
=
'
.
0)
y
g
'(o>
(2.16)
1/A 2 (0)
Assuming proper convergence differentiate Vi^x)
= ii^i(x) =
of the integral in (2.13), let us thus obtain, U(x), n times.
f J
We
e rt t"Yi(t)dt
.
THE THEORY OF LINEAR OPERATORS
344
Substituting this function in the expression F(x,z) > u(x)
we
get,
F(x,z) *vi.i(x)
(2.17)
^Y
f
i
(t){A Q (t)+A 1 (t)x]\A p (t)x'}dt
.
Jii
Let us
now
define a
new
function,
t
U*(A)=
( Jo
Y
i
(8)A(s)(trs)**ds/(kl)l
.
Recalling the identity [see equation (6.4), chapter 2], f'...
fV(*)
Jo
we note
n
=
t
( Jo
that
Making use
we
es
definition of Ik (t),
we
of this relationship and integrating by parts,
tablish the identity
f Y (t)A(t)e"dt l
l
Employing
={U
t
1
this identity
(A)xU
a
(A)
and recalling the
obtain the following expansion
:
f
(2.18)
A
different integration by parts results in the following identity
JJ
Making use
of this expansion
t
^'(YiA)^^
and recalling the
of HJ W (t) we obtain the following:
.
:
(2.19)
definition (2.11)
THE LAPLACE DIFFERENTIAL EQUATION
345
(2.20)
h x**Hp
H
Combining expansions
<={2
We
(0
<
(2.18)
+ ^ r"#, V> (t)dt a'
} e?*\
7
and (2.20) we get the equation:
D^/^^^ + i^/^HO^
(
must next
.
1
(2.21)
}^'!^
establish the identity, f
2)
!
.
Jo (2.22)
To prove
this consider the integral,
r
f
\
H
{l)
(s)ds
=
Jo
From is
H^ (0)
equation (2.15) we see that i 2. Similarly we have
=
established for
?=
and the identity
f
J
From is
this integration
established for
an
i
^
the identity and the observation that 7/ 2 (3) 3. The extension to the general case follows in
manner. Returning now to equation (2.21) and recalling (2.22) we see that the integral of the right hand member vanishes identically since for every #o (i) (s) is identically zero. Furthermore we have A (0) (i) k and from (2.15) #, (i) (0) Hence i. 0, j making 1, #, (0) to the assumptions of theorem 2 that there exists a path ^ from identical
=
some point lim Hj
(i)
ta
a such that lim
(t) e
rt
=
0,
j =
t
7/;
(i)
n
i,
i
+
^
=
1,
(t)e
rt
0, ,
p
,
^
k ^=
then
it is
1, 2,
,
i
1,
clear that Uti (x)
by (2.13) is a solution of equation (1.1) in which f(x) ^= x*The negative sign is chosen for the integral of (2.13) since we have arbitrarily assumed that zero is the lower limit of the integradefined 1
.
tion.
Corollary: If the conditions of theorem 1 are satisfied and if there exists a unique limit a for all the paths ? satisfying the condi4
THE THEORY OF LINEAR OPERATORS
346 tions of
theorem
2,
then the resolvent generatrix of equation (1.1)
takes the form,
X(x,z)
= el7(aO %X
t
(s)
i^i
+
fiX
<'
Ja
l
(z)Y
l
(t)e*
.
il
(2.23) If,
moreover, a
=
oo,
second term in (2.23) can be written
J0) \7 ( r
ivhere
we
V V
\ (V^iv
f\f>~ rl+ O xt W/t^ (/ y
(V 9<1\ yij.^JT:^
,
abbreviate as before
W(z,t)
=X
1
(z)JM*)
+X
9
(z)Y a (t) H\X,(z)Yp (t)
.
3. The Homogeneous Equation. In theorem 2 of the preceding section the functions u,(x) which satisfy equation (1.1) for f(x) x l were determined to within a solution of the homogeneous equation,
=
F(x,z)
={A
(z)
+A,(z)x
+ A,(z)x* Q
.
(3.1)
It is the purpose of this section to show the efficacy of the Laplace transformation in the solution of this equation. The present status of the problem may be briefly summarized as follows. The classical case where all the A (z) are polynomials was first systemt
in 1885 and 1886.* Numerous memoirs have since contributed to the extension of these original ideas.f The case where the A,(z) are polynomials in ez is that of difference equations and this has been the subject of extensive study by H. Galbrun, J. Horn, R. D. Carmichael, G. Birkhoff, N. E. Norlund, and others.J atically treated
by H. Poincare
*Sur les equations lineaires aux differentielles ordinaires et aux differences American Journal of Matiwmatics, vol. 7 (1885), pp. 203258; Sur les integrales irregulieres des equations lineaires. Ada Mathemaiica, vol. 8 (1886),
finies.
pp. 295344. fSee L.
Schlesinger: Handbuch der Theori? der linearen Differentialgleichungen, vol. 1 (1895), pp. 409414; E. L. Ince: Ordinary Differential Equations, London, (1927), chapter 18. JGalbrun: Sur la representation des solutions d'une equation lineaire aux differences finies pour les gran des valeurs de la variable. Ada Matheruatica, vol. 36 (1913), pp. 116; Compfest Rendus, vol. 148 (1909), p. 905; vol. 149 (1909), p. 1046; vol. 150 (1910), p. 206; vol. 151 (1910), p. 1114. Horn: Integration linearer Differential gleichungen durch Laplacesche Integrale und Fakultatenreihen. Jahresberickt der Dentschen Math.Vereinigung, vol. 24 (1915), pp. 309329; Laplacesche Integrale als Lbsungen von Funktionalgleichungen. Journal fur Mathematik, vol. 146 (1916), pp. 95115. Carmichael Linear Difference Equations and their Analytic Solutions. Trans. Amer. Math. Soc., vol. 12 (1911), pp. 99134. Trans. Amer. Birkhioff: General Theory of Linear Difference Equations. Math Soc., vol. 12 (1911), pp. 243284. Norlund: See Bibliography. :
THE LAPLACE DIFFERENTIAL EQUATION
347
A
comprehensive discussion of this problem together with an incluis given in the well known treatise of N. E. Norlund: Vorlesungen uber Differenzcnrcchnung, in particular chapter sive bibliography 11.
The first example knowln to the writer from the point of view of the present chapter was given in 1908 by T. Lalesco [see Bibliography: Lalesco (1)], who solved the special case (e z ,r) > u(x) =0, and the domain of solutions was further extended by E. Hilb in 1920 [see Bibliography: Hilb (1)]. 0. Perron (see Bibliography), employing results in the theory of equations of the form oo
2 n

(
o
+ &<)
#*!+
=
m = 0,
'
1, 2,
gave an existence theorem for solutions of equation



(3.1)
which
satis
fy the restrictive condition that they shall be of finite grade. Returning to the problem of this section let us first observe that
B(z)F(x.z) *u(x)
= {F(x,z)
^B(z)} >u(x)
.
This follows immediately from an application of (3.1) of chapter 4 since
we have F(x,z) >B(z)
= [FB]Cr,z) =F(x,z)B(z)
.
We thus conclude that if B (z) is a factor common to all the A (z), then the set of solutions of equation (3.1) may contain as a subset l
the solutions of
=0
B(z) *u(x) If
we
(3.2)
.
designate by v (x) a solution of the reduced equation,
F(x,z) >u(x) ^=0
,
clear that corresponding solutions of the unreduced equation are immediately obtained from
it is
=v(x)
B(z) >u(x)
(3.3)
.
This nonhomogeneous equation with constant coefficients has been discussed in chapter 6.* Proceeding to more general considerations we now write,
u(x)
=f
L
e"Y(t)dt
(3.4)
,
*Let us note that the general solution of (3.3) which contains also the solutions of (3.2) in its complementary function may not always furnish a solution of the unreduced equation. Thus the equation )x
may
+
(z
I)
2
* 2 } *u(x)
+ (z*l)x
I2sx 2
But the unrestricted solution of (z 1) which satisfies the reduced equation,
tion
2
be written >
u(x)
=
2
}
> u(x)
v(x), where v(x)
is
a func
will not satisfy the original equation.
THE THEORY OF LINEAR OPERATORS
348
where L is a path in the complex plane, and substitute inF(x,z) >u(x). We then get
F(x
9
z)
this function
>U) = JX'Y(0{Ao(0 + At(t)x
Employing integration by the definition
oH
(t),
F(x,z) ^tt(ar)
(2.11)
={!(*)
,
parts, formula (2.20), and recalling easily reduce this expression to
we
+ xH
2
\
(t)
x^H,(t)}
\
e
xt \
L
(3.5)
If
we now
define Y(t) to be a solution of the equation
H
(t)^0
(3.6)
,
and choose L to be a path in the complex plane at the extremities of which we have simultaneously,
(3.7)
then u(x) as defined by (3.4) is a formal solution of (3.1) If we note the following identity:
= X(z)
>/(#)
1^0
= const.,
form than possible to give the solution u(x) a more symmetric set of 2 (z), that of (3.4). Let X^(z), p (z) be a fundamental , Y p (t) solutions of the resolvent equation (2.4) and Y l (t), Y 2 (t), the corresponding adjoints. Let L be any path in the complex plane it is
X
,
X
for which equations (3.7) hold. We may then write the solution of (3.1) in the form,
(x)z=e**
fL e"W(z,t)dt*f(x)
,
(3.8)
X l (z)Y l (t) + X p (z)Yp (t), X*(z)Y 2 (t) where W(z,t) hand memthe of existence the hight to within is and f(x) arbitrary
+
+
ber.
The case of constant
coefficients is
immediately included in this
formulization by writing,
W(z,t)=F(z)/F(t) where we use the notation of section
2,
,
chapter
6.
THE LAPLACE DIFFERENTIAL EQUATION 4.
Some
Particular Examples of the Solution of
349
Homogeneous
Equations. Before proceeding to more general considerations, it will be useful to examine some particular examples of the application of the theory set forth in the preceding section. It is a problem of much difficulty to formulate general theorems, since these theorems depend in
an
essential
manner upon the singularities of Y(t) and these sindepend upon the singularities of the adjoint resol
gularities in turn
vent generatrix equation which defines the function Y(t). The examples which are treated below illustrate many of the difficulties in the situation and will serve to introduce the more general discussion.
Example {(I
1.
Let us consider the equation,
+ Ax) + z/a\z~/a
2
3
\z /a*{
}
^0
*u(x)
.
(4.1)
The
adjoint resolvent
at once seen to be,
is
aY(t)/(a which has the particular
AY'(t)
t)
solution,
=
Y(t)
= (t
,
a)
x ,
X
Hi (t) =AY(t).
also find
Let us xt
first
assume that
zero provided H^(t)e closed path around the point L is
I is
L
In this case the solution
a.
=S
L
e
=
n.
We
Then
a closed path and, in particular, a
is
\
u(x)
a negative integer, i
a/A.
is,
n
xt
(t^a) dt
It is immediately obvious that when n c= 1, this function renders the original equation divergent. A similar conclusion is reached for all other values of n. We shall now assume that /. is positive, but we shall not restrict it to integral values. Also assuming that the real part of x is greater xt than zero we see that L is zero for the Laurent circuit of l (t)e
H
figure
1.
Hence we
\
obtain,
FIGURE U(X)
=
(1
C 2irU) e ox
1
/^D SX 6
I
Jo
It is proper to inquire in what sense this function of the equation since it is obviously of infinite grade.
is
a solution
THE THEORY OF LINEAR OPERATORS
350
We observe first the
operational identity, [see (12.9), chapter 2], a*  X~ m d n X' m
=
[F (m)
^ + n(n
l
Employing Borel summability we replace the factorials in the numerator by the integral /oo n
t e~
t
dt
= r(n\l)
.
Jo
The
identity then becomes, z n > e a *x~ m
= e*x
f
V^
1
(a
t/x)
n
dt/r(m)
.
Jo
Hence
if
*e af x m
=
a power series in either z or 1/2,
e?*x m
f Jo
e f t m
^(a
we may
t/x)dt/T(m)
,
write,
(4.2)
provided the integral exists. Applying the operation defined by (4.2) to our solution u(x)
we
get,
r Jo If
Jl
>
0,
the integral converges and the right hand
member
of
may
be
the equation becomes,
Since the application of the operator defined in (4.1)
open to some objection in the present instance, it is worth while justifying the solution on other grounds. z / a + (^AO 2 f 1 is equivalent Since the operator B (z) to 1/(1 z/a) and since this function has the Laurent expansion
= +
2 ) we may consider the relationship of B^(z) i= (a/z f a/z fthe latter to the differential operator. But we know from theorem 1, chapter 6, that (B^z) B(z)} *f(x), where f(x) is arbitrary to within the limits of the existence of the operator, is a solution of the equation B (z) =0. It may be proved without difficulty that this solution is identically zero provided the integrals of B l (z) are taken over a proper path to infinity. In the present instance this path may be oo to x. The essential equivalence of B (z) and B(z) chosen from is then assured. l
THE LAPLACE DIFFERENTIAL EQUATION Operating with l/z
n
B
v
351
upon u(x) and noting that
(z)
*u(x)<=
{*
(xt)*u(t)/('nrl)ldt
,
Jf/D
we
get
+ a(x
C* [l
)a
J 'JO 2
/2
+
!
}
T
e ffr
(e
^ x
at
1
t)
/t
^
JGO
= Example
2.
ea
(1
' ri;L
)r(A)e' 'A I
2
we
=
}*w(a)
=0
(4.3)
)=<(*
l
2
found to be,
is
Y (0=l
1),
2
<
2
terms of which w e compute the four functions, r
where we have adopted the notation of In order to determine a point, t c= 0, y t= 1, 2, we consider 7(0 tion is substituted in the equations
=
(2.11). a,
for which
^^ (0 + H
}
(t)
we have
a 2 Y*(t).
=
0,
we
When
H
e= } (a) this func
obtain the system,
A
is
.
becomes,
0,
fundamental set of solutions
Y in
.
have,
the equation, #<>()
A
>
A
,
Let us examine the equation,
a;
Since
x
necessary and sufficient condition for the existence of l) that the determinant, #/ (0 shall vanish, 

:0
.
a, l9
THE THEORY OF LINEAR OPERATORS
352
In the present case this gives, (1
*
2
2
)
,
1. Corresponding to these which yields the values t E= 1 and t a o, and values we compute a x x 2 a^ respectively, and hence reof solutions the fundamental set original by the following place
=
=
:

= (HD i) = 1 and = for which Hi(t) and H equal zero when = = lim e y when R(x) spectively. Since lim e y^(t) i()
1) (t
(*
2
2
1),
2/2
2
(0
(*
*
(t)
t
1,
xt
xt
2
t= uO
we
,
tuQ
(t)
re
>
0,
obtain as solutions of (4.3) the two functions, 1
f
2
!
(
/ uO
(4.4)
It is to be observed that neither of these solutions is a function of finite grade. Hence it will be instructive to see in what sense these functions may be said to furnish a solution of the differential equation of infinite order since obviously a direct substitution leads to a
divergent series.
Applying the operational identity defined by (4.3)
(4.2)
to equation
we have,
/oo e t t z x*{2/[l(lt/x)*']
= (4072!)
Jo f 4z(l
fV**
(6c73!)
3
t/x)/\l
ar*{2/[l
(I
x
t/xy\
2
}
dt
x"}
dt
dt
=
(1
Jo
(lt/x)
=
r(*6*7 Jo
4
)
{2x
2
+ 4(xt)x* f x(t
2
2 '}
2te)
}
.
We thus see that Ui(x) is a solution of equation (4.3) in the sense that if it be substituted in the left hand member of the equation, a divergent series is obtained which is summable to zero by the method of Borel. A similar interpretation applies to the statement that u 2 (x) is a solution of equation (4.3).
THE LAPLACE DIFFERENTIAL EQUATION Example
Let us
3.
now
353
consider the following equation
:
3)
(z
(4.5)
We
derive a particular value of Y(t) to be,
=
2Y(t). corresponding to which we have the function, H^t) For the purpose of discussing this equation we introduce a circuit which was first studied by L. Pochhammer and which may be described as follows:*
*
designate by L a a circuit in a positive direction around a ~l L and by a a circuit in the opposite sense, then it will be clear that if z traces the path L 6 s= L a L b L a ~^L b 1 a function of the form Y(t) x b)' 7/()e^ will return to its initial value. (t d) (t The integral of this function, if / and /< are fractions greater than 1, will have the value, If
we
=
l
u(x)
=
e 2 "^') (1
(1
e
(4.6)
FIGURE 2
The function u(x) can be given a
useful formulization in the fol
lowing manner Let us write, :
rb l
e~" >)
w)(e*^
u(x)
(at
Jo,
= _ 4 sin ^M sin Noting the abbreviation,
JI
f (art)*(bt)ty(t)dt
identity, sin na
K=
4^ u/F(l j
= n/[r(a)F(la)], ,a)r(l
A),
u(x)
.
and using the
may
be written,
*See L. Pochhammer: Uber ein Integral mit doppclten Umlauf. Annalen, vol. 35 (1890), pp. 470494
tische
M at he ma
THE THEORY OF LINEAR OPERATORS
354
^(a)/r (/Hl)}
.
(4.7)
Returning now to the problem, it is immediately seen that the following three particular solutions of (4.5) can be written down by properly specializing (4.6). Hence three particular solutions of the equation are, 2)(t
2(ff)
Uz(x)
=J =
(tl) (t2) \
1) (*
f ''[(*
2)
But these
solutions are obviously not independent since f i6^(^). That they do in fact furnish two independent solutions of the equation is proved by direct substitution and observaaT ax > e e
= ^(x)
l/(tl)
f
+ !/(*
2)
+ l/(t
3)
*^ i
One integration by parts
of the last term yields,
2
f
{
1/(_1)
+ i/(
t
2)
+ l/(t
3)
(3^
2
12*
Ji
tl) which
A
(t
2) (t
3)}[(<
2) (<
1) (*
is identically zero.
third independent solution of the equation
is
given by the in
tegral,
Joo
Example
4
Let us consider the following difference equation
(x+2) (x+l)u(x)
=0
.
(4.8)
:
THE LAPLACE DIFFERENTIAL EQUATION Employing the symbolic identity write this equation in the form, (x
2
2z
+ 1)
2e z
(e'
4^
f x (e
The adjoint generatrix
e"
y
>
= u(x{a,), we may
u(x)
+ 2} >
3)
j
355
=
(x)
.
then derived to be,
is
with solutions,
l)*,Y z (t) =e
I)
2 .
Particular solutions of the original equation are immediately obtained from the following integrals :
KI(X) 2*3(0;)
JL
=
{e
j; {e<*+i>
where the path of integration
n=0,
1,2,.. These solutions are
(M >V(e
f
I)
y(e'_ I)
3
}
d
,
}
dt
,
2
about any of the points
is
2nni
t
,
.
u^(x) =jiie
shown n * ix
(x
2
to be, znif **x x) ,u2 (x) =27iie
.
Since linear combinations of these functions are solutions and since sums with respect to n are solutions, we readily derive the fol
lowing more general solutions:
=
where n(x)
We
is any function of unit period, n(x{l) n(x). can readily show that {f/ (x), U 2 (x)} forms a complete set t
of solutions as follows
:
Hl
(t)
= YI() + AY =_ '{(2A l)e* + 2(l
H
(t)
^e^^ + l
Let us write, Y(t)
If
=1
2
(a>)
^
0.
2, it is
and hence compute,
(^),
C
R(x) exceeds
We now
0.
=H
2
2
A)}/
1)
l}/(e<
clear that lim tjr,
=
a in the seek a point t If we designate e^ by m,
=
+
H^(t)e
=

.
xt
,
=
lim
H
2
(t)e
xt
t^jrj
plane such that H^(a)
find that
the equations: ra 1 l)w 2(1 I/A, (2^ from these equations, we obtain (A 1)/A Since the oo and a 2nni, n <= 0, 1, 2,
m
2
.
finite
we
I)
(et
=
A) 0,
first
=
?/^
must
satisfy
Eliminating which yields a point is not in the 0.
=
plane and the other points have been used in obtaining U(x) and U?(x), no path L other than those previously employed can be found to yield a third solution of the equation. Hence {U (x) U 2 (x) } forms a complete set of solutions.
finite
l
,
THE THEORY OF LINEAR OPERATORS
356
Example 5. Let us now consider the following three equations, the last one of which (C) we have previously discussed in example 2. (A)
2
{(30 2
6)
+
+
I2z) x
(I2z*
(z
2 2 1) # }
2
=
u(x)
,
4z
(B)
{(62
(C)
{2/(l
+ 4rrz/(l
2
)
s2 )
* 2 } *u(x)
=
.
It will be found upon examination that these equations share same adjoint generatrix,
two independent solutions of which are 1
Y
l
=
t(t
2
Y
and
1)
the
2
=
t*.
Considering equation (A)
we compute
the values of
#\ 0)
(t)
and
thus obtain the following expressions:
=
1. both of which obviously reduce to zero for t Hence differential equation (A) has four solutions, two of which are given by the integrals,
Ul (x)
=
\P
(tl)c*'dt
1)
e
= (_4 A + 8 A 2
r
+
3
(4
6
A + 8A + 6A
A
2
4
)
3
+
(4A
d
4 )
^*
+ 6A
4 )
.
&*
The other two solutions are obtained by integrals to infinity and are easily found to be, u a (x)==
2
p
(t
f
V
l)(t
l)6 dt Jff
= e*(4A'
6A
1
4
)
>
Jco
4
(ar)
=
2
1)
(*+l)c"d*
=
e*(4/*
s
+ 6/a;
4
)
.
J00
The distinguishing difference between the two sets of solutions u2 (x)} and {u 3 (x), u 4 (x)} is found in the fact that the for
(Ui(x),
of finite grade while the latter have poles at the origin. (j) Proceeding to equation (B) we obtain from the values of Hi (t)
mer are
the expressions,
THE LAPLACE DIFFERENTIAL EQUATION
2
1)
(M
a2 )
Upon explicitly substituting: the values t we see that they reduce to zero provided d
sions
Y
=
357
.
1 in these expresx
=
cu,
that
is
2
1). Hence equation (B) has one nonsingular 1) (t (t the and two singular solutions u^ (x) and u (x) ?^ tion, (x) In the same way we find for equation (C) the expressions,
for
solu
.
,
=
1)
(t*
cu)
(
~
which cannot be simultaneously reduced to zero by the values t 1. Hence equation (C) has no nonsingular solutions and only the two singular solutions u3 (x) and n(x). We may note that the essential difference between these three equations is to be found in the rank of the following matrix:
Y(a), 1
A
(a),
A A
s
(a)Y 2 '(a)
2
(a)Y 2
=
(a)
=
One may easily verify that for a 1, ft 1, the rank of A is zero for equation (A), one for equation (B), and two for equation (C). It may also be shown by explicit substitution that the function ii 1 (x) is a solution of the first two equations, but is in no sense a solution of the third.
PROBLEMS 1.
Solve the equation
2.
Show
that the following equation:
(_4 X 2 __ 2x
f 6) it(x
+ 2) +
(8x
2
f
12x
14)
=0 has as special solutions two quadratic polynomials. 8.
Show
that the equation ii(x)
has as
its
=^xu(x
general solution the function
u(x) =ir(x/r) r*/T(l
where
r)
TT(Z) is a function of unit period.
+ x/r)
(Pennell).
THE THEORY OF LINEAR OPERATORS
358 4.
Show
that
= r(#)
u^x)
~ l/r(#)
and u 2 ( x )
are particular solutions
of the equation jc
u(x + 2)
1)
+
2 (a; 2
+
(#f2)
x
+ 1)
it (05)
+
2
(a
u(x+l)
1)
(Wallenberg and Guldberg).
.
Solve the following equations: 5.
6.
2x
(3o;2
2
+
5
(a*
7.
o;
8.
Show
(15x
1) tt(o?t2)
4)
w(a;+2)
(4^
tt(ttf2)
+
2
+
2
= m*
that u^(x)
(So:
,
2
+ l) + (12x* + 16x) + 8) w( + l) + (2x* 6x)
4z
2
IBx
u(x + l)
1)
^2 (a*)
=
2a] a
+
aa; 2
4) tt(s
4
(3cc
+
bx
2
+
+
6x
c
u(x)
~0
u(x)
~0
f 3) w(oj)
^=0
.
.
.
are particular solutions
of the equation:
[b(m 1) f
{(m 2
x2
1)
+
m) a ^2,
[6(m
__
[(
2
m
c(w& 4a]
l)
2
m )&
+ (m 9.
a^
^. (^ 2
a
1)
+
m
2
6}it
4a
2
(m l)c 4m) a] x
2m) 6
+
(m
(m2
f
2
4m) a
m)c}?^(cc)
=rO
.
If in the equation
A(x) u(x+2) +B(x) u(x + l) +C(x) u(x)
=0
the functions A(x), B(x), C(x) are quadratic polynomials, determine their coa x'2 f bx c and u 2 (x) ^:ax 2 + efficients so that MI(X) 'Y are par
=
px +
+
ticular solutions 10.
Express the integral equation
+ 3t(t
x)
(t
a?)
J*[t j in the
form
From
this obtain the general solution
u(x)
=
c
xx \ ,
=
V3
.
5. The Homogeneous Equation Case of Degree One (p=l) with Polynomial Exponent. Proceeding now to more general considerations it will be convenient first to discuss the case where the gen
eratrix is an equation of first order. If we let p 1, then equations (2.4) and (2.5) become,
THE LAPLACE DIFFERENTIAL EQUATION with the explicit solutions
359
:
Moreover we have H,(z)
= A,(z)Y(z) = ^
If
now we assume
A
(z) /A,, (z)
e
we may
H
l
(z)e
xs
.
the following expansion
= d + a,z f a
:
\ a n z*

2 z~ \
zZ*)
then
{A (z)/Al( ^ }d *
i

+ a /(zZ i3
i
3
)
write H^(z)e xz in the form,
^=e P (s) e^^(z t
Zi)
a
(z
z 2 ) a "(z
z,)
ai3

(z
where we employ the abbreviations, cp
(z)
= (a + x z + )
2 [eWd
2
fti
i)
(
+ ou^ /3 ^O^ + a^/Cl 3
/2
\
,
1
i) (2
2
)*
For convenience we shall refer to the function cp(z) as /ie exponent of Hi(z) and shall limit ourselves in this section to the case for n > s. where
=
^
,
a]}
Let us set cos{ (s[l)^+ a } "^
0,
from which we
get,
For values of d such that d 2w _! < # < i) 2n we see that the real +1 part of (W /(s+l) will be negative. Hence if we choose a path to ,
infinity lying within one of these intervals, it is clear that H^(z) will xt approach zero. Evaluating the integral Jfe Y(t)dt over a path which z m and which approaches does not enclose any of the points z l9 z 2 , intervals shall have a formal integral we in two of the above infinity of the equation. In this manner we obtain s f 1 independent solu
,
tions of the equation. If
z,
Zi,
,
z m are simple zeros, that is to say if
ly
a n a i2 a lm are fractions, then we can obtain , tions of the equation by means of Pochhammer integrals.
and
if
,
,
= m
0, i
>
1,
1 solu
In case any
THE THEORY OF LINEAR OPERATORS
360 value of a l}
hammer
is
a negative integer this factor is excluded from the Pochand a Cauchy integral is substituted. If all the a.,;
circuits
are negative integers no infinite branch is necessary to obtain the totality of solutions. If any two a kj are positive integers, for example n, a 12 then modified Pochhammer circuits, ,
3 "Y(z)^ dz
,
z
can be employed. If only one a tj is a positive integer then the path infinite branch terminating in z In considering y>(z) let us assume that there exists an integer z by r e ei Si such that a 8 i 0, a wl 0, n > Si. If we represent z
must be taken with one
}
=
^
and a
/(I
By means
R
by
Si)
of an
e
ai ,
it is
clear that
we
shall have,
argument which does not
we
viously employed
.
from the one pre
differ
find that the real part of
<
will be negative for values of $ within the sectors: # 2 wi #2W where we define $ n to be, n
=[(2n
1)*/2
SO,
a]/(l
tt
= l,2,. .,25!
2
.
Hence if a path be chosen which does not include any of the z m and which consists of a loop emerging from z^ in one values 2 2 , of the sectors defined above and returning to z in another of the sectors, then the integral taken along this path furnishes a formal solu1 intion of the equation. It is thus clear that we shall obtain Si z m in similar fashion tegrals. Considering the other values z 2 z 3 ,
L
,
,
+ S + + S m integrals Combining these solutions with those previously obtained we thus  S m formal integrals. obtain a total of s + Si + S
we
thus obtain
S
2
x
2 ]
An
example
{l/(Z
I)
2
is
.
OT
\
supplied by the equation
:
+[1/2(3+1)
where the operators are written for convenience in their closed form, but are to be regarded as equivalent to their power series expansion within the unit
circle.
It follows readily that
AJAi=
we have
:
(z+1) (2+2) (2+3) / (2
I) z
2
I)
2 ,
THE LAPLACE DIFFERENTIAL EQUATION
361
from which we compute, H! (z) ex*
=
>
(z
(z+3) It will
be immediately perceived that there exist three formal Y (z)c^ over the three
which can be obtained by integrating paths indicated in figure 3. The axes of X and axes of reals and imaginaries. solutions
.
Y
Y
Y
y
are respectively the
FIGURE 3 Case of Degree One (p 6. The Homogeneous Equation 1) with Transcendental Exponent. An extension of the argument of the preceding section is possible if
r (z)
=
(6.1)
where g(z), h(z), and k(z) are polynomials. Let us assume that the degrees of these polynomials are p, q, and r respectively and that the coefficients of the highest powers of z in each are G e si H e hi and K e** R e 01 we should then have the For sufficiently large values of z .
,
,
=
asymptotic value, (>0f0)}
HR
q
[cos{q&
GR?
+
&
+ (qoth)i
I
GR P sin (p\)\g) }
Xe
+
i
sin
KR
RP <**<**
Let us define a family of curves in polar coordinates by the equation : q#

h f
GRp sin(ptf+0)
= (2nI)n/2,nt=\
means
1,
2,
3,.
1,
2,
3,
f

of
THE THEORY OF LINEAR OPERATORS
362 If
we use
written,
Rp
the abbreviation,
=
{
(2nl)np
A
= p* +
2q (ig)
0> this
2ph}/2Gp
equation sin A
may
be
.
(6.2) It is evident that the curve represented by (6.2) has infinite branches in the direction X 1 2, p and these nn, n 0, branches divide the plane into regions within which the real part of the coefficient of e QRP ros <^< is of one sign. Let us designate by A any region within which the real part is positive and by A 2 a region within which the real part is negative.
=
,
,
.
made by means of the lines then cos (p f g) will be of p<0 f g 1, 2, 2p, (27il)7i/2, one sign in each region so constructed. Let us designate by /?! a region within which the sign is positive and by B 2 a region within which the sign is negative. If still a third partition of the plane be made by means of the lines rft \ k 2r, we see that the real (2n \)n/2, n t= 1, 2, r KR of i sin(r# ) A:)} is positive in one set of part {cos(rfr { k) \Let us call such regions Ci and in the other. and regions negative If a second partition of the plane be 7i
=
,
=
C2

,
respectively.
Let us designate by the symbol L (C6) a path between two points a and b, by A L (utt} a path from the point a to the point b which lies wholly with the region A, and by B L (f7&) a path between a and 6 which lies wholly within a region common to A and B. .i
Returning now to a consideration of the problem of evaluating the integral,
clear that a formal solution of the differential equation is given means of this integral provided the path of integration avoids the
it is
by
singular points of Y(t) and consists of the following branches:
There may in particular exist an infinite number of such and hence an infinite number of formal solutions. Two examples will illustrate these principles.
Example
1.
Let us
first
consider the classical equation,
u(x\l)
xu(x)
=0
,
in particular the gamma function, UQ (X) Symbolically this equation can be written in the
which defines
ferential equation of infinite order, (e~
circuits
x)
w(#)
,
=
F(x). form of a
dif
THE LAPLACE DIFFERENTIAL EQUATION from which a simple computation
c
s
yields,
+ xz
y(z)*=0
,
1,
K=x
;
g
& = fc
h
0,
=
.
=r
we have p
Referring to the notation of (6.1)
G <= H =
363
Equation
n.
1,
q
= 0;
(6.2) becomes,
3)jicsc
The partition of the plane is made by a series of lines parallel to the axis of reals and spaced at intervals of n, the first at a distance 7i/2 from the axis of reals. The second and third partitions are coincident, being merely a division of the plane into two parts by the axis of imaginaries.
Axis OF IMAGINARIES 5TT
2
3TT
sir
2
FIGURE 4
One path L can be composed
OA
of the segments
and
We
01*7.
thus obtain,
u(x)
/oo ' e e +xz dz~\
Jo
/*o
e e *+ xz dz
Changing variables by means of the equation
which
is
part of x
the familiar integral representation of
>
0.
.
J co t
e
z
we
get,
F(x) for the
real
THE THEORY OF LINEAR OPERATORS
364
choose for a path of integration the segments which ev (s} vanishes. We thus
It is of interest to
AO
\OB\BC
at the extremities of
obtain, *
fV ew cfe4
u(x)
J (
(jo
l+ * vi
integral
+ e wi
idy
+
f "V'i^di J1
e z*i*_l)
The second
e e
f Jo
Ce~ e *^dz
,
Jo
C** e^'^idy
(6.4)
.
J<>
may be evaluated we find,
as follows
:
Integrating by parts *
f
=e
eenwiidy
l
2 * lx
(l
e'
)
+
(x
f Jo
1)
Jo
=
e*
e* vif )
(I
+
(I
\
(x1)
(s_l)
l)(x
f (x
2)
(a
e~ eyi+(x ~^^idy
..
(xn)
This series is obviously divergent but may be method. Thus employing the Borel integral
2)
+.
+ .
}
summed by
,
Borel's
^eWdt^nl
Jo
we
get,
f Jo
=.
e ^(l
c'
W
^ xut>
idy
f e*{l+ (x Jo
+
(x
4_
(xl) (x2)
fV
e*)
6 2irix )
cr^(\
f
(1
Jo
l)(x
2)tV2! ...
+ ty^dt =
+
(jcn)t n /nl
(1
e 2Ti *)
+
}
dt
fV^'^dt
.
Ji
Substituting this value in equation (6.4) we find that u(x) is identically zero for this path. Employing still a third path (AOBD) and noting the results just obtained, we compute, /0
u(x) t=
e es
^"dz+
(ie
x c't *dt { e 2
)
e
Ji
Jco
e~* +**dz
,
feWidt
,
Jo
a
=
f
e
<
f l (/t+
(1^^) fV^Mt
Ji c==
/*JQ
^QO 2
_
e ^ir
*/i
rv^'^t
e"*
1
*
Jo
.
Jo
Since the coefficient of the integral is a function of unit period and since the employment of paths parallel to BD in higher admissible strips serves to multiply the exponent of e by n, we are able to infer
THE LAPLACE DIFFERENTIAL EQUATION
365
that the most general solution of the original equation for values of x in the right half plane is
u(x)
= n(x fV
f
*'
1 c
Jo
where n(x)
is any function of unit periodicity. The solution of the equation for values of x in the left half plane for the poles at the negative integers is obtained from a well known generalization of this
integral.*
Example
A
2.
more complex example
is
furnished by the equa
tion, l
(I
[4iz
+ 2z'
2
2
) e'^
=
x} *u(x)
\
,
from which we have,
(z)
=
ze z2
~f iz* f
xz
,
t/>
we
Referring to the notation of (6.1)
(z)
c=
.
find,
Equation (6.2) then becomes, R* c={(2n
3) 7i/2
ft}
/sin
2ft
.
=
R cos ft, y <= by 2 sin ft cos ft and let x ft. Taking the tanR sin #, this may be written 2xy 3) n/2 (2n gent of each side and noting tan ft y/x, we reduce this equation to If
we
replace sin
2ft
=
=
(6.5)
Introducing the parameter parametric form,
x
=
H=
(/*
cot 2/0
xy,
JLI
*
T/
,
we
=
obtain equation (6.5 )in
(/i
tan 2/0 *
.
The graphical representation of this function is given in figure the branches being numbered (1) for identification. From the fact 5, that the derivative, x[I =.
vanishes
when
/<
=
0,
cot
we
2jii
2(x* (sin
find one
+
4/j
2
?/
)]
A/[l
+2 (a
4/0 / (sin4/
maximum and
2
f
one
minimum
at the
points of tangency with the circle of radius 1/2. The branches shown to 5^/4. Other similar in the figure correspond to a range of /* from the parameter. values of for exist not larger shown, branches, second partition of the plane is given by the equation ft t= third par(2n l)ji/4. These lines are marked (2) in the figure.
A
A
*See for example, Whittaker and Watson: (1920), 3rd edition, p. 243.
Modern Analysis, Cambridge
THE THEORY OF LINEAR OPERATORS
366 tition is
(n
made by means
of the lines,
marked
(3), given
by #
=
l)ji/4.
Referring to equation (6.3) we see that two formal solutions of the equation exist provided the path of integration L is chosen to be either AOB or AOC.
FIGURE 5 7. Perron's Theorem. We have seen from the examples of the preceding sections that the various solutions of the homogeneous equation
F(x,z) >u(x) which appear
in the
form u(x)
where Y(t)
is
=
I e*
JL
1
Y(t)dt
(7.2)
,
a solution of the equation
=0 may
(7.1)
,
(7.3)
,
be conveniently
paths
L
(1) (2)
classified according to the characteristics of the of the integral. These paths of integration may be either
Pochhammer Cauchy
circuits
circuits
around two zeros of
around single zeros of
A
p
A p (t)
(t) ;
;
THE LAPLACE DIFFERENTIAL EQUATION Laurent circuits (see section
(3)
A p (t)
zeros of
7,
367
chapter 2) around single
;
(4)
Circuits not included in the first three classes.
The
first
as regular
may be conveniently referred to their relationship to the regular singular points of
three types of circuits
from
equation (7.3) the other circuits are accordingly irregular. If in equation (7.3) the function A p (t) has n zeros (multiplicity counted, but in no case exceeds p) and if the other coefficients, Ai (t), satisfy the conditions of regularity,* then it is easily shown that possible paths for L are in general the following: ;
,
(1)
n
(2)
s
Cauchy
(3)
p
s
p Pochhammer circuits
Laurent
circuits;
;
circuits:
The regular singular case
of (7.1) was studied by 0. Perron [see Bibliography: Perron (3)], who stated the following theorem:
Theorem
A
If in the equation
3.
(t)X(t)
+A
1
(t)X'(t)+A (t)X(t) 2
()=0 the functions A (t) are analytic t fg q , and ularity in the circle
all
t
I
*We (x
recall the following definition:
a)ny()( x )
f (x
a)iP 1 (a5)
and if
A
(7.4)
,
satisfy the conditions of reg
within the domain considered differential equation of the
#<i>
(a;)
form
H
P n (x) are analytic in the neighborhood of x P l (x) t P 2 (x), a, is , a. Such an equation may be called said to have a regular singular point at x a Fuchsian equation since the character of its solution was first indicated by L. Fuchs in 1866. The theorem in question states that, for a linear homogeneous differential equation of ?ith order to have n independent integrals which possess no singularities other than poles and branch points in the neighborhood of the point a, it is necessary and sufficient that the coefficients be of the form indicated above. The solutions of the equation are then of the form
where
y(x)
= (xa)*[g
1
(x) f
ff 2
(x) log (a
a)
+g
3
+ ff^(x) where g l (x), g 2 (x),
The value
X
is
a) H
(x) log*(x
logP(x
a)]
,
g ^ l (x) are functions analytic at the point a. a root of the indicial equation (fundamental characteristic ,
r)
equation)
F
1
(a)X(X
1)
..(X
n+2)
+
If the roots of this equation are distinct and if the difference between the real parts of any of them is not an integer, then the logarithmic singularity is absent from the solutions.
THE THEORY OF LINEAR OPERATORS
368
n
p zeros (multiplicity considered and in all instances or equal to p) of A p (t), then there will exist than assumed less n p__s linearly independent integrals of (7.1) the grades of which do not exceed q, where s is the number of linearly independent integrals of (7.4) analytic within the circle t 5^ q* there exist
:
\
Proof: By assumption the n zeros of A p (t) are regular singular a, there will points of (7.4), so for any one of them, for example, t exist p linearly independent integrals of (7.4) of the form
=
X(t)
where
=
X is a root of the indicial
JiW
(t
,
equation (see footnote to this section)
U p+i)&PMW
1)
a)^P(t)
U
i)
^& oc =0 and P(t)
analytic at
is
t
=a
(or has at
Vi
iH2) ,
(7.5)
most a logarithmic singu
larity).
Similarly the adjoint equation (7.3) will have a set of solutions of the form
= (ta)K}(t) (or has at most a logarithanalytic at the point = Y(t)
where Q(t) is mic singularity) and
,
t
/*
is
a,
a root of the equation
4.
^0
(_!)"&
.
(7.6)
obvious upon substitution that the relationship between the roots of (7.5) and (7.6) is given by the equation It is
Y (t) has a pole of any order at t <= a, the corresponda of (7.4) will be analytic at t ing solution Let us first consider the case where the real part of /* is negative, but where ft is not equal to a negative integer. Then the integral (7.2) will be a solution of (7.1) provided L is a Laurent circuit about the Hence
if
=
~
.
In assuming the existence of a Laurent circuit a, point t the use of following theorem due to A. Liapounov.f .
we make
the theonegative, then s integrals satisfying the conditions of number of linearly independent integrals of (7,4) analytic within the circle. vol. 9 tProbleme general de la stabilite du mouvement. Annales de Toidous, in 1893 in the Communi(series 2), (1907), pp. 203474. (Originally published cations of the Math. Soc. of Kharkov). *If
rem
n
p
is
will exist provided s is the
THE LAPLACE DIFFERENTIAL EQUATION // the coefficients P,(x) in the equation
are bounded in the interval
y(x)
that, if
all
is
)
we assume
*
oo
that n
(x)
+ y^(x) ^0
then there exists a value z such
,
any solution of the equation,
tend to zero as x If
(0, oo
^
369
the following functions:
.
>
p,
it is
clear that there will be in general
n such
now
We
integrals, but of these only p are linearly independent. remove the restriction that the real part of ju is negative. If the
real part of
/<
is
positive or zero, then the Laurent circuit about
=
t
=a
a to 1 <= co The path of intereplaced by a line integral from t gration avoids the singular points of the differential equation and approaches infinity in a sector which makes the integral converge. is
.
The remaining n p integrals may now be found by combining the singular points in pairs to form Pochhammer circuits. If fi is a negative integer, then the Laurent circuit is replaced by a Cauchy circuit about the singular point. If there are s such values of /i, and if the corresponding integrals have no other singularity than the
pole,
circuits.
But
the p Laurent circuits are replaced by Cauchy no singularities other than poles then the has Y(t) ad that is to say, the corresponding solutions of corresponding joints, will be (7.4) analytic. If n < p, then all the solutions of (7.1) will be derived from Laurent circuits, unless s of the characteristic values are negative in of the case s which Laurent circuits are integers, replaced by s
then
s of
if
jit
Cauchy
circuits.
We now examine the character of the ly
show by the methods of chapter
solutions
5 (see
problem
and we may readi5,
section 2, chap
ter 5) that the following are true: If in the integral (7.2) L is a
Cauchy circuit about the point a of is if L is a Laurent circuit about the then a, u(x) grade a and t<=b, then the grade of u(x) is equal to the the two points t if L is a Laurent circuit b larger of the two numbers a and the then of about the point t grade a, u(x) is infinite. we see that we have established the Combining these results theorem. t
=
= =
;
;
\
The Nonhomogeneous Equation of First Degree (p t= 1). Having in previous sections discovered conditions under Which formal solutions of the homogeneous equation exist we now turn to the problem of solving the nonhomogeneous equation. Before proceeding 8.
THE THEORY OF LINEAR OPERATORS
370 to
more general considerations we
first
examine the equation of
shall first
degree, 00
2 ((*+b n x)uw(x)*=f(x)
(8.1)
.
o
We
assume
shall
A
(z)
beginning that the functions,
in the
= 2X,
n
and B(z)
=2 b z
n
n
n~o
,
w=o
are analytic in a circle about the origin of radius not less than q, that 6 =7^ 0, and that f(x) is an infinitely differentiate function but not necessarily of finite grade.
we have
Equations (2.4) and (2.5) thus become,
+ B(z)X'(z)*=0
A(z)X(z)
A(z)Y(z)^{B(z)Y(z)} The
solutions of these equations
X(0)=l
,
,
=Q
.
which satisfy the conditions
X(z)Y(z)=l/B(z)
:
,
are clearly,
Referring to equation (2.23)
we
see that the resolvent genera
trix becomes, ~ lg
X(x,z)=e* X(z)U(y)\c*:X(z) where U(x)
is
&'Y(t)dt
,
(8.2)
a solution of the homogeneous equation and a
is
any
point such that
As an example
consider the equation,
xu(x)*=f(x)
u(x+I) which in the notation of
this
(8.3)
paper becomes,
x)
(e*
,
*u(x) =:f(x)
.
The values of X(z) and Y(z) are determined from the equations,
X'(z)
e*X(z) t=Q
X(0)=l, whence we
find
:
X (z)
=e
e
*/e,
Y(z)
=
6
,
THE LAPLACE DIFFERENTIAL EQUATION The general
solution of the
homogeneous equation has been shown n(x)F(x), where n(x) is any
=
in example 1 of section 6 to be, u(x) function of unit periodicity.
Obviously the point a and introducing the
may new
latter
be either variables
^X(z)n(x)r(x)
X(x,z)
371
p~
)co
u x
=
or e
f
,
o>.
p =
Choosing the
e\
we
have,
Cu^e^ Jo
The second tion
integral
recognized as the incomplete
is
gamma
func
which has the expansion, y (p)
=
f u*^p u du =. p*/x 4 p'+ l /x (x+l)
we have
Since n
p >f(x)
the operational identities,
= f(x+n)
and e*X(z) *f(x)
= a constant
,
the solution of equation (8.3) becomes,
u(x) *=n(x)r(x)
{f(x)/x
+ f(x+\)/x(x+\) }
If
ployed,
a second expansion of the incomplete
we
gamma
.*
(8.4)
function be em
obtain,
Operating upon f(x) with this new operator
we
get,
(xl)f(x2) + (xl) (x2)f(xB)
+ }
(8.5)
.f
This solution is usually though not always divergent. is furnished by the equation,
An
ex
ample
xu(x)
= r(
This is a well known result due to H. Mellin: Zur Theorie dcr linearen Differenzengleichungen erster Ordnung. Ada Mathematica, vol. 15 (1891), pp. 317384.
fSince
pV2! /(O)
we have
x p e? ~> f(x)
p~
x
e?
* f(x)
= p
* (1
4
THE THEORY OF LINEAR OPERATORS
372
we merge
If
the divergent coefficient of the second F(x)
(10.5) with the arbitrary function n(x) we
f(xl)
+ (xl)f(x2) +
in
find,
(xl) (x2
and hence obtain, u(x)
z=~n
Borel summability will also serve to rescue a solution in many divergent cases. If as before we merge the constant coefficient of F(x) with the arbitrary function of unit periodicity, n(x), and apply the Borel integral to (8.5) we obtain,
m
+ er*{f(x 1) + (xl)f(x2)t/l\ + (xl) (x2)f(x3)t*/2l +  dt
n(x) z=ji(x)F(x)
}
An
example
is
furnished by the equation,
xu(x) for which
we
.
=1
,
obtain,
u(x) =7i(x)r(x)
f
9. The Nonhomogeneoiis Equation Expansion of the Resolvent in Powers of z. Three different forms for the resolvent generatrix can be given if the expansion is made: (1) in powers of z, (2)
in inverse factorials in .r, (3) in negative powers of x. The first of these expansions has been extensively studied by Hilb and his results will now be attained through the methods of the present chapter. Hilb's actual method of solution by means of the theory of bilinear
forms
is given in section 13. Neglecting for the moment the contribution geneous equation let us consider the operator,
made by
the homo
where we employ the abbreviation S
0(z)=X(z)
( e?'Y(t)dt
.
Ja
The expansion of X, (o?,z)
this operator in
powers of
z is obviously
+ vi + V^ /2 Hh ^nZ m /m + 2
1/'
!
,
(9.1)
THE LAPLACE DIFFERENTIAL EQUATION
373
where we abbreviate,
0(m>(o)
_# 0(^(0)
m(ni
f
l).
Explicit calculation then gives,
Vo
=*>){
(a?)
fVY(0
Vl
(ff)
=
0(0)
?fo(*)
If
we adopt
=X
+ X(0)Y(0)
the notation,
(IH)
wzaX (wl  1)
(0)
m(m
(0)
l)x
2
X (m
2)
x m X(Q)
(0)/2! 
,
(9.2)
=Y
(w)
+ mxY
(0)
(m ~v
(0)
+ m(mrI)x' Y< m * (0) /2! the general term
yi
(iP)
If
we
W (w
\x
j
Y(Q)
,
be conveniently written,
may
 Wu () [X] +
,_j_
m

2
7w
{
Y}' [X]
W
1){Y}[X]
2
/2
!
 1
+ + [Yj^CX]
.
(9.3)
place in these symbols the explicit values of X(z) and the preceding section and represent by A (i)
Y (z) as calculated in the numerical and B
A
(l)
(i) (0) and Z> (0), this solution of the coefficients of the original equation.
(l)
values
be exhibited in terms thus obtain to four terms the expansion,
X
(x,z)=u*(x)
+{1
Bx+
(A
2
+ Bx)Uo(x)}z/B+{AB' BA' + AB' + 2xAB (A
+ + (A
2
2J3A'
2
ff'JS )
+ 3AB'
NO
(a?)
2 }^/^ 2
BE" + 2B
/2
!
(9.4)
may
We
THE THEORY OF LINEAR OPERATORS
374
+
3A 2 B' + 2BB'A'
[3AA'5
S A" 2
+ ABB" B
3x (A~B
2
A'
+ ABB')
3AB
A
2AB' 2 2
3
x2
where we use the abbreviation f Jo
The complete operator
is
obtained by adding U(x)e*~X(z) to
this expansion.
As an example
consider the equation, 2 xn(x) ^=x
Since
(9.5)
.
we have
A(z)=e*,
B(z)=l, f e xt+l
u (x)*=
 et
U(x) t=n(x)r(x) fV'e 1 "^
dt ^z
Joo
,
,
Jo
the operator becomes
X(x,z) =n(x)r(x)e"X(z)
Cu'Wdu [I
+ (lx)z
Jo _j_
2x
(2
+^
2
2
)^ /2
+
(5 (3
Applying e~*
s
X(z)
we
2o;
=
+ 3^' ^0^/6+ + )c /6 + a;
2
J


}

.
}
2
and merging the constant obtain for the solution of (9.5) the func
this operator to /(#)
* x with n(x)
6x
.r
tion,
u(x)
/i
=n
u*Wdu
.
Jo
We
shall refer to equation (9.1) as Hilb's solution of the differential equation since the y m may be identified with those obtained by this writer in a different manner. Consideration of the convergence
properties of the operator will be postponed to another section. 10. Expansion of the Resolvent in a Series of Inverse Factorials. second form of expansion for the operator of the last section may be obtained as follows Let us integrate by parts the integral in the operator,
A
:
THE LAPLACE DIFFERENTIAL EQUATION xt Recalling that lim e Y(t)
=
0,
we
375
thus obtain,
ta
Writing the integrand in the form e (J+1) *Y'()e and extending the original assumption as to the value of a to include the limit, lim e xt Y'(t) 0, we obtain the following formula by means of a secf
=
td
ond integration by parts:
Employing the abbreviation,
w n (z) *={ w^(z)}
w (s)=Y(z)
,
,
(10.1)
and assuming that we have, ]ime<*+n)t w n (t)
=
(10.2)
,
ta
for all positive values of n, we are able to continue the process outlined above and thus derive the following factorial expansion:
X
(x,z)
t==
X (z) {w w
3
(z)
(z) e**/x
ff
w, (z) e /x (xf1)
/x
(x+l) (x+2) (x+3) +
.
(10.3)
To obtain the complete operator we must, of course, add to this the operator e~Jfz X(z) U(x) where U(x) is a solution of the homogeneous equation. It is of some interest in particular applications to obtain the condition under which (10.1) is a finite series. An examination of the function w n (z) shows that we have, Wi (z)
where
D
represents the operator d/dz
w* (z)
and
= e*Y'(z) t= e*D
=
er**

,
(Dl)D * Y
in general,
w n (z) =e(D If
we
n+I)
...
note from explicit values that
(DY)/Y*=(A(z)
B' (z)
(D
THE THEORY OF LINEAR OPERATORS
376
is
=
n e~ *R n (z), where R n (z) write w n (z)/Y(z) expressible in terms of R(z) and its n 1 derivatives. For example,
we
observe that
{
we may
(D2) (D1)D 
Y}/Y
=
/J3 _j_
3ft>#
_
3/e?
_3
/>/
_j_
#// ^_ 2ft
.
It is clear that these expansions furnish a criterion for the termination of the factorial expansion of X(x,z) since we need merely assume that the coefficients A (z) and B(z) satisfy the equation,
n+l^
(D
...
l)D> Y(z)
(D
=0
.
Thus we reach the conclusion that will terminate
(10.3) satisfy the eq^lation,
the expansion of the operator with the nth term provided A(z) and B(z)
n+1)
(D
(DI)D>Y(z)
.
=0
(10.4)
,
which expresses a
differential relationship between them. explicit derivation of the equations corresponding to the first cases will be of interest, the first case in particular applying to
The
two
the theory of singular operators. The condition D * Y is equivalent to A(z) Hence the equation,
=
W(z)+(z)x}*u(x)=f(x) where its
an
infinite
B'(z)
=
0.
(10.5)
,
operator with constant coefficients, has for
resolvent generatrix the function,
X The
condition:
(D
1)D 72'
The
t
(x,z)
>
F
2 R}R r=
solution of this equation
I is
leads to the equation, .
is,
R(z) =le*/(l
where
=
+ ie?)
an arbitrary parameter. From
,
this
we
obtain the rela
tionship,
A(z)^B(z){^/(l
+ W)} + B'(z)
and hence we may conclude that the equation,
has for
its
resolvent generatrix the function,
,
THE LAPLACE DIFFERENTIAL EQUATION
377
A
11. Expansion of the Resolvent in Negative Powers of x. third expansion of the resolvent generatrix is possible if we assume that a value a exists such that
lim for
values of
all
e ft
t=
n.
assumption we formula to the resolvent
Under
Y^ (t)
this
may
apply the integration by parts
:
r>rt V/f \ /7f JL e \ i) ciu
Car
..
and thus obtain,
X
(x,z)
=X(z)Y(z)/x
{e**X(z)/x}
fV'Y' Ja
= X(z){Y(z)/x
Y'
+ Y"(s)/a3 + ...} An
interesting special case is obtained when Y(z) is a polysince the resolvent generatrix then consists of a
nomial of degree finite
(11.1)
.
number
m
of terms,
X,(x,z)=:X(z){Y(z)/x
Y'(z)/x*

.
+
Y (m} (z)/x m ^}
(11.2)
.
special form of this operator immediately suggests a simple of obtaining the solutions of the corresponding homogeneous equation. If we assume that A (z) is of the form
The
way
where P(z) is any polynomial, we obtain Y(z) in the desired form, namely Y(z) =P(z). Since X(z) l/[Y(z)B(z)'] has poles at the zeros of Y (z) B (z) we can apply the method of Cauchy circuits as explained in theorem 1 of chapter 6. Hence designating by X (i} (x,z) a Laurent expansion of
X
Q
(x,z)
regarded as a function of z within one of the annular re
gions bounded by concentric circles through successive zeros of Y(z)B(z) and by X (x,z) the expansion of the operator in the region about the origion, we obtain a solution of the homogeneous equation from the function, ,
(11.3)
THE THEORY OF LINEAR OPERATORS
378
where f(x) is arbitrary to within the limits of existence of the right hand member. An example of this method is furnished by the Bessel equation, (x
2
z
n
2 4 xz 4 (x
2
) }
> y (x)
=
,
for the special case where n is half an odd integer. ttx", this equation becomes, By means of the transformation y
=
{xz
We is
2
+
(2n+l)z +
x] >u(x)
2 (1 f z )"*
then find Y(z)
half an odd integer. For n *=
which
we
!/2
=
(11.4)
.
is
a polynomial when n (x,z) the func
obtain for
X
tion,
we have two zeros on the unit circle in this case we must modify the method outlined above by considering each term
Since slightly
of the expression in brackets separately. in (11.3) by setting f(x) 1. Then since
=
I/ (1 4
iz)
 1
we
4 l/z  ) > 1
2
cos x
i
sin x
specialize the function
have,
= (i/z 4 1/z + i/z* 1
we
We
,
obtain as one solution of (11.4) the function, M! (x)
cos x
(
i
sin x) / (2x)
.
=
1 cos x \ i sin x Similarly since we have 1/(1 iz) > 1 for one expansion and 1 for the other we get as a second solution,
u2 (x) t=
cos x
(
f i
sin x) / (2x)
.
Linear combinations of these obviously lead to the solutions in customary form, }/x
For n
,
v 2 (x)
^
= 3/2 we have the operator,
where we abbreviate,
S= s
e=
i/2x
2
2
t'/2a:
4
(
l/2x
a ,
T
= l/2a? 4
THE LAPLACE DIFFERENTIAL EQUATION
379
Proceeding as before and regarding the operators separately, we obtain, Ui (x) >=
cos x/x^
sin
x/x f i(cos x/x
=
cos x/x 3
sin
x/x
u
2
(x)
As l
12.
2
f i (
2
sin x
cos x/x'2
)
sin
before linear combinations of these yield the solutions,
v (x) i=
now be
2
A (cos x/x* j sin x/x
2 )
,
v 2 (x) :=
B (cos x/x
2
sin
A
X/X
)
.
Extension to Include Solutions for the General Case. It will desirable to formulate the results exhibited in the last three
=
sections for the differential equation of first degree (p 1) so that will the In include case. order to this end we achieve they general
return to the statements of theorems 1 and 2 from which derive the following theorem
we
shall
:
Theorem 4. which we have,
If there exist*
point a in ihc complex plane for
a,
then the resolvent (2.23) can be expanded in the following series:
X(x,z) t= U(x)cr**X(z)
(A)
,(r) +^y it H
M
/ml
,
(12.1)
where we abbreviate
y n (x)
=u m (x)
mxu,
HL
(x)
+m(m

x m ii<> (x)
l)xu m  z (x)/2l ,
m
5=i
p
1
m>pl in which
we
write,
f Ja
,
THE THEORY OF LINEAR OPERATORS
380
// the additional conditions are fulfilled,
then the resolvent has also the expansions,
X(x,z)
= U(x)e
fy
X(z)
+ {(l)'W/x(x+I)
(B)
+ ?*Vi (z,s) e V 2
(a;

f p)
(x
+p
1)}
+ p + 1)
(a:

}
,
(12.2)
where we abbreviate,
w n (z,
t)
=A.^ w n _i(s,
t)\e*> w*(z,
t)
= Pf
(z,
)
J
t=SX (2)^(0
(12.3)
t
X (WV.(~,2)
(C)
^(S,2)/*
+W
ptl (
(12.4) in
which we use the notation,
= V 0V
( ~\ ^ <& \Z ) i
V JL
/19 f^\ l.'d)
//\ i
V
{l; )
The derivation of the formulas contained in this theorem proceeds from the formal expansion of the operator (2.23). If Proof.
C 'c rt Y>(t)dt Ja
and account taken of the defining conditions (2.9) and (2.14) we obtain (12.1). (A) is thus the power series solution of the generatrix equation (2.2). xt (m) (t) t== 0, Assuming next that a point exists such that lim e Y
be expanded in power series in
z
l
wle
can perform the following integration by parts
:
S
C
,t)
/dt}e*dt/x
THE LAPLACE DIFFERENTIAL EQUATION f Ja
VW
(z,t)
dt
=
e*~~{w (z,z)
where we use the abbreviations
From
/x
381
Wi (z,z) e*/x (x+l)
(12.3).
however, we observe that the
first p 2 terms are and we thus obtain expression (12.2). Finally the operator (C) is derived by means of a similar integration by parts and the employment of (2.8) to eliminate the first 2 terms. p It will be of interest to consider an example
(2.8)
identically zero
:
+
4.x (z
The homogeneous section 4,
where the
+ z* + z* + .)
%*}

u (x)
= / (x)
.
case has already been dsicussed in example
2,
solutions:
6/V)
,
U,(x)
=
were found. From the resolvent,
and the adjoint
resolvent,
with account taken of conditions fundamental sets of solutions:
Y,(z)=z^
z
(2.9)
and
Y
,
2
(2.14),
(z)
=
we
**
derive the
+!
.
Making use of (12.1) and neglecting the contribution of the first term of the operator which is easily constructed from the values of t/i(#) and U 2 (x) given above, we can write,
f
R(x)
>
,
Jo z2 )
.
(12.6)
This is obviously expansion (C) of theorem 4; expansion (A) easily made through multiplication of the expression in braces by the series 1 j z 2 f z* . is
+
In order to obtain series (B) tion
:
we
resort to the following calcula
THE THEORY OF LINEAR OPERATORS
382
=
W<>(Z,Z)
W The
'"(z,z)
=
2
6/(l
)
?<;
,
W
,
Q '(Z,Z
=0 w>3
<"(z,z)
,
derivatives are here taken with respect to t in the resulting functions.
.
and then
t
z is
substituted for
From
these values
we next compute:
2
4)/ (1
^
Adopting the notation: x
we
~ n}
(
z")
,
^2 )
186) /(I
=^ l/[#(xl) (^f2)
(x^n
1) J,
then obtain the resolvent operator in the form
X
(x,z)
*= {(
+
(4
+
(186
2
12^f2^ )^
This expression following identity
(
 4)
200+242; is
+
2
(
44z+6z
(30
c >
)a;
+
2
5
z2 )
.}/(!
>
)a,'<
(12.7)
.
easily verified
by substituting
number
m
in
(12.6) the
:
where
Z> s (m) ,
the Bernoulli tained from the series:
l)}"'
{t/fe''
of order
= 2t'B,<
M
and degree
/s!
f<2;T
,
ob
is
s,
.
.s_0
For values interesting
B w==l
,
B
to us here ^w 1
m (15ra
n
>= 30m
B
m/2
2
(m}
is
found explicitly
to be,f
,
+ 5m + 2) /240
,
_ 2) /96 2 Substituting the explicit expansions of 1/x (12.6) we obtain series (12.7).
*See N. E. Norlund: Differenzenrechnung, Berlin, fNorlund: loc. cit., p. 146.
,
l/%
.
?>
,
and
(19124), p. 243.
4
l/,r
in
THE LAPLACE DIFFERENTIAL EQUATION
383
PROBLEMS Solve the equation
1.
u(x + l)xu(x)=T( x + l)
(Boole)
.
Discuss the equation
2.
(6 f x?)
2xu
u
r
+ 6?*"
2jeM<3)
+
6tt< 4 >
2x
u<">>
+ 6tt<> Note that a particular
P 2 (z)
given by
=
set
of solutions of the
y2 (3^_l), Q 2 ( z
)
=
y2 P 2
(
2.rtt<7>
+...= f(x)
specialized resolvent
z)
log[(s
+ l)/(*
.
(2.4)
is
3*/2
1)]
.
Solve the equation
3.
(acosz
+ #)
> ?t(se)
/(#)
.
Obtain the solution of the equation in problem 2 when f(x)
4.
=x
3
.
5. Employing the method of the illustrative problem of section 11, compute the explicit values for the Bessel functions of orders n 5/2 and n 7/2 .
Limited Bilinear Forms section In we this shall consider two methods of Appcll Polynomials. The first of the method of limited bithese, solving equation (1.1). linear forms, is due to E. Hilb (see section 6, chapter 1), who has thus furnished an excellent example of the practical application of the theorems relating to these forms. Because of the complexities of 1 the general problem the discussion will be limited to the case p other details. and the reader is referred to Hilb's paper for The second method, that of Appell polynomials, is due to S. Pincherle who applied it to the case p 0, that is to say, the case of differential equations of infinite order with constant coefficients. The extension to the general case was effected by I. M. Sheffer. A brief account, together with a bibliography, of Appell polynomials is given
Other Methods of Inversion
IS.
=
=
in section 6, chapter 1.
The Method of Limited Bilinear Forms.
We p
1,
shall consider the solution of equation
that
is,
(1.1)
for the case
of the equation
,
wo
(13.1)
where the functions
A(z)
=2a
n
2"
,
B(z)
=
are analytic in a circle about the origin of radius greater than
where
6
q,
and
THE THEORY OF LINEAR OPERATORS
384
an infinite number of times, system of linear equations is obtained, which is a special case of the general system discussed in section 13 of chapter 4. Referring to (13.5) of chapter 4, we see that the matrix of the coefficients of the unknowns u (n) (x) is given by If equation (13.1) is differentiated
an
infinite
+ (kl) &
K n k 
where am and
n
m<
+
ft***
(13.2)
, 
0.
system 1] t=
^
[/c
q
b m are zero for
If the fcth equation of the 1 for k 2, [1 1] t= k
w **i
is 1,
divided by
and
if
u
(n)
1] q*
[}c
(x)
is
1 ,
where
replaced by
v (n) (x), then (13.2) becomes
[fc_ 1]
The problem resolves itself show that the bilinear form
g*i
essentially into three parts:
(1) to
00
V
A \) e*== 2^ U/ n n fc JUn T v, \S1 yk (
is
limited; (2) to find a matrix
,
which
satisfies the
#.A = A# t =/ and
(3) to
show that the
bilinear
equation (13.3)
;
form
,
is limited.
To show that (A)
(1)
This chapter
problem
^2 2

o k
/>/
Since [k
(A)

+
, I
i
1] i GO
I
disposed of by means of theorem 11, k by we consider
is easily
m
Thus replacing n
3.
(A)
I
first
is limited.
1,
we
1) &i4l
+ &m
obtain the inequality
GO
^2 2
{
a*
00
But since
(&
2
^*
~
m
I
+
I
OO
and
1)10
circle of radius q, limited.
& I
theorem
2 6w m0
+ ^w
I
&
I)
!
*
converge absolutely within the
11, chapter 3 applies
and the matrix
A
is
THE LAPLACE DIFFERENTIAL EQUATION To
(2) If
we
find the matrix
designate by
.
the matrix
<
nk
then
we must determine If the
385
q
n~l
1
[n
so as to satisfy equation (13.3). equal to the backward inverse,
forward inverse
is
we
are
led to the equation

2
n  m+1
<
+
b n . m X\ q n '*
nk
(13.4) It will
now be observed
that the function
?*() satisfies
= 2
the differential equation
^ [A(z)

(z)+B(z)
(13.5)
Conversely, the coefficients of the series development of each integral of (13.5) satisfy the matrix equation (13.4). It is also clear that the singular points of (13.5) are the values of z for which B (z) c= 0. Moreover, for z < Zi\, where z l is the first zero (i. e., with smallest modulus) of B(z), there will exist an ininfinite number of solutions of (13.5) and hence there will exist an infinite number of inverse matrices. In order to get a single inverse we and must limit z to the annulus between 2 [, where z 2 is the first zero of B(z) of modulus greater than Then the series
converges only
if
the integral of (13.5),
i.
e.,
(13.6)
where we abbreviate
M(z)
=exp
N is
k
(x
9
regular at the point z
dz
[
z)=z ZL
,
THE THEORY OF LINEAR OPERATORS
386 Let us
now
write
then follows that, for values of
It
>
A
0,
the expression inside
the brackets of (13.6) will be
where P(z Zi) is a series in positive powers of P(0) 0, and K is an arbitrary constant.
zj such that
(z
^
From
these considerations the value of y k in (13.6) can be easily If A is a positive integer, one sees that y k *= 0. If this is not the case, then let z make a circuit C about the points and z and the with We thus new get identify integral

fc
f Jc
and
N
k
(x,z)dz
follows by comparison that
it
(3)
We
To show that (0)
must show
limited.
is
finally that the bilinear
form
*rr ,/C
is
limited,
where x k and
are subject to the limitations
2/ w
2 ^ I
(/
1
^
2 I
1
>
First, applying the
^ ^ 2
21
1
I
wi
ki
Schwarz inequality (see section
chapter 3),
9,
we have GO
./>
I
2 v,* 9 A
1
*
**
I
[
2 !
A1
1
But
'JO
^ 2 kn
<jpu.
= r/^(0) = A =.=
(
4 5*
2 /v
>'^
<
Q
/")
2 I
1
and we have
f 1^
^c
affc
I
1
!
^ 2 [
^=1
2 I
Vik
I
9
2 2
!
"]
THE LAPLACE DIFFERENTIAL EQUATION
387
= 0[h(k)~\
where we employ the customary notation that g(k) g(k)/h(k) is limited as k  oo plies that
im
.
Hence we
see that
I
2 <Tin Q
l ~k
x*
^ M( 2 i/#) = M n 6i
I
We
next consider a form (0')> which
cept for the deletion of the term in y and (13.5) we see that
Since d
member must
1
.
(13.7)
7ci
k\
(z)/dz
is
From
v .
regular at the point
is
also be regular at this point.
=Ak(s)
identical with
()
ex
the expansion of
~
<=
c,,
We now
the right hand write
fflrU^)
r/^
where we
define
x
(z)
in
a(z
which we use the abbreviations
z
=
2^ and
if
Since both MZ) and g k (z) are analytic about expanded into the following series
It follows
from
(13.6) that if
remains bounded, then
fc
(
)
*In deriving this identity use was
then
A(zJ ^(z^
zf~\
since
g
i>
B(zJ
^
= O(q /k) k
10
made
1/^ e
z
and consequently
of the fact that .
they can be
if z
z
in (13.5),
THE THEORY OF LINEAR OPERATORS
388
We
know, moreover, that* 2V
(
\
g* (q e")
= 2n^\ g,
\^d^
From
j
q*^
(13.8)
.
and the fact that
this
9k(q

we deduce
2
n
H2
t/
^Mq
to
c'
)
1
that GO
Returning to the main problem, namely, to show that limited, we obtain from the Schwarz inequality en I
I
on
y,
*i
^ ^L
n n ~2
/j
I
i
<2j
I
'v ^fc
/2fc f L
1
*/
I
V ^L
is
c/3
I
I
//
i/fcn
12 (r/ 2 l
V
(^~ 2 )
.Zj
I
I
o/
/n
I
l

2 !^
J

cr> i
i/n
*^/c
jjrr;
^* ^
co
v>
i//cn V
(<[>')
XT ^y
\
\
/C
M *'*
Q*" 1
II
^f
f
I
^n
Irt
I
n/n'>\Tl
(13.9)
where d.
is finite.
way we can show
In a similar
that
(13.10) (1
where C 2
is finite.
By combining proved that (<) *Thus
\z,
is
inequalities (13.7),
(13.9)
and (13.10), we have
a limited bilinear form.
if

so that 00
= Vg2
[o^ n C os
(712
W2
+
z
29
n2
[6 fcn cos (n
2) e 4
7;n
sin (n
n=2
equation (13.8)
is
readily deduced from the fact that
f*2TT I
/*27T
m^
sin ne sin
5
wm
is
mo
cos w^ cos
I
JG
Jo
where
do
the Kronecker symbol, and that
PTT I
Jo
sin no cos
mo
do =.
.
do
5
wn
,
2) 0]
,
THE LAPLACE DIFFERENTIAL EQUATION
We
389
can now state Hilb's theorem on the basis of the three results
obtained in this section:
Theorem 5. If q is a number such that z < q < z2 where and z2 are the two zeros of B(z) of smallest moduli, and if f(x) is a function of grade not greater than q, then there exists one and only one solution, u(x) of equation (13.1), whose grade is not greater than q, and this solution is given by ,
ZL
The Method of Appell Polynomials. In section 6, chapter 1, we have already defined Appell polynomials by means of their generatrix to be the polynomials Pn (x) in the expansion
V *P n\&)*' ii>\+n
/1Q 11\
2u
\JLO.1.L)
NO
Well known examples of such polynomials are the Bernoulli polynomials of rath order, B n (m} (x)* and the Hermite polynomials, h n (,z) .f In the first case we have m A(t) <=t /(e
and
l)
w
Pn (x) t=B n (m) (x)/nl
,
in the second
A(t)<=e }t The which fer,
l
*
Pn(x) t=h n (x)/n\
,
.
admits of a slight extension Thus, adopting* the notation of Shefshall refer to the polynomials G n (x) defined by the expression definition given in
(13.11)
will be useful to us here.
we
[A,(0
+ xA
1
^
(t)
n=o
(13.12)
as generalized Appell polynomials. Identifying the functions A r (t) with those given in section
2,
*For further information concerning the Bernoulli polynomials consult N. E. Norlund: Vorlesungen uber Differenzenrechnung, Berlin (1924), or H. T. Davis: Tables of the Higher Mathematical Functions, vol. 2 (1935), pp. 181272. fTwo forms of the Hermite polynomials are in common use connected by the relation fc
We
B
( !B
)=2H ,(jB/V2)
.
1
use the definition
h(x) See example
3,
section
1,
=
chapter
dn (
12.
l)"ew
dx n
ew
.
THE THEORY OF LINEAE OPERATORS
390
we
G
see that the polynomials GO
G n (x)
e=
2 xm m~o
%n
["
flow il\
\_
tl
by the formula
(x) are explicitly given
+

a lm 
X n~l
\7l
J.
Hh a* )
1
1 .
J
Conversely, the values of the functions A r (t) can be expressed in terms of the Appell polynomials by means of the relationship
A,
no
X^Gj^o)
(G;;>(0)
+ ,C
2
G
(0)

where r C m
is the 7nth binomial coefficient. Let us adopt the notation
xA Then
l
(t)\
since the functions are analytic at
\x>A f (t)
t
^
we may
.
obviously write
where C is a path about the origin. Turning now to the solution of equation (1.1), we shall first con0. In section 3, chapter 6, the solution of sider the case where p
=
the equation
A was obtained
where we
in the
>u(x)=f(x)
(z)
(13.13)
form
define / /t
and where C
+ / /^ + / / + 1
2
f
3
... >
/n
= /()(o)
a path about the origin. Operating now with A (z) upon (13.14)
,
is
we
obtain formally "
2*.G.(J!)
where the
(13.15)
(13.16)
,
coefficients are defined
W "" *
by
THE LAPLACE DIFFERENTIAL EQUATION
391
xt then, in (13.14) the function e is expanded as a power series and if account is taken of (13.17), the following solution of
If,
in xt y
(13.13)
is
obtained: (13.18)
The convergence of series (13.16) may be readily established provided the coefficients are subject to the limitation sup lim
1
k
3/n tl
=R
71 oo
A
=
within the circle t ?, where r > If A represents the maximum value of A n (t) on t= r, and if C is the circumference of this circle, then, count of Cauchy's inequality [see equation (2.1), chapter tain
and
Q
(t) is analytic

p
i
2m
J
>
R,
Hence, since r
Not only does (13.16), but
it
R. the circle
taking ac5],
we
ob
x
dt
we have
this equality establish the uniform convergence of shows that the series represents an entire func
also
tion.
case
The method which we have developed above for the elementary may be generalized in the following manner: Let us assume that u(x) may be expressed as the integral dt
where C
is
a suitably chosen path about the origin.
We
shall then
have dt
f(x) <=F(x,z)
(13.19)
where we
define
?=
(
J
n U(t)t dt
.
(13.20)
THE THEORY OF LINEAR OPERATORS
392
As in the previous case u(x) has the expansion (13.18) in which the coefficients of G n (x) are defined by (13.20). The problem of inverting the original equation (1.1) in any given instance is, of course, the determination of the function U(t). Although this determination might be made by inverting (13.20), a more effective method of procedure is the following: Let us assume that u(x) may also be written in the form
where V(t)
the function defined by (13.15). Operating upon u(x) with F(x,z) we obtain
f(x)
is
= F(x,z)
from which
it
u(x)
follows that
=^ JV(<) [F(x,z)
*X(x,t)] dt
we must have
F(x,z) ~>X(x,t)
=c"
.
(13.21)
But this equation is precisely (2.3) and hence X(x
have
X
(x,t) is a particular solution of equation (2.3) in which z has been replaced by t. Let us observe that (x,t) is not uniquely defined since to any particular solution of (2.3) there must be added
where
X
the general solution of the homogeneous equation (2.4). The restrictions imposed
by theorem 1 will be assumed here. happens that XQ (x,t) has the following expansion:
If finally it
then u(x)
may
be written
Since f(x) has the expansion (13.19),
<=TT Lni
( Jc
it
follows that
THE LAPLACE DIFFERENTIAL EQUATION
393
PROBLEMS Prove that
1.
where G n (x) are the generalized Appell polynomials given by (13.12), defines an entire function of grade not greater than R, provided the functions A r (t) are R, (Shelter). analytic in \t < R, and provided sup lim k n p/n \

n
Prove that L(x n )
2.
=P
n
(x),
S)
where we
...+
=
If the operator L(w) is defined as has a polynomial solution if and only
3.
bers X n
p nw
(
in if
define
1) no;*
/>(*)]
problem
3,
.
(Sheffer).
show that L(u)
X is one of the characteristic
\u num
(Sheffer).
.
H. Validity of the Solutions. We shall now investigate the character of the solutions which have been formally defined in the preceding sections. We have seen that the general resolvent operator for equation (1.1)
is
the
sum
of terms of the form
X(x,z)=X(z)F(x,z)
(14.1)
,
where we abbreviate
Ce^Y(t)dt
F(x.z)=^c*~
(14.2)
.
Ja
Let us
first
observe that
X(x,z) *f(x)=F(x,z) > \X(z)
/(*)]
,
(14.3)
a result immediately obtained by forming the Bourlet product [F X~\ of section 3, chapter 4. But from theorem 6, chapter 5, we see that the grade of the function g(x) X(z)  /(.v) is the same as the grade of f(x) provided X(z) satisfies certain mild restrictions. These conditions we shall assume are fulfilled by X(z) Hence we need consider only the function
=
,
.
u(x) =F(x,z) ~*g(x)
There are several cases
to
(14.4)
.
be studied, which
may be
listed as fol
lows: (1)
The function g(x) The limit
The
is
of finite grade.
of the integral in (14.2) is finite.
limit of the integral is infinite.
THE THEORY OF LINEAR OPERATORS
394 (2)
The function g(x) is of infinite grade. (a2 ) The limit of the integral in (14.2) The
(6 2 )
The case
is finite.
limit of the integral is infinite.
has already been covered by theorem 7 of chapter where it was proved that the grade of u(x) 5, docs not exceed the larger of the two numbers q and a , provided
and
is
3
)
analytic throughout the interior of the circle Q larger than q and a . The following two theorems pertain to case (b^ is
Y(t)
R
(a
its first corollary,
=
R, where
\
:
Theorem
Y(z) and g(x) are functions of then u(x) exists in the region x q respectively, general of infinite grade. 6.
If
> Q
F(x,z) ^=Y(z)/x
to the case
Y'(z)/x*
and
we employ formula
Proof: To establish this theorem
which for simplicity we limit
grade
finite
and
is
Q in
(12.4),
p =1,
+ Y"(z)/x* 
.
(14.5)
From theorem 6, chapter 5, we know that Y(z) > g(x) is a function of grade q and hence, from the assumptions regarding Y(z), n 1. Hence we where lim A/ n 1/n we have \Y (n) (z) * g(x) nQ
^M
I
'
1
find
which, from the fact that lim
=
,
nj~,
a majorant for F(x,z)  g(x)
M
in the series
l/n
=
1,
is
seen to converge for
H'_T}
\x
> Q. We thus
establish the existence of a function in a region does not include the origin in its interior. Since, in general, a lar point is included in the region bounded by the circle Q function u(x) is in general of infinite grade. If we admit the validity of the semiconvergent series
which singu
= Q, the
of the
form r(x){a n
+ a /x + a /x* +
2
,
(14.6)
as an asymptotic representation of u(x), where the expression in braces satisfies the Poincare criterion (see section 4, chapter 5), then a further extension of the region of definition of u(x) is possible. This we state in the following theorem :
Theorem
Y (z)
a function of unbounded grade but other1 and if theorem 4 for the case p g(x) is a function of grade q, then F(x,z)  g(x), where F(x,z) is defined by (14.5), yields a semiconvergent scries which is the asymp7.
If
is
ivise satisfies the conditions of
totic
representation of the integral
=
THE LAPLACE DIFFERENTIAL EQUATION
395
Q

u(x) *=F(x,z)
(
e t
{Y(zt/x) >g(x)}dt
Jo
(14.7)
in the sense of (14.6), where r(x)
than
is
a fiinc+ion of grade not greater
q.
Proof.
Representing by S n (x,z) the operator
S n (x,z)
=2
l)
(
m ^Y ()n ^
mi
let
us consider the remainder operator,
R n (x,z)
S n (x,z)
Making: use of equation (3.7) of chapter
5,
we
obtain the in
equality,
R
x\
n
(x,z)
fe Y^ xt
J
By
*f(x)\
P(0)
jo
Y (n) (t)
hypothesis
is
bounded by M,,A n nl, lim
M
l/H
=
1
and
njn
satisfies the
assumptions of theorem
4.
Hence the inequality may be
reduced to
R n (x,z) P(0) e *M A n
x\
g
{ei*M n An\
+
w
9
w
where T(x) is a positive function of zero grade which dominates both \x\ and S(x) \\x\ We then conclude that the function r(x) of (14.6) is dominated by a function of grade q and hence is a function of grade which does .
not exceed q. In order to attain the integral of which (14.6) is the asymptotic representation, we apply Borel's integral (see section 4, chapter 5) to (14.6) and thus obtain,
F(x
9
z)
= (I/a;)
^er^Y(z
Jo
The
integral representation of ii(x),
(14.7), follows as
an im
mediate consequence. As an example illustrating Theorem 7 consider the equation
:
THE THEORY OF LINEAR OPERATORS
396 Since is
we have X(z)
=1
Y(z)
z,
= 1/(1
z), the solution
given by the operator,
u(x)
=
s
e* (I
z)
(e
ft
/(l
3!/(l
t)
}dt
3 *) s*
+
}
=
If we set f(x) 1 we replace z by zero and thus obtain the asymptotic development of the integral,
C
u(x)
Jo If,
however, we set f(x)
u(x)
^ e**{\/x
This series (14.6),
u(x))
= =e
c
1 !/ (1
is
where g(x)
=
r
<
ax
a)
,
a:
2
a
<
1,
f 2 !/ (1
we
get
a)
V
the asymptotic representation in the sense of ?"*, of the formal solution
tu
(l
a) Jo
We The
turn next to a consideration of cases (a 2 ) and (& 2 ) above. numerous theoretical difficulties, but a
situation here presents partial solution is furnished
by theorem 10, chapter 5, where condiwhich are sufficient to secure Borelsummable equivalents for u(x) when g(x) is a function of infinite grade. In general the grade of n(x) is infinite. The reader is referred to theorem
tions are stated
10, chapter 5, for further details. If the limit of the integral (14.2) is infinite [case (6 2 )], this is
equivalent to adding
roo
to the solution obtained in case (&2 ).
Y(t) by theorem
(14.8)
The conditions imposed upon
chapter 5, assure the convergence of the integral in (14.8). If, then, g(0) exists, the function (14.8) may be added to the solution obtained in case (a 2 ). The grade of u(x) is infinite. Finally let us consider the convergence of the factorial form of the operator (14.1) as given by the expansion (12.2), which for con1. venience we shall specialize for the case p 10,
=
THE LAPLACE DIFFERENTIAL EQUATION Since the nth term of the series
w n (z)e n *=
(D
n+
1)
...
readily seen to be of the
is
D=
form
l)D> Y
(D
_ D)(l where
397
D/2)
d/dz we are led to a consideration of the operator,
A n (D)=D(l
D)(l
D/2)...\_l
Product operators of the form,
^
where
converges, have been extensively studied by J. F.
l/\a, n
Ritt,* but it is clear that the condition
in the case of the operator lim
A
n
(D)
imposed by him
is
not satisfied
.
n(j~,
We may
then proceed as follows: Introducing convergence factors e D/m into the product
A
n
(D)
A n we have,
D
^[D(lD)e(lD/2)e /*..{lD/(Vrl)}e" n ^] \y e fl{lH/2...l/(ni)}
f
(14.9)
For sufficiently large values of n the product in the brackets may e CD /r(D) f C CD e n (D), where C is Euler's constant be written, and s n (D) is a function that tends to zero as n > oo.f Similarly the Cn D in the exponent of e may be written log n coefficient of where the difference C n C becomes vanishingly small with increasing n.% We can then write (14.9) in the form,
+
A n (D)
,
= [l/F(
chapter 5 and recalling that l/F(D) is analytic in the entire plane we see that A n (D) > Y(z) is a function of grade less than or equal to Q provided Y(z) is of grade Q. If in 3 where Q is a positive number, we shall have e*? particular Y(z)
Employing theorem
6,
=
A n (D)
~>
from which we
Y(z)={ infer that,
A n (D) *See section
6,
chapter 1 and section 10, chapter
fWhittaker and Watson, Modern Analysis, 3rd d.: p. 235.
6.
ed. (1920), p. 236.
THE THEORY OF LINEAR OPERATORS
398
More generally let us consider the case where Y(z) is any func~ tion of grade Q. Operating upon Y(z) with 1/T( D)e D(C Cn) and recalling theorem 6, chapter 5, we obtain a new function R(z), which D}0 * n we is also of grade Q. get Operating upon R(z) with e
R (z
log n). Since R (z) is of grade Q it is dominated for large values of the argument by a function of the form e^P(z) where P (z) is a suitably chosen positive function of genus 0. Hence we can write R(z log n) rg n Q eV~P(z ~4 log n) for n sufficiently large. But since P(z) is a function of genus zero, P(z log n) is dominated by n8 where d is an arbitrarily small positive constant,* and we have ,


+
,

R
(
Z
log n)
We
<

n Q+t
e<*
s .
are thus able to assert that
d>
A n (D) *Y(z)=o(n<>+*), provided Y(z)
The
a function of grade Q. much more complicated
is
situation is
infinite grade.
if Y(z) important special case, however,
An
(14.10)
,
is
a function of
is
furnished by
the series,
Y(z)
= + o,
where a n <= O(\/n
dtC*
!),
+
o.2 e
2
~H^a,.c
wj
+
.
(14.11)
,
which from the example of section 8 is seen xu(x), that is when a n u(x ~j 1)
to
=
include the classical operator
Operating with
n shall
1, and have
(
l)
w l
A (D) and noting that A f(m + l)^ //^^)/^^ n
5
= (_i)i mn r^{l + a +a Under the hypothesis tion within the braces
is
(w f 1) (w
nf2
that a n ^=
entire
O(l/w
nfl
We now first
=
mr (D) * G w 1) if
+
(w f l)f VI n
!)
it is
we
!a w ... }
.
clear that the func
D!]

(14.12)
turn to a consideration of series (12.2) when p applies.
*This follows from the fact that f
n,
and bounded with n and hence that
examine the case where (14.10)
limP(z
m^
if
w^
+ 2)6^/2! a +
A H (D) *Y(z) =0[c"V(w Let us
n
log?i)/w 6
= limP(s
H oc)/e 3 *
.
=
1.
THE LAPLACE DIFFERENTIAL EQUATION
From
the operational identity
w we
399
n
=
(z)e
(w1)
(_l)
lA n (D) > Y(z)
write equation (12.2) in the form,
X
(x,z)=X(z){Y(z)/x+ (A^>Y)/x(x
+ l)
(14.13) + (A >Y)ll/x(x + l)(x + 2) + (A > Y)2!/ff(s + 1) + 2) (x + 3) + l
(a;
2
}
.
We now recall (see section 11, chapter 6) the fundamental convergence fact associated with a factorial series
namely, that this series converges with the exception of the points 0, for values of x the real part of which exceeds a value, 2, 1, ,
CD
A,
called the abscissa of convergence.
verges then A
^
and
If the series
determined by the
is
A
*=
^
a,Hi di
limit,
p
/logp
lim sup log If the series
A
converges then
A ~
.
and
is
determined by the
limit,
lim sup log /'t^j
The abscissa
of convergence of (14.13)
twlo cases already discussed:
is
equation
(1)
easily obtained in the
(14.10);
equation
(2)
(14.12). on
In the first case the series
^ no
Hence employing the abbreviation
i
tt''=p*'V(tf
o
From
this
we
+
nQ + * diverges for q
=Q
f
(3,
Q
f
d
>
1.
we compute,
i:
obtain
We are thus able to conclude that the abscissa of convergence for the case where Y(z) is a function of grade Q is not in general smaller m e ~, where is a posithan Q ^, although the special case Y (z)
=
+
tive integer,
shows that
it
may
be
GO.
m
THE THEORY OF LINEAR OPERATORS
400
00
In the second case, equation (14.12), the series
2 n=o
e nz /(n
1)
!
obviously converges. Hence from the function, 00
/
e/
(
!)!
p !}{!
+ e*/ (p + 1)
n=p+i
we compute
the abscissa of convergence to be,
sup log / (p ) /log p 

=
liml (p '
/>=GO
We
+ l)^
log p! l/log p
=
oo
are thus able to state the theorem:
Theorem 8. If Y(z) is a function of grade Q then an abscissa of convergence exists for series (14.13) which is in general not smaller than 1 Q if Y (z) is a function which has an expansion of the form oo. (14.11) then the abscissa of convergence is k
=
15.
;
Operators with Regular Singularities.
we have made hitherto of the inversion of which we shall write conveniently in the form, that
{A*(z)
+xA,(z) +x*A,(z)
H
In the discussions the Laplace equation
\x'A p (z)}>u(x)*=f(x)
p> where the
A
%
(15.1)
(z) are the functions,
A
%
(z)
=I>*i2 n 71
it
,
,
>
=0
has been necessary to impose the restriction A V (Q) ^ O. Perron has formally removed this restriction in the following .
manner:* If in equation (15.1)
we make
e ax v(x)
u(x) then,
from the fact that
z n >
the transformation,
u(x)
,
&*(z
+
a)
n
> v(x), equation
(15.1) becomes,
SzM^z + a)
*v(x) ^e^f(x)
*See Bibliography: Perron (3), in particular p. 351.
p. 41.
.
(15.2)
See also Sheffer, (1),
THE LAPLACE DIFFERENTIAL EQUATION
401
Hence in the transformed equation the coefficient of x p has the value A p (a) when z and the auxiliary parameter a is then to be so chosen that A p (a) 0. The resolutions of the difficulty in this manner, however, is only apparent as may be seen from the following elementary example
=
=
:
If
we make
>u(x)
xz)
(1
=x
use of (15.2) and (2.24)
.
we
get
/tOO
u (x)
e
{
(z+a) e**/ (z\a
s)
2
}
ds >
lx
xe~'
.
Jo
But
we
if
recall the operational identity,
F (x,z)
> xe
=
e
nt
xF (x,a)
f
F/ (x,a)
,
and apply it to the present problem, we observe from the singularity of the integrand that the solution cannot be attained in this manner. The nature of the difficulty is revealed, however, if we sot a t= and observe the expansion X(x,z)
=
C{ze**/(zsy}ds
=l
e$li(e<>)
,
Jo 2
/2 2!
+ # /3 3
3!
+
}
,
= xz and C
.5772157 (Euler's constant). The appearance of log z in the inverse X(x,z) suggests that the resolution of the problem is to be found in an interpretation of the logarithmic opera
where $
tor
which was discussed
The
in section 11, chapter 2. object of this section, therefore, is to discuss the operational
theoretic problem involved here for the case where the multiplicity does not exceed p. It will be necessary, of the zero of A p (z) at z also, to make the assumption that the generatrix equation (2.4) has
=
0. only a regular singularity at z We shall find it convenient to begin by proving several lemmas
Lemma
1.
The operator ra
^~eX8 Jo
where
v is
:
any constant and
(zs)~*ds
,
R(x)
>0
any function analytic about the
origin, is equivalent to the operator
where
z~
chapter
v
2.
is
the fractional integration operator defined by (7.1) of
THE THEORY OF LINEAR OPERATORS
402
Proof In order to establish this identity let us consider the solution of the following differential equation of infinite order, :
{z*Q'(z)+xz* where Q(Z)
is
e
(z)}>u(x)=f(x)
(15.3)
,
a function analytic about the origin and not vanishing
there.
Since this is an equation of the singular type excluded from the general theory, we apply the transformation of Perron and obtain the limiting case where a =0. We thus easily find
arid
hence the resolvent operator
^(zsY'ds To obtain a second form
for this operator
(15.4)
.
we expand
the integral
formally as a series in l/z and thus find,
X(x,z)
= (1/0(3)}
("
= (X/Q(Z) }z'(l + y/xz + + When
zi
1
is
replaced by
its
v
041)
operational equivalent,
this series reduces to
X(x,z)x={l/ Q (z)}
.
f*e^'*>{(xt)**/tr(v)}dt
In order to effect a further transformation z
where
X
Q
v
v
>u(x)^z *{X
(x,z)
(x,z) is the operator (15.5).
X
>f(x)}
We
(15.5)
we now compute ,
thus obtain:
403
THE LAPLACE DIFFERENTIAL EQUATION d r* dx J
0~ v
(X
dt
I
<:
where we use the abbreviation &(x) Making the transformation, (# tain the last equation in the form d
r*{>(s)
=
[l/e(2)]
O/U
s)

~
/(.r).
y)/y,
(1
r
rum,)"
'
(15.6)
.
this
we deduce
ob
^
*
J,
From
we
that u(x)
=z~
v
> [ft(x)/x}
9
and hence
that (15.7)
Let us verify this conclusion directly. We the Bourlet product (3.1) of chapter 4 that
*Z V
= Z*Q'(Z)
note by use of
first
+xz* Q (z)
(15.8)
.
Computing likewise the operational product,
we
get,
A(x,z)
=
(Q'
which establishes the fact that l/[(>(z) Q'(Z)

+X Q (Z)
x]
is
the inverse of
.
Combining this with (15.8), we immediately derive (15.7) as the inverse of the operator Z V Q'(Z) f XZ V Q(Z) If we finally replace I/Q(Z) by
Lemma
2.
The operator
A, (x,z)
= r (z) Ce **log(zs) (zs)* ds <
,
Jo
R(x) is
>
,
equivalent to the operator,
B (x,z)=z l
v
]vgz+{
,
a function analytic about the origin and operator defined by (11.1) in chapter 2.
where
(p(z) is
Proof: Employing the abbreviation,
O=
*er*
Jo
z~ v
log z is the
THE THEORY OF LINEAR OPERATORS
404
we may
=L
write,
L=
f
e xaf
f
M, where we v
define,
v s/z) ds
log32 (l
and
,
Jo
M = z fV" log(l v
v s/2) ds
s/z) (1
.
Jo
Limiting our attention to the
first
element,
we
shall
have
/^co I
v
l
s/z) 'ds>{z \ogz
e"(l
Jo
> {2" log
<=
=
Z^ f(x)}
,
f{! + "(* OA + v
f'f(8)dS
*/ c
rWxY*(t
S )"*{ V (v
1)]
,
i)]
,
1)
*^s
where y(0 is the psi function denned by v(v) (t Employing the transformation 1 y
=
= r'(j>)/T()>). s)/(*
s),
we can
write this equation, as follows: ]
r
,
*'
1
(i
u^^
J log
j ( J
JX
Q\I
\&
(
\
Q \I
/y X/
re F(v
<>
/"*
1
I
Vx J
T\ 1)
where we abbreviate p
[(^6)
V1 ft Q tx/o
(x s) /x Similarly for the second element
yY~*
(1
y)]} dy/x
W^ (I
w)" 2
l
S(1
.
we
get
C
f
V*(^)
tn
(l
v s/z)~ ds
,
y}dy
THE LAPLACE DIFFERENTIAL EQUATION
1
x
f
n,,,,,. I2!
(**>"
I
^w
J
y
omHi
ds
I
r c i x
n^F
J,
(&)
^
T(r
gg 1)
.f
~ Y t
J,(
)
X
= r
r(r+l) 2!X /H42
(*Q"V
I
r _L(?) ,
405
8)
v
(i)
(t^)"
fiog(^)(^/o"(*
x
/
.
v2
2
^
rf*
,
.
Js
Making the transformation 1 the last equation to
y
=
(i
s)/(x
s),
we reduce
where we abbreviate as before (x a)/x. Now combining L and M we obtain >
O
>/(*)
= (L + M)

r(v where we abbreviate,
For the evaluation of I(p) we note that
= ~Tvd
*
p* (1
2
t/)"
^
J {!/(.
!)}
.
=~ __
d
_
P) ("
D
(15.9)
THE THEORY OF LINEAR OPERATORS
406
Recalling the functional equation
y(y
+ 1)
=y(v)
+ l/v
,
and introducing (15.9) into 0, we get
 W
c *f( S )(XS)vi (v(r) ^=7^
^
Jc
ds
log(x
s)}
, >
The lemma follows from this fact, since
Lemma
The operator
3.
A n (x,z)
=
f
V" logn (z
s)
(zs)*
ds
,
>0
R(x) is
in
,
equivalent to the operator
B n (x,z) where
of chapter
about
z~
z
v
log"
=
z>[
and
z~
v
,
log" z is defined
by (11.10)
2.
Proof: The equivalence stated in the lemma can be justified by an analysis which differs from that given above only in the introduction of a greater complexity of detail. If, however, we note the principle introduced in the derivation of (11.8) of chapter 2, namely that differentiation with respect to the parameter v is justified when the operator involves ~ v we derive from the operational identity ,
A
(x,z)
=B(x,z)
=
2~
v

the desired result
We
now
are
Theorem
9.
in a position to prove the following
If the differential equation (2.4)
theorem
:
has the origin as
a regular singular point and if the numbers of the sequence i e= 1, 2, p, where the /U are roots of the indicial equation,
{At}
,
,
D/2I
+
do not differ by integers, then the formal resolvent operator for equation (15.1) is given by
THE LAPLACE DIFFERENTIAL EQUATION
X
=
(X,Z)
z*'
1
lh (Z) >
407
[Z^X^Z^X]
+ z~^ y
(z) *
2
&*x
2
(z) /x]
+
...
(15.10)
,
where the
z Xi x,(z) are a fundamental set of solutions of equation l v z~^~ the (2.4), iji(z) are their Lagrancje adjoints, and z~ is the fractional operator previously defined,. If r of the values {hi} differ by
integers, then in general r
members
of (15.10) will be replaced by
terms which contain operators of the form ZP log n z y (z) > [z x
where
v
8
1
log**
z
x (z) /x]
,
w
and
71
integers,
defined by (11.10) of chapter 2 and y(z), x(z) are the origin and do not vanish there, about functions regular z~
log z
is
Proof: In order to prove this theorem
let
us
first
note that the
function,
which appears
X
(x,z)
in the resolvent operator,
=e* s
fV W(z,t)
dt
,
R(x)
>
(15.11)
,
J on
may
be written in the form, (15.12)
are roots of the indicial equation. We assume that the fa are not integers. The functions Xi(z) and yi(t) are regular about zero and do not vanish there. z s, and substituMaking in (15.11) the transformation t for we member of obtain the (15.12) W(z,t) right ting
where the
differences
fa
A*
=
,
(15.13)
Finally taking account
X
(x,z)
lemma
oi'
^i.z ii
A
' 1
1,
y (z) t
we
get
from
this equation,
\z^ x>(z)/x\
,
which establishes the first statement of the theorem. Xj If we now remove the restriction that the differences A are not integers and assume instead that r of the characteristic numbers differ by integers, then the fundamental solutions X (z) and t
t
THE THEORY OF LINEAR OPERATORS
408
v contain terms of the form log" z x (z) and t log'' 1 y (t). r Hence terms of W(z,t) will be replaced by a function of the form
Yi
(t)
may
'
= z t^ 2 # mn x
(z,t)
where the functions & mn
(z,t)
1
1
log" z
(z,t) log"' t
,
are regular about the point (0,0).
Furthermore the functions # mn
have the form
(z,t)
2
#r(2)
7" r
(),
r_i
from which zx
Z
r
it
follows that terms will appear in (15.13) of the form
f *e* 8 log" (zs)
(z) log" z
(zs)
1
*
T (zs)ds r
.
Jo
But by means of lemmas 2 and 3 we sion is equivalent to the following z 1 * log'7 z
T
r
see at once that this expres
:
x (z) > [z
Z
r
(z) log" z/x\
.
These results combine into the second statement in the theorem.
We
shall
now
apply the foregoing results to obtain the formal
inverses of two linear equations. The first is a singular integral equation of Volterra type, the second a difference equation which is used by G. Wallenberg and A. Guldberg (see Bibliography) as an illustra
example throughout their treatise. We have previously examined the homogeneous case of the latter example in section 4.
tive
Example
1.
The
integral equation,
/?!) x\ lap
+
(2a
+
2ft
+2
e*<
y)x
+
3
y)x
2a;
2
2( r '> 
]e
,
has as
its
(15.14)
resolvent equation the hypergeometric differential equaz), z) (2 z) (3 (1 by the reciprocal of n(z)
=
tion multiplied
namely, ;*)
+
!>
<
In order to simplify the discussion p 7 resolvent equation, gle specialization, namely, a
we
= = =
1,
shall consider only a sin
for which
we
obtain the
THE LAPLACE DIFFERENTIAL EQUATION
A X,(z)
fundamental set of solutions for
= I/ (1.?), X
this equation is
z/(lz)
log
(z)
2
409
found
to be
.
Solutions of the adjoint,
v
dt* Sire
'
n(t)
{
v
dt (
computed from the
'
n(t)
[
n (t)
I
relations,
Y,(t)=.X,(t)/\W(t)
Y
2
(t)
^X,(t)/
W(t)
W(t) is the Wronskian of the solutions X,(0, ^ 2 (0. The values of the solutions of the adjoint equation are explicitly,
wlhere
In terms of these functions, the Cauchy function becomes,
Hence the formal inverse cialization,
of (15.14), subject to the
assumed spe
becomes, C^e* 8 Jo {
W
[log z/
(
(z,
zs) ds > f (x)
lz)
,
C^e " n (zs) dx
]
Jo
**n(z
s)
o
Now
invoking theorem
we may
9,
write this equation in the
form,
u(x) t=n(z) > {[z
1

Z
(15.15)
We now
note that zn
/(lz)
>
g(x)
^
f
6
s
(n) (y
Jo
and we
shall
employ the abbreviation, 9(x)
=
J
(
Jo
e 8 f(x\s)ds
.
(15.16)
THE THEORY OF LINEAR OPERATORS
410
Returning to (15.15) and making use of (15.16) and (15.17), we may write u(x) in the form,
n(z)
n (z) >
Making use
1
{
[2
log z > g' (x)
]
log> {g)x)/x}
,
/x}
of formula (11.1), chapter 2,
we
then obtain
(*)=*<*) >
,<,)
.
j;
a
(/() /
log(x
where we abbreviate
*)] h(x,t)dt
= g'(t)/x
fe(a:,t)
g'(t)/t
,
+ g(t)/t
2 .
In order to interpret this last result it will be necessary to employ theorem 4 of chapter 6. By means of it we are at once able to compute,
z~* Jc
[^(1)
s)] h(x,s)ds
log(.r
\_h(x 9 c)log(xc)1/c
[V(^s)
(,s)
+ &/(*,*)]
log (a;
5)} d5
.
The derivatives ^ 2 and z which also appear in a(z) are similarly attained and the formal problem is thus completely resolved. 3
Example
2.
Let us consider the formal solution of the difference
equation,
x(x+l)u(x+2)
2x(x+2)u(x+l)
+
(*+2) (x+I)u(x) =f(x)
which, replacing n(x\n) by
e
ny
,
~> u(x), can be written
(15.17)
THE LAPLACE DIFFERENTIAL EQUATION [
f
x 2 (e**
+ l)\x(e*
2e?
z
4e z
f 3)
+ 2]
>
411
u(x) ==f(x)
.
(15.18)
Since wte have already considered the homogeneous case of this equation in example 4, section 4, we shall content ourselves here with the determination of a particular solution of the nonhomogeneous problem. The generatrix equation, s
2e*
(e?
+ I)X"(z) +
(e
w
+ 3)X'(s) + 2X(z) ^0
4c*
,
(15.19)
has z
=
X
1)
=
1,
A
as a regular singular point, and from the indicia! equation, 2 2A 0, we obtain the corresponding indicial numbers
+ =
jlt=2. set of fundamental solutions of (15.19) Xi(z)
=
l)*/e**
(er
X
,
(z)
2
=
seen to be,
is
(c*
Computing the corresponding adjoint functions, we get
Y
t
l
(t)=c /(e
ir
t
,
r (t)=
I)
c'/(e'
2
2 ,
and from them the Cauchy function,
W(z,t)=X,(z)Y (t)+X l
=
er*
s
e t (e s
Employing this expression, solvent operator to be,
X
(x,z)
= (e
1) (c
we
z
(z)Y,(t)
,
3
l
l) (e
*')
find the explicit
form of the
^00 8
e"{e(l
1)
.
re
3
!) }
c)/(e'
,
Jo
>
.
Since s z appears as a cubic singularity of the integrand write the operator in the form,
X
n
(x,z)
=
(e~
1)
we
f{e"
Jo
+
(s
4 (e*
2) (e
1)
M
f 2*
{
(e
e
^CO
c
3
[e*V (s
)
)
]
ds
Jo
_
/tOO
(#**
xs
e *)
[e
Jo
where we abbreviate
=
(6~
28
vV
e"*) (6
s
/2 (sz)
2
]
ds
,
THE THEORY OF LINEAR OPERATORS
412
With attention
to the identities,
Af(x)r=f(x+l)f(x)
,
we
,
get as the formal solution of (15.18) the function,
u(x)
e
(c**
**[
("(e
y
)
Jo
+
(sz)
+ f [{(*
2
*)
/J
+ e')/2]/(*
(e**
2
/(*
+
2)
zy] ds
)[/(<
(x
A f(x)
>
2)

(15.20)
=A
where Q(x)
()
A^x
(
+
A,x 2
The
.
Q (x)
two terms of
last
con
stitute the general solution of the homogeneous case; the constant is to be so determined that (15.17) is satisfied at the point x *= c.
A
()
Let us evaluate (15.20) for two special functions, namely f(x) c= x.
~ 1 and f(x) In the
first
we
case
The constant A
is
get,
determined from (15.17) by noting that u(Q)
In the second case, where f(x) 
we
x,
=
get
/U}
u(x) <=
I
e x *{
c
e*/(l
s
2
)
{~L/s
}ds
xlogx~\x
Jo
Integrating by parts
we then
obtain, x.OO
'JO
Jo
o
/ 1*
"j
We now
"I
//
7*
loo*
1*
I
'Y _l
(~)
( fy*\
(
~\
^\ ty~\
\
note the definition,*
~
x
JO
Jo
where y(x)
satisfies the equation,
^(x~{l)
y*(x) z=~L/x
,
v'(l)
=
C
(Euler's constant).
Hence (15.21) becomes, u(x)
=
y%
x y(x)
f
x

*See Davis: Tables of the Higher Mathematical Functions, p. 277.
vol.
1
(1933),
THE LAPLACE DIFFERENTIAL EQUATION
We
must now determine the value
We
original equation is satisfied. 0. Noting that lim x y>(x)
G
in
Q(x), so that the
notice that this requires that u(0) 1, we obtain for the desired solu
=
=
A
of
413
.r=o
tion the function,
u(x)
=
x y(x) f x
1
A
\
L
x
\
A
2
x2
.
It is possible, of course, to obtain these particular results from the expansion (x,z) as a power series in z. This may be accomessential without plished difficulty and to four terms may be shown to equal:
X
X
(x,z)
=%
+1
I> y(x)
+ 2x y(x)

f 2 f 2x
j
x
PROBLEMS Show
1.
that the equation
(2x + 7)
(2^fl) (3^+4) u(x + l)
(3ar
+ l)
u(x)
= 2x+l
has the solution
X(2x + I)/(Sx + l)
u(x)
In this solution
'Tr(x)
is
+ ir(x)[(2x + l)
(2x*H3) (2a
+ 5)/(S
a function of unit period.
(Markoff and Selivanoff). 2.
Express the integral equation f*[t2
+
3t(t
x)
(f
a?)2]tt(t) dt
= f(x)
Jo in the
form
Hence show that the equation has the particular solution u(x)
=
(
V3?6)
where we write X x == 3
Show that 3. general solution
if
{z****
+
f(x)
V3*
=
* (z^/x > /) ,
xi
r
\=3 ,
ft
i1
r(/i 4.
2)
2,
V3~
>
tx) +
(t
/)
,
.
then the equation of problem 2 has the
#f1 3
02
(Vx 
60
+
X
, '
= V3
6
Express the integral equation
/'< t/0 /0 in the
>
l
s**+2
2 x) /2!] u(t) dt
= f(x)
form
=/(*>
.
.
THE THEORY OP LINEAR OPERATORS
414
Hence show that if f(x) is a function which vanishes together with its detwo orders at x Q, then the general solution of the integral equation is given by
=
rivatives of first
u(x) 5.
=
(1/cc)
sin
/'Jo
[log(zA)] /(*) dt
Derive lemma 2 of this section from lemma 1 by invoking the principle
introduced in the derivation of (11.8) of chapter 6.
The ordinary (a
is called
=zac 3
*u(x)
+
a l xz
%xz
+x
2
z*)
>u(x)
~x
.
2 .
differential equation
+ a2 x* z*/2l
~\
+ an x
n
z n /nl
Eider's equation (see next chapter). Discuss
methods of 9.
+ xz + x*z*)
Find the solution of the equation (4
8.
2.
Solve the equation (1
7.
.
> u(x)
its
this section.
Discuss the solution of the equation
+\u(x)=f(x)
)
.
=f(x)
general solution by the
CHAPTER
IX
THE GENERALIZED EULEK DIFFERENTIAL EQUATION 1.
The Functional Equation. The
OF INFINITE ORDER
following rather general type
of functional equation,*
u(t)
<
tgv
where f(x)
1
(1.1)
,
known
function of the form x*~ v g(x), g(x) being about x where the c, and //, are any set of constants, and analytic 0, leads to the integration of what is referred to in the literature of differential equations as the Euler equation:'^ is
a
F(x) >=a u(x)
4 a^xu'(x)
x)
where the
(1.2)
,
=
>
r)
and
..
co and F(x) is analytic about x bounded as n be seen that (1.1) generalizes the functional equation,
o n are
It will
+
+u(x)\
also includes as a special case Abel's well
known
,
0.
(1.3)
integral equa
tion,
/' ^o
[u(t)/(xty]dt = f(x)
(1.4)
.
In order to exhibit the connection between (1.1) and (1.2), we n(ii.x) can be developed into the following Tay
assume that u(t) and lor's series
:
u(t)=u(x)+
*This equation
(tx)u'(x)
+
~
u(x)
related to a functional equation studied by P. J. Browne: (3), pp. 63198. See also C. Popovici: Comptes Rendus, vol. 158, pp. 18661869. Also, American Mathematical Monthly: Question 34 (1917, 134, 341; 1920, 114, 301, 405,460; 1921, 19) and problem 3076 (1924, 254). fFor the finite case see: E. L. Ince: Ordinary Differential Equations, London (1927), pp. 141144; pp. 534536. is
Atmales de Toulouse,
vol. 4
THE THEORY OF LINEAR OPERATORS
416
When these values have been substituted in equation (1.1), we x v ~ l f(x) obtain a differential equation of Euler type where F(x)
=
and
2 Ml
w
iO

]
t=0
In equation (1.3) these values of a* are seen to reduce to
(_l)[l/(n+l)
l/(32^)]
1/6
= 0,
a
,
and for Abel's equation we have, v)
(1.5)
,
It now happens rather curiously that all the formal aspects of the solution of equation (1.2) are preserved in the discussion of a considerably more general equation which we shall call the generalized Euler differential equation of infinite order. This equation we shall write in the following form :
4 (&!
+ b}X)u'(x) 4/2! + d 4 c
(a2
+bx 2
~\
c z x 2 /2l)u"(x)
2
,
4
b,x
3
8
(1.6) It will
be convenient to abbreviate this equation as follows
G(x,z) >u(x)
where G(x,z)
2.
will
is
=g(x)
:
(1.7)
,
the generatrix function,
The Homogeneous Case of the Generalized Euler Equation. It first to examine the homogeneous case of equa
be desirable for us
tion (1.6),
G(x,z) *u(x)
Let us
first
(2.1)
.
assume that the solution can be expanded
in
a series
of the form,
#A
u(x) !=zx^
+ It will
/ (A)
A (A)
+ A
(

2
f.^/^4...)
\
be convenient to adopt the following notation
=a + M + c*M
= M 4 &
2
A (A
1)/2!
1)
1) (A
2) /2
1) 4 c 3 2 (A
+ d A (A 4
/ 2 (A)
^MU
1)
+ WU
U
+ cU(A 1) (A
1) (A
A_1)U 2)U
2) (A
(2.2)
.
:
+
2)/3!
.
!
3) /3
!
+

,
2)
3)/2!
+
,
(2.3)
THE EULER DIFFERENTIAL EQUATION or, if
we employ
the abbreviation,
A<>^U /(/)
We
1)U
2)
= cu<> ,+
ft,,^
...
(n
w
(A
^+
1.
+ 1)
W
can then state the following theorem
Theorem
417
n+2)
,
/2!
+
:
If the equation,
/oU)=0
(2.4)
,
has a set of roots, A,, / 2 i r which do not differ from one another by integers, and if for each one of these roots the infinite matrix of i 1, 2, quantities /\ (1 exists, then equam) 0, 1, 2, ,
,
,
,
~


,
m=

,

tion (2.1) possesses r formal solutions of the form,
u (x)
=x
x
'
(^o f
yjx
+ ^/o* H
Wz + n
h
'
'
>
(25)
>
where the q> n are explicitly calculated from the formula,
3) ../
(A_w)
^
.
(2.6)
Proof: The proof of this theorem
is
obtained from an explicit
substitution of (2.2) in (2.1) and the subsequent use of the follow
ing identity
:

=
=
(im)
(lm1)
(/I
^^
2)
..
(A
m
n (2.7)
where the
MC
t
are the binomial coefficients."
"This is referred to by G. Chrystal in his Algebra, vol. 2 (1889), p, 9 as Vandermonde's theorem, although it was probably employed before the time of C. A. Vandermonde (17351796) who stated it in his memoir: Sur des irrationnelles des differens ordres avec une application au cercle. Histoire de VAcademie Royale des Sciences (1772), pt. I, Paris (1775) pp. 489498.
THE THEORY OF LINEAR OPERATORS
418
This identity is easily established as follows: Taking the nth derivative of the function x x x~ m and setting x 1 we get by the rule of Leibnitz
=
:
m x (x x )
lim
=l
But an expression lim*=1
dn .r
x
m
We now
(n)
w
C wA (n 
1)
4
1
n
identical with this
= (lm)
m
(A
C,m( obtained as the limit,
is
U w
1)
substitute (2.2) in (2.1)
(lm
2)
and arrange
w+1)
coefficients in the
following manner:
G(x,z)
u(x)
=
(2.8) !
+ cU
c 2 ;./2
(3)
/3!
+
)
f 3
!
+^
+ + (a, + 26 A + 3c A >/2 4 4
4_
( a,^
(2) 4 6 2 A
+ c.a
(3)
/2
!
(2
!
c4
>
4
(2M43W
4 (., 4
(2)
<
4
6rf,A
>/2! 4
(2)
3)
44c^< /2!45
3W 4 6c,A 4 (6 3 4
(2)
4^/2!
(3 (4) 44. (a 3 A 4 & 4 A
104/ (3) /3!
/2! 4
+ 10cU
4 (c 4 /2
>
!
!
3
(2) 13 44. (a,A 4 6 3 A
/3
)
(3)
3
2
(4)
c B A (5) /2!
!
4
4
<2)
4
/3! 4
54A/3
!
4
.
)
(x) /x
^ ()
)x
)ar
9/"(a:)
THE EULER DIFFERENTIAL EQUATION Noting the fact that the mth derivative of
+
(ra+2)
!
m* 93/2 x !
+
419 is,
m>
],
,
the columns of the above array for the coefficients of x m and find that the multipliers of &,, &,, c are sums of the form given in the left member of (2.7). Transforming these by means of the identity we obtain for the expansion of (2.8) the following series:
we sum
t
G(x,z) *u(x)
Equating the fntt)
=x
x
{/o(/0'/>of [/iU)
coefficients of x~ n to zero,
+ /nlU
D^i
+ /n2
2)^Hh /O (^1^)^ = 71
and solving for
+ /o(A
qr n ,
n
>
0, in
terms of
/oU)=o and the determinant
(2.6).
= 0,1,2,..
r/\,,
we
,
,
(2.9)
are led to the equation,
,
These results are expressed in the state
ment of the theorem. As in the analogous theory
of the Fuchsian equation of finite order (the linear differential equation with regular singular points)* as it relates to expansions about the origin, we see from (2.6) that the formal solution (2.5) cannot be attained if two of the characteristic
/oUi
numbers differ by an m) would vanish.
To avoid this difficulty the form, u(x)
=x
x
log x
0/>o
integer, /
t
we assume
+ ?/!/# +
A,
=
m, since in this case
the existence of a solution of
V'2
Substituting this function in equation (2.1) we find after a tedious calculation similar in detail to the one given above the follow
ing identity
:
*See E. L. Ince: Ordinary Differential Equations, pp. 365375.
THE THEORY OP LINEAR OPERATORS
420
G(x,z) *u(x) =rMoga{/,, (;.)'/'+ [A(A)i,'
aa {/o'(A)v'o
+
[A' (A)
V'o
+
[A' CO
+ // (A
+/
( ,
/'.
+
A,'
Ul
1)
vi
+
A,'
l)
(A
Vi]/a;
) 2
2)
(A
y. 2 ]
/Z f
}
be the smaller of the two characteristic numbers A and /j in terms of iy n by the algorithm stated we can compute in (2.6). And since A f p is also a root of /<>(/) 0, we get for the determination of
=X+
p,
i/>,
=
,
Dvi
f,(l+P)
+
//i
UHP
+/o(A+p
1) yi
+
(2.10)
=
If A is not a double root of /,,(A) 0, this system can be solved If A is a double root, however, and ty uniquely for
.
,//
m+
=
u (x)
+
.r
x'
x
1
Iog
.r (
log x (
+
+ ^/a; + * v,/ar + i/'sA +
2
.K"
2
f
2 /a;
/>
2
+
(


)

H
)
I
)
2
If this function is substituted in equation (2.1)
=*Mog
G(x,z) +u(x)
+
/o
(A+
f /,.,
1
(A+m
h
+
i
2
)
a{/,(x)tf
v,] 1
)
+
[AU)^ f /o(A
A Mh
Vl
f
A (A+?+l
)
f
q>^
+
[/
(A+w)
(
A+n) vo
fo'
(A+n)

}
get
!)/
'/'
A (Af1) vi + /o' (A)
+ + A' (A+l) v.i = f /o"(A)i?o] /*'" + /o'
we
.
i/
]
/x'"
THE EULER DIFFERENTIAL EQUATION
421
that the # can be computed in terms of # by the alcomputed in terms of # and t/> ,
It is clear
gorithm of theorem 1, the y\ can be and the
.
,
This conclusion can be stated with easily established generality as follows
:
Theorem
2.
If the equation,
has one root of multiplicity A!
=
^, A 2
X
+ mi,
 *
,
Ar
r,
or r single roots differing by integers where the m, are integers, then r :
= +w A
.
3 ,
equation (2.1) has r formal solutions of the form,
u2 (x)
~x
x
u r (x) =^^ x
where the
It will
log x
log
r
a*
(x)
x "n
+x
(pn(x)
f
*
q> 22
(x)
%^ mi log r ^
expansions about
infinity.
be convenient in illustration to consider the hypergeometric is a special case of the generalized Euler equation,
equation which (x
x*)u"(x)
+
apu(x)
(a+/i+l)]tt.'(#)
[7
=0
.
(2.11)
Since in our notation )
we
shall have,
Hence we obtain for the root
=
AU
from which we get the
n) //o (A
A
w
solution,
a (l_f_ a
7)
1)
,
62
=1
,
C2
=
2
,
THE THEORY OF LINEAE OPERATORS
422
which, in the customary notation of the hypergeometric function,*
In a similar acteristic root, X
way
=
the second solution ft,
and we
is
is
obtained for the char
find,
a ^= p is a positive integer or zero we must employ the result If jS of theorem 2 since the denominator of the coefficient of x~ a p in (2.12) is zero. The solution of the hypergeometric equation will then become,
Ui(x)= xt logxF( p,
+1
y
/J
+
or*'
;
a
ft
1
(^o
+1
l/x )
;
+ q\/x +
v'o 2
<jr;
2 /a;
+



)
,
where we have,
+
in
which
y> tn
or what
is
12/5
2wi)
the coefficient of or w in the expansion of
is
A second
[(}'
solution can be obtained
equivalent,
by setting
r/? p
^r constant ^L
0,
by adding
to this function.

a negative integer or zero, a similar expansion is obtained by interchanging the role of (5 and a. These solutions will be found to accord with those obtained in another manner by E. Lindelof.f If
p
is

*For ready reference we

,
4
recall the definition:
l!y 2!y(y+l) fSur ^integration de 1 'equation Svientiarum Fennicae,
vol. 19
 # 3 4y (y+1) (y+2) Ada Socictatis differentielle de Kummer.
x2
,
~r
3!
(1893), pp. 331; in particular, pp. 1617.
THE EULER DIFFERENTIAL EQUATION
423
The distinguishing feature of
these solutions, it should be especially noted, is found in the fact that we have attained directly the solution of the equation for the regular singular point x oo. The solution about the origin belongs to the theory developed in the next
chapter. 3. Excursus on the Factorial Series /(A). In the last section we exhibited the fundamental role played in our theory by the function defined by the factorial series
a,
We
+ ad +
1)U
2)/3!
few
shall, therefore, set forth briefly a
which we
+

(3.1)
facts about this series
important in a later section.* considerable history the modern epoch of whicli shall find
This series has a began with Abel's
investigation of the convergence of the binomial series, that is to say, the special case, a n a n .\. The series is frequently referred to as
=
Newton's series because his interpolation formula
is
another special
case.
We have already shown in section 11, chapter 6, that the region of convergence of (3.1) coincides with the region of convergence of a Dirichlet's series and hence the theorems of that section apply to /(A). If the coefficients {&,} are real numbers then the region of convergence of the series (3.1) line of convergence.
of reals cally
is
a half plane bounded on the
is
The point
(A
)
where
called the abscissa of convergence limits
by the following
left
by a
this line crosses the axis
and
is
defined analyti
:
711
A
Jo*
~
{lim sup log
v
(
i)
(
I)

HJJ
10
= (Km sup log
2
'
'
^
I
/log n}
&< /log 
n}
1
l
,
.
n Y,
(3.2)
*The author is indebted to an admirable account of this series by N. E. Norlund: Vorlesungen iiber Differ enzenrechnung, Berlin (1924) pp. 222240; also: Sur les formules d'interpolation de Stirling et de Newton. Annales de I'ficole Normale Superieure, vol. 39 (3rd ser.) (1922), pp. 343403, vol. 40 (3rd ser.) (1923), pp. 3554. The reader is also referred to the following: J. L. W. V. Jensen: Oin Riikkers Konvergens. Tidsskrift for Math., vol. 2 (5th ser.) (1884), pp. 6972; J. Bendixson: Sur une extension a Finnni de la formule d'interpolation de Gauss. Acta Mathematica, vol. 9 (1886), pp. 134; E. Landau: Tiber die Grundlagen der Theorie der Fakultatenreihen. Sitzungsberichte der Akad. Munchen (math.phys.) vol. 36 (1906), pp. 151218; S. Pincherle: Alcune spigolature nel campo delle funzioni determinant!. Atti del IV Congresso dei Matematici, vol. 2 (1908), pp. 4448; also: Quelques remarques sur les fonctions doterminantes. Acta Mathematica, vol. 36 (1912), pp. 269280. fN. H. Abel: Werke, vol. 1 (1881), pp .219250. ,
THE THEORY OF LINEAR OPERATORS
424
The second di
first limit is to be used if * 1 is negative.
+
J1
+
if ^
1 is positive
or zero and the
The theorem can be illustrated by the binomial theorem, where t= a\ If a < 1, the abscissa of convergence is negative and hence
we compute
2
IVa^
(
Ao*
=
(_i)" n
{lira
sup [log a / (1f a.)] /log %}
1
=
<*>
.
N.UO
The
series thus converges for all values of 1. If \a\ 1, the abscissa of convergence is positive
>
and we com
pute
2
(
1)'
a'"
i^O
= [1 +
Ao
(1)' a]/(l+a)
= {Hm sup
(n log a/log n)
}
,
1
=
oo
.
HOO
The
series thus diverges for all values of
=
A.
=
1, we find that A 0, and Similarly for the case where a hence the series converges for all values of I such that R(l) > 0. Also for a fi= 1, Ao 1, and the series converges for all values of A
=
such that
R (A) >
1.
Factorial series of the type under discussion have one rather unfortunate pecularity, namely, that they include the development of zero.
Hence the expansion of a function
in such a series is not, in
general, unique.
For example, the
where \C n
is
series
the nth binomial coefficient, converges identically to zero
R
in the half plane, U) > 0. Similarly the function
F r+l U)=xC F a r
r
T)
2
(
D^nCrxCn
converges identically to zero in the half plane Thus the null development
R (2) >
r.
can be added to /(A) proc a , Ca, , C M are arbitrary constants, 1. vided the abscissa of convergence exceeds n It will thus be clear that the expansion (3.1) is unique only when the abscissa of convergence is negative or zero.
where
THE EULER DIFFERENTIAL EQUATION
of
The following two expansions are Newton series
425
useful integral representations
:
(A)
f \t/a) *y(t)dt^ f \ t/a) x (a f Jo Jo
= 9>i(o)
X
()/
<jr>2
where we use the abbreviation n
(a) =
+IU
aj
D'fi
f
M
2
f
.
.
)
dt
()/
(3.4)
+
(3.5)
:
f V(0
Jo
and assume that for
all
values of n. /GO
(B)
^e' j
o ./o
= ^ _ ^A + #
1)
2l ( A
7U(A where we write
/2
!
1)(A
2)/3!
..,
:
rv
# nC=
Jo in
which
L n ()
is
the nth
Laguewe polynomial:
and ^(0 is a function expansible in terms of these polynomials.* Expansion (3.4) is obtained by means of a suitable integration by parts; expansion (3.5), however, is not so obviously derived but can be obtained as follows :
(^
Jo
e<{
2 wo
(
1
n
Cm
t n /ml} dt
,
But we have the identity! *See R. Courant and D. Hilbei*t: Met/ioden der Matkcmatisclie Plujsik, vol. Berlin (1924), pp. 7779. These authors define the Laguerre polynomials to be nl Ln (t). See also J. Shohat: Theorie generate dcs polynomes orthogonaux de Tchebichef, Memorial des Sciences Mathematiques, fasicule 66, Paris (1934), G9 pp. fSee I. J. Schwatt: An Introduction to the Operations with Series. Philadelphia, (1924), p. 48, prob. iv. 1,
THE THEORY OF LINEAR OPERATORS
426
2
l)
(
and hence we can write
i/O O
We
thus see that
=
( w0
/
w0
But we also know that #(0 can be expanded in the form :* if
0(0 =2
is
a properly defined function
#(*)
it
(3.6)
wo
to oo, and recalling the Multiplying by e~ L n (t), integrating from the of Laguerre polynomials, we have orthogonality properties l
rV0(OMOd*=n
r'V'L n (Od* = *n 2
Jo
Jo
The Nonhomogeneous Case of the Generalized Enter Equa
4.
now to the formal solution of equation (1.6) we seek the resolvent generatrix, Y (#,), which satisfies the equation, tion.
Proceeding
Y
(x,z)
>G(J,3)=1
(4.1)
.
purpose we differentiate (1.6) an infinite number of times, or casting this statement in the language of operators, we form
For
this
for the convergence of (3.6) have been specified the Development of Arbitrary Functions in Series. Annals of Mathematics, vol. 28 (2nd ser.) (192627), pp. 593619, in particular, p. 618:
*The following conditions
by
J.
V. Uspensky:
On
r*> (1)
That the integral
e*[0(t)]' dt exists for a certain value of a.
I
Ja
(2)
That the integral
C Jo
b
tv*\e(t)
\
dt exists for a certain value of
(3) That 0(t) is of limited variation in a certain interval absolutely integrable in any finite interval.
Under these 0)
conditions the series
+ *U
(3.6)
t
d, t
converges and has for
b.
+
d and
its
sum
0)}.
See also: G. Szego: Beitrage zur Theorie der Laguerreschen Polynome. I. Entwickhmgssatze. Math. Zeitschrift, vol. 25 (1926), pp. 87115; E. Hille: Proc. of National Academy of Sciences, vol. 12 (1926), pp. 261269, 348352.
THE EULER DIFFERENTIAL EQUATION
427
=
the operational products, z n  G(x,z), n and equate 0, 1, 2, to z n We thus obtain a set of equations of the form
them
,
.
:
= g^ + 023^3 Hh ^n + = 033^3 Hh a H = On 
f a i:,u6
\
[
a ln u n j
'
,
92
Jntt
=
=
where we have in particular, g 1, # 2 and where the matrix of the coefficients, l
follows
a
,
,
a,
a
,

;
,
2%
(/ 3
(4.2)
,
= ^ "% n
... ,
tfn
explicitly given as
is ,
:
+ +
a, 4
,
0,0 As we have learned from sections 6 and 7, chapter 3, the solution of system (4.2) can be explicitly obtained either by the method of the LiouvilleNeumann series or by the method of segments. In the first case, the
expansion
is explicitly
the following
; _ 1
4
1
:
7v
In the second case, the solution appears in the more convenient form,
Uigt/du
ZV
?)
^+i/aam m f
(4.3)
where we abbreviate,
n
xy u <*)
.
\)
,
,
Cv42
*
lt2
"
*
,
,
^1+2, i+n
tl^ n ^n
According to the theorems of sections 6 and 7, chapter 3, both formal solutions converge and represent a solution of the original
THE THEORY OF LINEAR OPERATORS
428
system provided the g are bounded and the t
coefficients are subject
to the restriction, JO
Jl+l
we
Making the proper specializations in (4.3), state the following theorem:
S
Theorem 3. If there exists a region the following inequalities hold
are then able to
of the variable x for which
:
'JO
where we use the abbreviation,
A n (p m
)
=p m + nq
n+l
in(nl)r n ^/2l _
t/iere exists
n
within the region S a resolvent operator for equaz < 1, which is given explicit
tion (1.6), convergent in the region ly by the following formula
\
:
+
Z> 2
(a;)z=Al(2)

(4.4)
,
where we abbreviate, 4(0)
=0o,
X
(Oo
/1(1)
ciotao
+ 2b, + cO, 
,
+ bt)
and
,
a.2
(2)
=00(00
+ 60
= {^o + nbi
^l(t)
A^) ^ + 6^
4
,
+
Oi
&ia:
,
2