Bocher, Maxime
An introduction to the study
of integral equations
Presented to the
LIBRARY of the
UNIVERSITY OF TORONTO
by
Mr. J. R. McLeod
Cambridge Tracts in Mathematics
and Mathematical Physics
GENERAL EDITORS
J. G. LEATHEM, M.A.
E. T. WHITTAKER. M.A., F.R.S.
No. 10
AN INTRODUCTION
TO THE STUDY OF
INTEGRAL EQUATIONS
by
MAXIME BOCHER, B.A., Ph.D.
Professor of Mathematics in Harvard University
Cambridge University Press Warehouse
C. F. CLAY, Manager
London : Fetter Lane, E.G.
Edinburgh : ice, Princes Street
1909
Price 25. 6d. net
Cambridge Tracts in Mathematics
and Mathematical Physics
GENERAL EDITORS
J. G. LEATHEM, M.A.
E. T. WHITTAKER, M.A., F.R.S.
No. 10
An Introduction to
the study of Integral Equations
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AN INTRODUCTION
TO THE STUDY OF
INTEGRAL EQUATIONS
by
MAXIME B6CHER, B.A., Ph.D.
Professor of Mathematics in Harvard University
CAMBRIDGE:
at the University Press
1909
Gr,
Cambridge:
PBINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS.
PREFACE
TN this tract I have tried to present the main portions of the theory
-*- of integral equations in a readable and, at the same time, accurate
form, following roughly the lines of historical development. I hope
that it will be found to furnish the careful student with a firm founda-
tion which will serve adequately as a point of departure for further
work in this subject and its applications. At the same time it is
believed that the legitimate demands of the more superficial reader,
who seeks results rather than proofs, will be satisfied by the precise
statement of these results as italicized, and therefore easily recognized,
theorems. The index has been added to facilitate the use of the book-
let as a work of reference.
In these days of rapidly multiplying voluminous treatises, I hope
that the brevity of this treatment may prove attractive in spite of the
lack of exhaustiveness which such brevity necessarily entails if the
treatment, so far as it goes, is to be adequate.
I wish to thank Professor Max Mason of the University of Wisconsin
who has helped me with some valuable criticisms ; and I shall be
grateful to any readers who may point out to me such errors as still
remain.
MAXIME BOCHER.
HARVARD UNIVERSITY,
CAMBRIDGE, MASS.
November, 1908.
CONTENTS
PAGE
INTRODUCTION 1
1. Some preliminary propositions and definitions .... 2
2. Abel's mechanical problem 6
3. Solution of Abel's equation 8
4. Liouville's introduction of integral equations of the second kind . 11
5. The method of successive substitutions 14
6. Yolterra's treatment of equations of the second kind. Iterated
and reciprocal functions 19
7. Linear algebraic equations with an infinite number of variables . 24
8. Fredholm's solution 29
9. The integral equation with a parameter 38
10. The fundamental theorem concerning homogeneous integral equa-
tions, with some applications 43
11. Symmetric kernels 46
12. Orthogonal functions 52
13. The integral equation of the first kind whose kernel is finite . 60
14. Equations of the first kind whose kernel or whose interval is not
finite 65
INDEX 72
AX INTRODUCTION TO THE STUDY OF
INTEGRAL EQUATIONS
Introduction. The theory and applications of integral equations,
or, as it is often called, of the inversion of definite integrals, have come
suddenly into prominence and have held during the last half dozen
years a central place in the attention of mathematicians. By an
integral equation* is understood an equation in which the unknown
function occurs under one or more signs of definite integration.
Mathematicians have so far devoted their attention mainly to two
peculiarly simple types of integral equations, — the linear equations of
the first and second kinds, — and we shall not in this tract attempt to
go beyond these cases. We shall also restrict ourselves to equations
in which only simple (as distinguished from multiple) integrals occur.
This restriction, however, is quite an unessential one made solely to
avoid unprofitable complications at the start, since the results we shall
obtain usually admit of an obvious extension to the case of multiple
integrals without the introduction of any new difficulties!. In this
respect integral equations are in striking contrast to the closely
related differential equations, where the passage from ordinary to
partial differential equations is attended with very serious complica-
tions.
The theory of integral equations may be regarded as dating back
at least as far as the discovery by Fourier of the theorem concerning
integrals which bears his name ; for, though this was not the point of
view of Fourier, this theorem may be regarded as a statement of the
solution of a certain integral equation of the first kind j. Abel and
* The term Integral Equation was suggested by du Bois-Reymond. Cf. CreUe,
voL 103 (1888), p. 228.
t Another extension, in which serious complications do not usually arise, is to
systems of integral equations. We do not consider such systems in this tract.
* Cf. the closing page of this Tract.
2 PRELIMINARY PROPOSITIONS [1
Liouville, however, and after them others began the treatment of
special integral equations in a perfectly conscious way, and many of
them perceived clearly what an important place the theory was destined
to fill*.
As we shall not, except in one relatively unimportant case, take up
any of the applications of the subject, it may be well to say explicitly
that like so many other branches of analysis the theory was called into
being by specific problems in mechanics and mathematical physics.
This was true not merely in the early days of Abel and Liouville, but
also more recently in the cases of Volterra and Fredholm. Such
applications of the theory, together with its relations to other branches
of analysis t, are what give the subject its great importance.
1. Some Preliminary Propositions and Definitions. In
order to avoid interruptions in later sections, we collect here certain
propositions of the integral calculus for future reference.
We shall have to deal with functions of one and of two variables.
The independent variables, which we will for the present denote by
x and (a, y) respectively, are in all cases real. In fact, in order to
avoid unnecessary complications we will assume that, unless the
contrary is explicitly stated, all quantities we have to deal with are
real.
The range of values of the single argument x is usually
/ a ^ x ^ b.
We shall speak of this in future simply as the interval /.
In the case of functions of two variables, two cases have to be
considered. Interpreting (x, y) as rectangular coordinates in a plane,
we sometimes consider the square
and sometimes the triangle
T a^y^x^b.
It should be noticed that the three regions we have just defined,
7, S, T, are closed regions, that is they include the points of their
boundaries.
In order to avoid long circumlocutions we lay down the following
* Cf. besides the article of du Bois-Reymond already cited, some remarks by
Rouche, Paris C. R. vol. 51 (1860), p. 126.
t Cf. , for instance, much of Hilbert's work.
1] AND DEFINITIONS 3
DEFINITION. \\'e say that the discontinuities of a function of(x, y)
are regularly distributed in S or in T if they all lie on a finite number
of '•/</•/>. < icitk continuously turning tangents no one of ivhich is met by
a line parallel to the axis of x or of y in more than a finite number of
points.
In order to make the enunciation of some of our results simpler,
we will assume once for all that the functions we deal with are denned
even at the points of discontinuity, at least in the cases where they
remain finite in the neighbourhood of such points.
The following theorem will be important for us. We state it first
for the case of the region >$'.
THEOREM 1. If the tico functions <f> (x, y) and $ (x, y) are finite in
S and their discontinuities, if they have any, are regularly distributed^
the function
is continuous throughout S.
The truth of this theorem becomes evident if we interpret (x, y, £)
as rectangular coordinates in space. It is then clear that the function
under the integral sign is finite throughout the cube
a^x^b, a^y^b, a^g ^b,
and becomes discontinuous in this cube only at points on two sets of
cylinders whose generators are parallel respectively to the axes of x and
y. Moreover these cylinders are so shaped that any line .r = .r0, y = yo
in this cube meets them at only a finite number of points. — The formal
proof, based on these or similar considerations, presents no difficulty,
and we leave it for the reader.
COROLLARY. If <j>(x,y} and \j/ (x, y) are finite in T and thtir
discontinuities, if they hace any, are regularly distributed, the function
is continuous throughout T.
This is merely a special case of Theorem 1. For if we define <£ and
</f to have the value zero everywhere outside of T, it is clear that they
satisfy the conditions of Theorem 1 throughout >S' and that the function
F(z,y) reduces to H(x,y}.
1—2
4 PRELIMINARY PROPOSITIONS [1
If <f> (x, y} satisfies the conditions of Theorem 1, the double integral
of <j> extended over S may be evaluated in either one of two ways as an
iterated integral* and we thus get the formula
rb rb rb rb
I \ $(x,y)dydx=\ \ $(x,y)dxdy.
Ja Ja Ja Ja
If, in particular, <£ vanishes everywhere outside of T, we get
DIRICHLET'S FORMULA!. If <£ is finite in T and its discontinuities,
if it has any, are regularly distributed, then
rb rx rb rb
I / <£ 0> y) fy&B = / £0*. y} fafy-
Ja Ja Ja Jy
This formula admits of extension to certain cases in which the
integrand does not remain finite in T. The most general case of this
sort which we shall have occasion to use is contained in the following
statement, for a simple proof of which we refer to the first part of
a paper by W. A. Hurwitz \ :
DIRICHLET'S EXTENDED FORMULA. If <f> (x, y} is finite in T and its
discontinuities, if it has any, are regularly distributed, and if A., /*, v
are constants such that
O^X<1, 0^/Kl, OSv<l,
then
fb I* ^(x,y}dydx = fb fb <j>(x,y}dxdy
Ja .L (x - yY (b - a>Y (y - «)" Ja Jy (x -yY (b - xY (y - a)" '
Finally we turn to some theorems concerning functions of a single
variable.
THEOREM 2. If <j> (x) is finite and has only a finite number of
discontinuities in I, the function
is continuous throughout I, including the point a, where it vanishes^.
* By a double integral we understand the limit of a sum obtained by dividing
up the region in question into pieces both of whose dimensions are small. By an
iterated integral, the integral of an integral.
f Cf. Crelle's Journal, vol. 17 (1837), p. 45.
£ Annals of Mathematics, vol. 9 (1908), p. 183. This result may also be deduced
from a general theorem of de la Vallee Poussin. Cf. the Cours d' Analyse of this
author, vol. 2, pp. 89—95.
§ We define the symbol / ^ (x) dx to mean zero, whatever the nature of the
J «
function \j/ may be.
1] AND DEFINITIONS
To prove this we introduce the new variable of integration
3=i-a
x — a'
Then
By replacing <£ by the upper limit of its absolute value, we see that
¥ (x) remains finite, and hence that 3> approaches zero as x approaches
a. Consequently <£ is continuous at a. On the other hand the same
substitution shows that the integral * converges uniformly when
b. For any fixed positive 8 < 1 the function
is continuous throughout the interval a < x ^ b, since the integrand in
^i is finite in the rectangle
a<x^b, O^s^l-S,
and is discontinuous only along a finite number of curves in this
rectangle each of which is met by a line .r = x0 in at most one point.
Since, as we have just seen, *i (^) approaches ^(X) uniformly as 8
approaches zero, it follows from a fundamental theorem in uniform
convergence that * (x) is continuous when a < x ^ b, and hence the
same is true of & (x), and our theorem is proved.
THEOREM 3. If, in I, <£(V) is continuous and has a derivative
which is finite and which has at most a finite number of discontinuities
in I, and /f<t>(a) = Q, the function
has a derivative continuous throughout I and given by the formula
For if we integrate the expression for 3> (.r) by parts, we have, when
we remember that <f> (a) = 0,
Applying here the rule for differentiating an integral whose limits are
variable, we get the desired expression for & (x}. Hence from
6 ABEL'S MECHANICAL PROBLEM [1, 2
Theorem 2 it is evident that <$' (,r) is continuous. It should be noticed
that when X > 0 the integrals with which we have to deal are infinite
integrals (i.e. integrals in which the integrand does not remain finite)
so that the application to them of the ordinary rules of the calculus
requires careful justification.
An alternative form of proof for this theorem consists in applying
Dirichlet's Extended Formula* as follows :
t
= L
' w
The differentiation of this last formula gives us the result we wish
to establish t.
In conclusion we point out by means of the following two examples
that if we replace the condition of finiteness for <£' by the condition of
integrability, or even of absolute integrability, 3> will not always have
a continuous derivative :
(i) 4>0)=O-«)A, <&(#) = &(#-«),
° (a = x = «') \
In both cases k is a positive constant, and if 0 < X < 1, <£ is con-
tinuous in / and has a derivative which is continuous except at one
point and absolutely integrable but not finite. In the first case 4>' is
continuous, in the second discontinuous.
2. Abel's Mechanical Problem. In one of his earliest
published papers % Abel showed how a certain mechanical problem,
* It should be noticed that we use this formula here under slightly different
restrictions on the function 0 (x, y) since <f> is now a function of y alone, and there-
fore if it is discontinuous at all, is discontinuous along lines parallel to the
axis of x.
t This method of reasoning admits of immediate extension to the proof of the
more general formula
which holds under suitable restrictions on \f/.
t See Collected Works, p. 11. This paper was first published in Christiania in
1823. Cf. also a second paper beginning on p. 97 of the Collected Works, and
originally published in Crelle, vol. 1 (1826), p. 153.
2] ABEL'S MECHANICAL PROBLEM 7
which includes the problem of the tautochrone as a special case, leads
to what ha.s since come to be called an integral equation, on whose
solution the solution of the problem depends. On account of its great
historical interest, we take up this problem in this section.
A particle starting from rest at a point P on a smooth curve which
lies in a vertical plane, slides down the curve to its lowest point 0.
The velocity acquired at 0 will be independent of the shape of the
curve. The time of descent T will however depend on this shape.
We take 0 as origin, the axis of x vertically upward, and the axis of y
horizontal arid in the plane of the curve. Let the coordinates of the
point of departure P be (x, y], and the coordinates of the point Q
reached by the particle at the time t be (£, rj), g the gravitational
constant, and .-< the arc OQ. The velocity of the particle at Q is
/— l'Q
Hence v 2g t = - I
Jf
If we express s in terms of £
•-•
the whole time of descent is then
^ 1 [* *>'(€)<%
l = —j=- I — . . .
V2<7 .'o x/.r - £
If the shape of the curve is given, the function v may be computed,
and the whole time of descent is given to us as a function of x by the
last formula.
The problem which Abel set himself is the converse of this, namely
to determine the curve for which the time of descent is a given
function of ,i: If we write
our problem is to determine the function v from the equation
(1).
The formula for the solution of this integral equation was obtained
by Abel by two different methods. The first depends on the use of
series proceeding according to powers, not necessarily integral, of the
argument ; while the second, of a more general character, is closely
related to the one we are about to give in the next section. Neither
of Abel's methods can be regarded as satisfactory although they lead
8 SOLUTION OF ABEL'S EQUATION [2, 3
to the correct result. Among other objections it may be said that both
methods omit the essential step of proving that the equation (1) has
a solution.
3. Solution of Abel's Equation*. Instead of the equation (1)
of § 2, Abel set himself the problem of solving a more general equation
which we will write in the form
where /is a known function, u the function to be determined.
In order to solve (1) we begin by establishing the general formula
(2) below. We start from the well known formula
dx ,
f*
= | f
/£ (z-
Let <£ (£) be any function which is continuous and has a continuous
derivative throughout 7. Multiply this equation by <£'(£) d£ and
integrate from a to z, which we suppose to be any point of /. This
gives
) - $ (a)] = f Z f -. -4^ - ^- dasdt.
ja js (z - a?)1 - ** (# - $y*
If we apply Dirichlet's Generalized Formula to the second member of
this equation, we get the desired result
sn/«r j
- - (2)
a formula which holds under the sole restrictions that z be in /, and <£
be continuous and have a continuous derivative in I, and that
0</t<l.
Theorem 2, § 1, shows us at once that a necessary condition
that (1) have a solution continuous throughout / is that /(#) be
continuous throughout /and that /(a) = 0.
Let us suppose that these conditions are fulfilled and that u (as) is
a continuous solution of (1). Multiply (1) by (z — a;)*"1 dx, where z is
a point of /, and integrate from a to z, thus getting
O) dx
,7
^
* Except for the method of deducing formula (2), the method we use is, barring
notation, that of Liouville in Liouville's Journal, vol. 4 (183!)), p. 233. Liouville,
who seems not to have been aware of Abel's work, had already published on this
subject in the Journal de I'Ecole Poly technique, Cahier 21 (1832), p. 1.
3] SOLUTION OF ABEL'S EQUATION
If in (2) we let
it will be seen that the preceding equation reduces to
f'f®** ' r M(0# (3).
JaO-tf)1'* Sin \irja
Since the second member of (3) has a continuous derivative with
regard to z, the same must be true of the first member, and this gives
us a further necessary condition for (l) having a continuous solution.
By differentiating (3), we get as the value of this solution
, . _ sin Xn- d r- f(x)dx , }
~~ -i-*
We thus see that u is completely detennined, that is that (1) does not
have more than one continuous solution. That the formula (4) really
does give a solution of (1) may be seen by substituting it in (1). The
second member of (1) thus becomes
sin XTT * 1 d
which reduces by means of Theorem 3, § 1, to
sin XTT d [x 1 [* f(x) dx , ,
~ J~ I ( ~£\\ I ft _ \T^X *'
TT ax jd \^tV ~~ ^ ^ ya \^% »c j
and this in turn reduces by means of (2), when we let
*<*)=(*/(*)«^
Ja
to
Thus we see that (4) is a solution of (1), and we have proved
THEOREM 1. A necessary and sufficient condition that (1) have
a solution continuous in I is that f (x] be continuous in I, that f (a] = 0,
and that
f
Ja
have a continuous derivative throughout L If these conditions are
fulfilled, (1) has only one continuous solution, given by formula (4).
An important case in which these conditions are fulfilled is that in
which / is continuous and has a derivative which is finite and has at
most a finite number of discontinuities in /, and f(a) = 0. This we
10 SOLUTION OF ABEL'S EQUATION [3
see from Theorem 3, § 1, from which we also see that in this case (4)
may be written
TT Ja(z-xy--*
Hence
THEOREM 2. If f(x) is continuous and has a derivative finite in I
and with only a finite number of discontinuities there, and f(a) = Q,
equation (1) has one and only one continuous solution, and this is given
by formula (5)*.
While this is essentially Abel's result, that mathematician did not
consider the integral equation (1) but rather the differentio-integral
equation
By means of the theorems just established and Theorems 2, 3 of
§ 1 we readily deduce the result
THEOREM 3. A necessary and sufficient condition that (6) have a
solution which together with its derivative is continuous throughout I
is thatf(x) be continuous in /, that f (a) = 0, and that
have a continuous derivative throughout I. If these conditions are
fulfilled, the general solution of (6) is
, N 7 sin A.TT P f(x\ dx
v(z) = k + -
* Ja(z-xf~
where k is an arbitrary constant.
* Goursat, in Acta Math. vol. 27 (1903), pp. 131—133, has shown that
equation (1) still has a solution, though not a continuous one, if we drop the
requirement that /(a)=0. This may be readily seen by bringing in, in place of
u, the function
sin\7T f(a)
v(x) = u(x)-- -Hll'
TT (x-a)1
Making this substitution, we find that equation (1) reduces to
Conversely, we see that a solution of this last equation corresponds to a solution
of (1). Consequently a solution of (1) is
sin\7r /(a) sin\7r f* f'(x)d
u(z) = - - ^ — --\ -- / — ;
TT («-a)1 A TT Ja(z-xY
x
r.
3, 4] EQUATIONS OF THE SECOND KIND 11
By letting X = i we get the solution of the mechanical problem
of § 2. If in particular we let /(#) = const., we get Abel's solution
of the problem of the tautochrone.
An easy extension of the results we have found is to the case in
which X is negative, cf. Liouville, loc. cit. Let us for instance suppose
that - 1 < X < 0. We see by differentiation that any continuous solution
of (1) also satisfies the equation
» ^ / AN t 1
(7),
and conversely, if f(&) is continuous in / and/(a) = 0, we see by
integrating that every continuous solution of (7) is also a solution of
(1). Hence
THEOREM 4. IfO>\>-l,a necessary and sufficient condition that
(1) have a continuous solution is that /(.r) be continuous together with
its first derivative throughout I, that f (a) -f (a) = 0, and that
hare a continuous derivative throughout I. If these conditions are
fulfilled, (1) has one and only one continuous solution, namely
, , sin XTT d izf (%} dx
u (.r) = — v ' .
*
In particular, the above conditions are fulfilled */*/(#), /' 0*0 ^"O2*)
are continuous in Iandf(a)=f'(a) = Q. In this case the continuous
solution may be written
The extension to other cases in which X is negative may readily
be made by further differentiation. "We leave to the reader the
enunciation and proof of the general theorem to be obtained here
as well as the consideration of the special cases in which X is an integer
negative or zero.
4. Liouville's Introduction of Integral Equations of the
Second Kind. After Abel, the next to use an integral equation was
Liouville who, in the year 1837*, showed that a particular solution of
* Liouville's Journal, vol. 2, p. 24. This reference has no connection with
Liouville's work cited in the preceding section.
12 LIOUVILLE'S INTRODUCTION OF [4
a certain linear differential equation can be obtained by solving an
integral equation of a type somewhat different from, though closely
resembling, Abel's equation.
To explain this we begin with the non-homogeneous equation
where p is a parameter, and <£ (#) a function continuous throughout /.
The general solution of the reduced equation
is a sin p (x - a) + ft cos p (x - a), so that by a well known formula* the
general solution of (1) is
1 fx
y (x) = a sin p (x - a) + ft cos p (x — a) + - I <£ (£) sin p(x — £) d£ (2).
p Ja
Now consider the homogeneous equation
g+[p2-<r(*)]</ = 0 (3)
where <r (x) is continuous in /. "We will denote by u (x) the solution
of (3) which satisfies the auxiliary conditions
w(a)=l, w'(«) = 0 (4)-
This function will also be a solution of the non-homogeneous
equation
Consequently by using (2) and (4) we get
1 [x
u (x} = cos p (x - a) + - I a- (£) sin p (x - £) u (£) d£ (5).
P Ja
Conversely if u(x) is any continuous solution of this integral
equation, we see, by applying to (5) the rule for differentiating an
integral whose limits are functions of x, that u (x) has continuous first
and second derivatives given by the formulae
rx
' (x) = - p sin p (x - a) + I o- (£) cos p (x - £) u (£) d£ (6),
Ja
rx
*cosp(x-a)-p I o-(|)gmp(a?-l)»(0^+°'(a?)f»(^) (7)-
Ja
* Cf. Forsyth's Treatise on Differential Equation*, § 66.
4] EQUATIONS OF THE SECOND KIND 13
Formulae (5) and (7) show that it satisfies (3), while from (5) and
(6) we see that conditions (4) are satisfied. Thus, since there is only
one function satisfying (3) and (4), we have proved the
THEOREM. The solution of the differential equation (3) which
satisfies the auxiliary conditions (4) is a solution of the integral
equation (5); and conversely this integral equation (5) has only one
continuous solution.
The possibility here illustrated of replacing a differential equation
together with certain auxiliary conditions by a single integral equation
is characteristic for many important applications of integral equations.
Liouville's object in introducing the integral equation (5) was to
enable him to obtain a development of u (x) in a series which converges
rapidly for large values of p. Such a series he obtained by solving (5)
by a method which will be explained in the next section.
Comparing Abel's equation (formula (l), § 3) with Liouville's
equation (5), we see that they come respectively under the following
types :
(8),
in which f(x) and K(x, £) are to be regarded as known functions
and u (x} is the function to be determined. Equations (8) and (9) are
spoken of as linear integral equations of the first and second kinds
respectively. K is called the kernel of these equations*.
In place of the equations (8) and (9) in which the upper limit of
the integrals is the variable x we often have to deal with equations
of exactly the same form in which the upper limit is the constant b.
These are also called linear integral equations of the first and second
kind respectively. It will be seen that equations (8) and (9) are
merely the special cases of the equations just mentioned in which the
kernel K(x, £) vanishes when £> x since it then makes no difference
whether x or b be used as the upper limit of integration.
* These terms were first employed by Hilbert, Gottinger Nachrichttn, 1904.
An equation of the form
fb
* (x) « (x) =f (*) + | * (*. *) « (fl <$,
which includes the equations of the first and second kinds as special cases, has been
called by Hilbert an equation of the third kind, but only a very special case of this
equation has so far been treated.
14 THE METHOD OF [4, 5
The special case of (9), or of the more general equation in which
the upper limit of integration is b, in which /(>) vanishes identically
may be called a homogeneous integral equation of the second kind.
It should not be confounded with the equation of the first kind.
5. The Method of Successive Substitutions*. The method
which Liouville used for solving equation (5) of the last section
applies with equal simplicity to the more general equation (9). In
fact we will consider at once the still more general equation referred to
near the end of the last section
« (a?) =/(#) + fjTOir, £)»(£)# (1).
We assume that the kernel K is finite in S and that its discon-
tinuities, if it has any, are regularly distributed there t.
If (l) is to have a solution continuous in /, it is clear (cf.
Theorem 1, § 1) that f(x) must be continuous in /. Let us assume
that this condition is fulfilled.
Assuming that (1) has a continuous solution, we now proceed to
find it. Substitute in the second member for u (£) the value given
by the equation itself, thus getting
«(*) =/(*)+ /Vfce, *)/(*)# + /V(>,£) /V(ft ft) «&)<&#•
Ja Ja Ja
Here we again substitute for u(j;\) its value as given by (1), and
thus get a four-term expression for u (#)• Proceeding in this way, we
get the general formula
«(#) = & (a?) + ^0*0 (2)
where
*)=/(*)+ f *>
Ja
... + (Vo, i) f jsrcft *o ... f
Ja Ja Ja
n (a) = f V(ar, 0 ibK(i;, ft) ... (Vftwi ft,) u (ft,) dft, ...
Ja ya ./a
* This method of solving an integral equation of the second kind is usually
attributed -to C. Neumann, whose work, however, is more than thirty years later
than Liouville's. The connection of Liouville's work with the theory of integral
equations of the second kind has been generally overlooked. For a formulation of
the method of successive substitutions which is convenient even in far more
complicated cases, cf. Mason, Math. Ann. vol. 65 (1908), p. 570.
f In the work of this section and the next it is immaterial whether the
functions / and K are assumed to be real or are allowed to be complex.
5] SUCCESSIVE SUBSTITUTIONS 15
We may regard this expression for u (.r) as a finite series of n + 1
terms plus a remainder, this remainder, however, involving the very
function «(.r) which we are developing*. This suggests to us the
consideration of the infinite series
~ (3).
From Theorem 1, § 1, we see that the terms of this "series are
continuous in /. The series therefore represents a function continuous
throughout / provided it converges uniformly there. We will prove
that whenever this is the case, the function u (#) represented by (3)
is a solution of (1). For this purpose multiply the series for u (£) by
K(JC, £) and integrate the resulting series with regard to £ from a to b
term by term, as we have a right to do on account of its uniform
convergence. This gives us precisely series (3) without the first term ;
that is
and this is the integral equation of which we wished to prove that
« (x) is a solution.
We will now obtain two different sufficient conditions for the
uniform convergence of (3).
Let us first consider the case in which
JT(.r,£) = 0 when £>.r (4).
This, as we have seen, is the case in which the upper limit of
integration in (1) may be taken as .r. Here the general term of the
series (3) may be written
, ()
Let M and JV be constant such that
\K(x,t)\<M, \f(x)\<N
Then
fX ft ft, •
>"
fX ft ft, •, I if _ /lV*
f ... >"*fe...4b*«JnP«Sj
Ja Ja Ja \n + 1) I
* Cf. the various forms of Taylor's series with a remainder.
16 THE METHOD OF [5
The series of which this last written positive constant is the general
term is convergent. Consequently the series (3) is absolutely and
uniformly convergent throughout /.
Without the special restriction we just made on K, the inequality
which we should get in the same way for the general term Fn(x)
of (3) would be
rb i'b r b
7? (r\\ < NMn+l I d£ fit-tit-— NMn+l (h — aVl+l
Ju n \it ) I = -i> JJJ. I • • • j M'Cjj . .-. U'£jU'£ — -it ±rJL {u ~ U) ,
Ja, Ja J a
and the series of which this is the general term converges only when
M(b-a}<l (5).
Thus we see that (3) converges absolutely and uniformly when
either one of the two conditions (4) or (5) is fulfilled.
We will now prove that in either of these cases the equation
(1) cannot have more than one continuous solution. For this purpose
we turn to formula (2) by which any continuous solution of (1) is
expressed. Referring to the formula for Rn (x}, we see that, if we
denote by N' the maximum of ] u (x) | in 1, the same reasoning which
led us before to the inequalities for | Fn (x} j now gives
| Rn (x} \ ± N' [M(b - a)]n+l if M(b -«)<!.
Thus we see in either case that
lim Rn (x} = 0.
«=oo
On the other hand, Sn (x} is simply the sum of the first n+l terms
of (3). Consequently the function u given by (2) is precisely the value
of (3), and we have the two theorems :
THEOREM 1. If K (x, £) is finite in T and its discontinuities, if it
has any, are regularly distributed, a necessary and sufficient condition
that the equation
have a solution continuous throughout 1 is that f ' (x) be continuous
throughout I, and if this condition is fulfilled, (6) tuis only one con-
tinuous solution, which is given by the absolutely and uniformly
convergent series (3).
5] SUCCESSIVE SUBSTITUTIONS 17
THEOREM 2. If K(JC, £) is finite in S and its discontinuities, if it
/«(* <()iy, are regularly distributed, and /(.r) is continuous in I, then
pm r ided that
//•//» re M denotes the upper limit of \K\ in S, equation (I) has one and
only one solution continuous in I, and this solution is given by the
abfuilutely and uniformly convergent series (3).
Besides the continuous solution whose existence has just been
established, the integral equation of the second kind may also have
discontinuous solutions. In order to show this, we will consider the
special case
« = 0, /(-r) = 0, K(z,t) = S*-*.
The integral equation (6) then reduces to
(7).
If we take b as any positive constant, the conditions of Theorem 1
are fulfilled. Equation (7) has therefore one and only one continuous
solution, and this solution is readily seen to be u = 0 ; as, indeed, is
the case for every homogeneous integral equation of the second kind.
By direct substitution, we verify that it also has the infinite number
of discontinuous solutions
where k is an arbitrary constant different from zero.
It should be noticed that these solutions become not merely infinite
fb
when x = 0, but become infinite so strongly that I u (z) dx diverges.
.'o
It may readily be seen that if the integral equation satisfies all the
conditions of Theorem 1 or 2, any discontinuous solution which it may
have will necessarily be nou-integrable *.
These non-integrable solutions have not as yet proved to be of any
importance and we shall not be concerned with them in this tract.
There are various extensions of the considerations of this section
which naturally suggest themselves. In the first place we may ask
under what conditions it is possible to assert that the continuous
solution of (1) or (6) has a continuous derivative. Let us assume that
both K(x, £) and dK/dx are finite in S and that any discontinuities
which these functions may have are regularly distributed. Also that
* We note in passing that an integral equation of the first kind may, under
similar conditions, have a discontinuous solution which is integrable. Cf. the foot-
note to Theorem 2, § 3.
B. 2
18 SUCCESSIVE SUBSTITUTIONS [5
the curves on which the discontinuities of K lie have continuously
turning tangents which are nowhere parallel to the axis of x or of £.
It may readily be inferred from this that, on the curves where it is
discontinuous, the function K joins on continuously to continuous
boundary values, — different values, of course, on the two sides of the
curve. We may then, proceeding with some caution, differentiate
equation (1) and thus arrive at the conclusion that under the con-
ditions just stated a necessary and sufficient condition that the
continuous solution u of (1) have a continuous derivative is that
f(x) have a continuous derivative. It would of course be possible
to arrive at a similar result with much less drastic restrictions on K.
A second question which may be raised is as to the existence of
continuous solutions of (1) or (6) when the kernel is not finite. We
will consider here only a special case which may serve as a sample of
others.
In the equation (6), let K be defined throughout T so that
0=. CO<A<I),
where G is finite in T and where its discontinuities, if any exist, are
regularly distributed. It may now be readily proved that if ^ (,r)
is any function continuous in 7, the function
I
Ja
K
is also continuous throughout 7.
If then/(#) is continuous in /, we see that the same is true of each
term of the series (3), in which it must be remembered that the upper
limits of integration are now variable.
For the general term Fn(x) of this series we readily get the
inequality
,. , ^ NMn+i fx L_ r* J_ f *"-' * & &
where | /(#) | ^ N,
In terms of the quantities
the (n + l)-fold integral last written may be readily evaluated, and we
thus find
If, then, we can prove that the series whose general term is
1*0*i • • • kn (b - a) <w + D( i - A) (H)
5, 6] VOLTERRA'S TREATMENT 19
converges, the absolute and uniform convergence of (3) follows. The
ratio of two successive terms of the form (8) is
if (ft -a)1-**..
The convergence therefore follows if we can show that
lim kn = 0,
n= oo
and this property of the Ks may be established with ease from their
definition.
Having thus established the uniform convergence of (3), the proof
that this series represents a continuous solution of the integral equation,
and the further proof that this is the only continuous solution of this
equation follow almost as before, and we thus get
THEOREM 3. The equation
in which f is continuous in I, and G is finite in T and such dis-
continuities as it may have are regularly distributed, has one and only
one continuous solution, and this solution is given by the method of suc-
cessive substitutions in the form of an absolutely and uniformly convergent
wries.
6. Volterra's Treatment of Equations of the Second
Kind. Iterated and Reciprocal Functions. "We will now start
afresh and approach the theory of integral equations of the second
kind from a new point of view due to Vblterra*.
Let us assume that K (x, y] is finite in S and that any discontinui-
ties which it may have are regularly distributed. From K we form the
iterated functions Kl, K.2, ... by means of the formulae
]£.f~ -A — ' V(~. t\ V ft *.\ Jt (
* Six papers come primarily into consideration here and in the following
sections,— four in the Atti of the Turin Academy, vol. 31, 1896 (Jan. 12— April 26),
arid two in the Rendiconti of the Accademia dei Lined, series 5, vol. o1, 1896
(March, April). These articles will be referred to as Torino I, II, III, IV and
Lined I, II, respectively. There is also a paper by the same author in the
Annali di Matematica, ser. 2, vol. 25 (1897), p. 139, which should be cited here.
It may be mentioned that Volterra's work on integral equations commenced in
1884 with a note " Sopra un Problema di Elettrostatica," Atti d. R. Accad. dei
Lincei, ser. 3, Transunti, vol. 8, p. 315, iu which a relation to the calculus of
variations was pointed out. For the present section, cf. Lincei I.
2—2
20 VOLTERRA'S TREATMENT OF [6
A reference to Theorem 1, § 1, shows us that all of these functions,
except possibly Klt are continuous throughout S.
By successive applications of (1) we get
Ki(*> y) = f ». /**•<*&)*&. 4) ... K(^,y-)d^ ... <fc (2).
Ja Ja
If, by means of (2), we express Kt (x, £) and K} (£, y} as (« - 1) and
rb
(j— l)-fold integrals respectively, we get for I Kt (x, I) Kj (£, y) rf£ an
Ja
(i+j - l)-fold integral, which, when we change the order of integration,
becomes, except for notation, precisely the value of Ki+J (x, y} given by
(2). We have thus established the important formula
= tbKt(x,VK,(t,y)dt ft/-l, 2, ...) (3)
Ja
which includes definition (1) as a special case.
If K(x, y) vanishes when y> x, we see from (1) that Kt (x, y) also
vanishes when y > x for all values of i. Consequently in this case (3)
may be written
= *i(ff,*)J}(&y)<# (ij=l, 2, ...) (4).
Jv
The case just mentioned is the only one considered by Volterra, and the
formula used by him is formula (4).
Returning to the general case, let us consider the series
'K! (x, y) + Kz (x, y) + K3(x,y)+... (5).
It will be shown presently that in certain important cases this series
converges uniformly throughout S. Let us assume that this is the case,
and denote the value of (5) by — k (x, y). Since every term in (5),
except possibly the first, is continuous in S, it follows from the assumed
uniform convergence of (5) that k (x, y) is finite in S and is discon-
tinuous only where K (x, y} is discontinuous.
Let us denote the sum of the first n terms of (5) by Sn (x, y) and
the remainder after this point by Rn (x, y), so that
,-6
R» W y} = 2 Ki-j (x, £) Kj (£, y) dt, (6).
i=n + 1 Ja
In this formula, the integer,/ may, if we please, vary from term to tenn,
being however always less than i.
Let us first assign to j the value n in all the terms. Then the
series in (6) is what we should get by starting from the series
6] EQUATIONS OF THE SECOND KIND 21
multiplying it by KH (£, y) and integrating it term by term. Since this
integration is allowable, we get
R* (*t y) = - f* (*, 0 K* (£ y) # (7).
/•
On the other hand, let us give to /in (6) the value i-n. Then the
series in (6) may be obtained by starting from the series
-k(t,y)= 5 JCi-.tf.y),
1=11 + 1
multiplying it by Kn (.r, £) and integrating it term by term. We thus
get as a second form for the remainder
R* (* y) = - f *. (*, *> * (*. ^) ^ (»)•
/a
Since we have
RI (*, y) = -k (•*•> y)-K to y),
it follows that in the case n = 1 the two formulae (7), (8) may be
combined into
(9).
This is one of the most fundamental formulae in the theory of
integral equations. We base upon it the following
DEFINITION. Two functions K(x,y) and k(x,y} are said to be
-•id if they are finite in S, if any discontinuities they may hace
are regularly distributed, and if they satisfy (9)*.
The relation here denned between two functions is, as the very name
" reciprocal " implies, independent of the order in which these functions
are taken, since (9) is not changed by an interchange of K and /•. On
the other hand, we see from (9) (cf. Theorem 1, § 1) that the sum of two
reciprocal functions is continuous.
We are now in a position to give Volterra's elegant solution of the
integral equation of the second kind
u (x) =f(x) + I K(x, £) u (£) d£ (10).
* We note, for later use, the following application of this definition, which was
given by Goursat (C. R. Feb. 17, 1908) : If K and k are reciprocal, and if r (x) is
continuous in I and does not vanish there, then
-7-r K (a-, u) and —7—, k (x, y)
r(y) r(y)
are also reciprocal.
22 VOLTERRA'S TREATMENT OF [6
Here we assume as before that K is finite in $ and that any discon-
tinuities it may have are regularly distributed. "We also assume that
there exists a function k (x, y) reciprocal to K(x, y).
If (10) has a continuous solution u (x), we may write
Multiplying this equation by k (x, £) and integrating, we get
fa £)•(£)<&
If we reverse the order of integration, and then apply (9), the second
term on the riht becomes
The second part of this cancels against the first member of (11),
while the first part may be evaluated by means of (10). Thus (11)
reduces to
u (as) =f(x} - fbt (x, £)/(# dt (12).
Ja
We see then that, under the conditions we have imposed, (10)
cannot have more than one continuous solution, and if it has one, this
will be given by formula (12).
In order to prove that the continuous function u(x) defined by (12)
really is a solution of (10), let us write (12) in the form
This may be regarded as an integral equation for determining f(x).
Since K is a function reciprocal to k, we see that this equation satisfies
all the conditions which we previously imposed on (10), so that,
by what we have just proved, its continuous solution f(x) is given by
the formula
This, however, is precisely the equation (10), which we thus see is
satisfied by u (x), and we have proved the
THEOREM 1. If K(x, £) is finite in 8 and any discontinuities it
may have are regularly distributed, and f(x} is continuous in I, then
6] EQUATIONS OF THE SECOND KIND 23
the equation (10) has one and only one continuous solution provided tht-rt
t\c/.<fs a function k (x, y) reciprocal to K(x,y); and in this case this
solution is given by (12).
We saw above that a function reciprocal to K will exist provided
the series (5) converges uniformly. Although this is, as we shall see in
§ 9 (Theorems 5, 6) by no means a necessary condition for the existence
of a reciprocal function, it will nevertheless be of interest to determine
certain cases in which this condition is fulfilled.
THEOREM 2*. If K(JC, y) is finite in S and any discontinuity •< >*
may have are regularly distributed, there will exist a reciprocal function
• by series (o) provided that
where M is the upper limit of { K(x, y) \ in S.
For under these conditions, as we see from (2),
and from this the absolute and uniform convergence of (5) follows at
once.
THEOREM 3. If K (x, y) is finite in S and any discontin uities it may
/'.'»•!• are regularly distributed, there ic ill exist a reciprocal function given
by series (5) provided that
K(*>y)-Q wheny>x.
In this case we can use formula (4), by means of which we easily
establish the inequality
Consequently the general term of (5) does not exceed in absolute value
the quantity
J/ (b-ay-1
(•-1)1
and since this is the general term of a convergent series of positive
constant terms, the absolute and uniform convergence of (5) follows at
once.
In either of these two cases, or indeed in any case in which the series
(5) converges uniformly, the solution (12) may be written
3
=l Ja
* For a theorem which in some cases is more far-reaching than this, see
Theorem 5, § 12.
24 LTNEAR EQUATIONS WITH AN [6, 7
If here we replace the iterated function Kn by its value (2), we get by a
mere change in the order of integration precisely the series which we
found in § 5 by the method of successive substitutions. Thus we get
a new proof of Theorems 1, 2, § 5. Conversely, from those theorems all
the results of this section can readily be deduced so far as they relate to
the cases covered by those theorems.
Finally we prove
THEOREM 4. There cannot exist two different functions ki (x, y) and
kz (x, y) both reciprocal to the same function K(x, y).
For, as we have seen, K + k^ and K + £2 would both be continuous;
accordingly the same would be true of the difference of these two
functions
o- (x, y) = fa (x, y} - k,2 (x, y).
By substituting in (9) first k = k^ then k = k% and subtracting the
resulting equations from each other, we find
This may be regarded as a homogeneous integral equation of the
second kind for determining a-, y being regarded as a parameter. Since,
by hypothesis, K has a reciprocal function, all the conditions of
Theorem 1 are fulfilled. Consequently (13) has only one continuous
solution, and this is, by inspection, <r = 0. We thus have ki = k%, and
the assumption of two different functions reciprocal to K is impossible.
7. Linear Algebraic Equations with an Infinite Number
of Variables. We come now to a very remarkable and important
relation between the theory of integral and of algebraic equations.
This relation seems to have been first noticed by Volterra, who pointed
out* that an integral equation of the first kind may be regarded as the
limiting form of a system of n linear algebraic equations in n variables
as n becomes infinite. Though not explicitly mentioned by Volterra,
his remarks make it at once clear that the same is true for equations of
the second kind.
It is Fredholm's great achievement to have seen how this observa-
tion could be utilized to pass from the solution of the system of linear
equations to the solution of the integral equation of the second kind,
which he was thus enabled to treat in a far more general manner than
* Torino I.
7] INFINITE NUMBER OF VARIABLES 25
it had ever been treated before *. This method was used by Fredholm
as a heuristic method for discovering first the facts, and secondly
methods by which they may be proved. Later Hilbertt showed how
the theory can be rigorously established by following out in detail the
limiting process of Volterra and Fredholm. We follow Fredholm,
giving in this section merely the heuristic part of the work, and
reserving the proofs for the next section. The whole of the present
section is therefore, from a strictly logical point of view, superfluous
and may be omitted.
Let us divide the interval ab into n equal parts by the points
.TJ = a + 8, .r, = a + 28, ... ,?•„ = a +n& = b f 8= — — ) .
If we replace the definite integral in the equation
ii (.r) =/(.r) + feQ*(&tt (1)
.'a
by the sum of which it is the limit, we get the equation
n (JT) =/(*) + 2 K (a-, a-j) u (q) 8 (2).
j=i
We shall see in a moment that this equation has in general one
and only one solution, and we may expect that this solution, which we
denote by un (.r), will have as its limit for n = oc the desired solution
Since equation (2) is to hold for all values of .r in ab, it must in
particular hold when >r = .r1} .?•.,, ....r,,. This gives us the system of
n equations
- 2 K(xit *j) un (.r,) 8 + Utt (.r,-) =/(.r,-) (/ = 1, 2, ... ») (3).
J=i
These may be regarded as n non-homogeneous linear equations for
determining the n unknowns un(.t\), ... «„(.?„). When these have
been determined, the value of «•»(.?•) may be found by substituting
* Sur une nonvelle methode pour la resolution du probleme de Dirichlet,
Ofversigt af Kongl. Vetenskaps-Akaderniens Forhandlingnr, vol. 57 (1900), p. 39.
This is the Swedish academy at Stockholm. A second paper, complete in itself,
and much more extensive, appeared in Acta Math. vol. 27 (1903), p. 365.
t Gottinyer Xachrichten, 1904, p. 49. This is the first of a series of papers in
which important contributions were made to the theory. The plan of deducing
the results as limiting cases of algebraic propositions as the number of variables
becomes infinite is consistently carried through.
26
LINEAR EQUATIONS WITH AN
[7
these values in the second member of (2). Consequently, if the
determinant
-8 K (#2,
of the system (3) does not vanish, we see that, as was asserted above,
there exists one and only one solution of (2). In fact, if we desire
not un (x) itself but merely its limit for n = » , we need not consider
(2) at all, for this limit, which we assume to be continuous, is
completely determined by the values of un(x^) obtained from (3).
The determinant Dn when expanded takes the form
- 2
K(xi,x^)K(xi,xJ}K(xi,
K(xk, Xi} K(xk, $j) K(xk, xk)
K(x^,xl) ...
(xn,xl} ... K(xn,xn]
If we allow n to become infinite, we get in this expression a larger
and larger number of terms, and the terms themselves vary and
approach definite limits. We are thus led to' the consideration of the
infinite series
r
=1-
Ja
!/•&/&
if-, - /
A]. J a Ja
We shall prove in the next section that this series converges. Its
value is called by Fredholm the determinant of the integral equation
(1) or of the kernel K. We shall not stop to prove that it is the
limit of Dn for n= oo, although the method of deriving it makes this
extremely plausible +.
* The factorials in the denominators of all but the last term are extremely
important. They are due to the fact that in the following 2 every term is repeated
the number of times indicated by the factorial in question. For the sake of
simplicity, the notation has been changed in the last term, so that the factorial
does not appear.
t The proof here required is very similar to the one which we meet in the
elements of analysis when we prove that the expansion of (1 + !/;«)"' by the
INFINITE NUMBER OF VARIABLES
27
Let us further consider the cofactor of the determinant Dn which
corresponds to the vth row and the nth column. We readily see that,
when /x =£ v, this determinant may be written
.*',) - 8
-28
i=l
K(xj,xr)K(xj,xi)K(xj,xj)
(-D"
2S"-2
(w-l)-rowed
determinants
If here we let n become infinite, and at the same time allow p. and v
to vary in such a way that lim (^>, xv) = (x, y), we get* as the limit of
8-1 Dn (XP, xv} the series
K(x,y) K(x,^]
1 f f
21 Ja L
» y) Kfr, ft) JT(ft , ft) rfftrfft - .-.
This series, the proof of whose convergence is reserved for the next
section, we shall call the adjoint of the kernel K+.
It should be noticed that we have not here considered the cofactors
of Dn which correspond to the elements of its principal diagonal.
These cofactors differ in form only slightly from the determinant Dn
itself, and it is clear that we shall get as their limits when n - x pre-
cisely the determinant D of the integral equation.
Let us now proceed to solve the system (3) of equations on the
supposition that Dn 4= 0. We have by Cramer's formulae
, . _/(.rt) Dn fa, .r,)
Da
0*= 1,2, ...»).
binomial theorem (HI a positive integer) approaches as its limit for m = x the
familiar series for e. Both of these proofs, and many others of a similar character,
may be easily carried through by means of a general theorem of Osgood, Annals
of Math. ser. 2, vol. 3 (1902), p. 138, Ex. 8.
* This statement of course requires proof. Cf. the preceding foot-note.
t Fredholm uses the termyir*^ minor.
28 LINEAR EQUATIONS [7
If we let n become infinite, and allow at the same time /* to vary
in such a way that x^ approaches x as a limit, we obtain as the
limiting form of this formula, when we remember that
(\ JV \ x I f / f-\ T\ / l-\ 7 f-
M/*» I f I rv* 1 I I I > I lilt'' 1~\ rt f-
jjj — J V") ' /-j / y \€y •*•' V1*) ^/ "'£•
•*/ Ja
This is actually the solution of the integral equation (3) as found
by Fredholm.
Instead of placing this solution on a firm foundation by justifying
all the steps we have taken, we prefer to establish it with Fredholm
ab initio in the next section. The method to be used will be suggested
to us if we recall how Cramer's formulae for the solution of (3) are
established ; namely by multiplying the equations (3) by the cofactors
Dn (#M, Xi) and adding them together. The essential point here is
that this has the effect of eliminating all the unknowns except un (&v).
This is due to the theorem which says that if we take the elements of
the /tth column of Dn and multiply them respectively into the cofactors
of the elements of another column, the sum of the results thus obtained
will be zero. This may be expressed by the formula
n
X •< If I ™ ff \ T) //». rn \ _l_ T) ( r V \ — O /A =b ll\
^ •**- \*^i') *^M/ ft \ A J if -*^n \** A > ^M/ — \ ^ r/*
If we here divide by - 8 and then let n become infinite, at the
same time allowing x^ and x^ to approach respectively the values
x and y, we get as the limiting form of this formula
I K(£t y) D (x, £) d£ + K(x, y)D-D(x,y) = Q (4).
Ja
This suggests that (4) is the essential instrument by which we shall
solve the integral equation (1) in the next section. This formula must
of course first be established more firmly than has yet been done.
By the side of this formula is another similar one* at which we
arrive by starting from the fact that if in the determinant Dn we
multiply the elements of the vth row by the cofactors of the cor-
* This formula, or something equivalent to it, is necessary in showing that
Cramer's formulae really satisfy the system of linear equations.
7, 8] FREDHOLM'S SOLUTION 29
responding elements of another row and add the results together, the
sum is zero. This may be expressed by the formula
-82 K(xv, xt) DH (xh #x) + J)n (av, *A) = 0.
i=l
By taking limits, as above, we are led to the second fundamental
formula
(*, I) D (t, y) dt + K(x,y)D-D (*, y) = 0 (5).
8. Predholm's Solution. We begin by establishing a theorem
concerning determinants due to Hadamard*. For this purpose consider
the following simple geometrical fact.
If a parallelepiped has one vertex at the origin and the three
adjacent vertices at the points (-TJ, yi, zj, (x2) y^ z?), (x3, yz, z3), it is
well known that its volume will be
A= x.2 y.2
.•TS ys
Let us suppose that the three edges issuing from the origin are of
unit length
*f+9? + *f-l (/=!, 2, 3) (1).
Then it is clear geometrically that the volume of the parallelepiped
will be a maximum when the three edges in question are mutually
perpendicular, in which case the volume is 1. Hence under the
conditions (1) we see that i A | ^ 1. This suggests at once the following
generalization :
LEMMA. If the elements a^ of the determinant
au ... «!„
are real and satisfy the conditions
«rt2 + «fc2+-..+a,v=l (i=l, 2, ...») (2),
then I A I = L
* Bull, des sciences math, et astr. 2nd ser. vol. 17 (1893), p. 240. Hadamard's
method is purely algebraic. "We follow a method of Wirtinger, MonatshefU f.
Math. u. Physik. vol. 18 (1907), p. 158. Both authors consider also more general
questions, in particular they consider the case in which the elements of the
determinant are imaginary. Cf. also Fischer, Archiv d. Math. u. Phys. 3 ser.,
vol. 13 (1908), p. 32.
30 FREDHOLM'S SOLUTION [8
In order to prove this, let us attempt to find the maxima and
minima of A under the conditions (2).
In the first place it is clear that A has both a finite maximum and
a finite minimum under the conditions (2), since these conditions
restrict the point (an, a12, ... «„„) in space of w2 dimensions to a finite
closed region, and A is a continuous function of these arguments.
Moreover since A has continuous first partial derivatives with regard
to these arguments, we may apply the ordinary method of the differ-
ential calculus for finding maxima and minima. Let us allow
#il> ai2> ••• ain
to vary, leaving the other a's constant. We thus get from (2)
a-adan + a-itflata + ... + aindain = 0 (3) ;
and when we remember that 3A/3«y. is the cofactor Ay of ay in A, we
see that a necessary condition for a maximum or minimum is
A a datl + Ai2 dai2 + ... + Ain dain = 0 (4).
Moreover since, when A is a maximum or minimum, every set of values
of the differentials which satisfy (3) also satisfy (4), the first members
of (3) and (4) are proportional to each other, and we have
Atj = kitty («',./ =1. 2> ••• »)•
Multiplying these equations by ay and adding, we get, on referring
to (2),
A = A,.,
so that* Ay = Aay (/, j=l, 2, ... ri) (5).
The determinant of the nth order of which Ay is the general element
has, as is well known, the value A""1. Forming this determinant from
the values of Ay in (5), we thus get
A""1 = A"+1.
Consequently, when A is a maximum or minimum,
A = ±l.
The maximum value of A is therefore + 1 and the minimum — 1, and
our lemma is proved.
We can now readily pass to a more general result. Let us suppose
that the elements of A are still real but that conditions (2) are not
fulfilled. Let us denote the value of the left-hand side of (2) by o-,-.
If, for the moment, we rule out the possibility of all the elements of
* Notice that, if A=f=0, this is precisely the condition that A be an orthogonal
determinant.
FREDHOLMS SOLUTION
31
one row of A vanishing, these cr/s are all positive, and we may consider
the determinant
U ll <?!»
V O'lO-.j • • •
a HI a™
This determinant satisfies all the conditions of our lemma, and
hence we have the inequality
A ^
If now we let M be a positive constant at least as great as any of the
quantities a^ , we see that
<r, ^ )i J/J
and thus we obtain the formula
A 5iVw"J/".
This inequality obviously also holds in the case we have excluded
in which all the elements in some row of A are zero ; and we have
proved
HADAMARD'S THEOREM. If the elements «<, of the determinant
A =
On ... alr,
«»!••• ««»
are real and satisfy the inequality
then [ A j ^ xV J/".
We proceed now to Fredholm's solution of the integral equation
(6)
in which we assume* that K(x, £) is finite in 8 and that any dis-
continuities it may have are regularly distributed, while f(f) is
assumed to be continuous in /. Furthermore we will assume that
K(x, x) is integrable throughout the interval /.
* We shall not consider the case in which K fails to remain finite. This case
is of considerable practical importance, and we refer the reader to Fredholm's,
Hilbert:s, and E. Schmidt's papers for various treatments of it.
82
FREDHOLM'S SOLUTION
We form the two series
"b \ 7C 1 f1 [b ' -^(£l> £l
-i) -1 + ^]Ja Ja | K(£.2,^
ft
= l-
Ja
_ 1 fb fb [
O\ Ja Ja Ja
(7),
fad&-... (8).
All of the multiple integrals which occur here obviously converge
and may be evaluated as iterated integrals taken in any order.
We will now prove that the first of these series converges absolutely,
and that the second converges absolutely and uniformly in 8. As was
already stated in the last section, we shall call the constant D the
determinant of K, and the function D (x, y) the adjoint of K*.
Let M be the upper limit of K(x, y] in & Then each of the
determinants which occur in (7) satisfies the conditions of Hadamard's
Theorem, and the general term of (7)
-i)Y'...f
W! Ja Ja
<&...
does not exceed in absolute value
The series of which CH is the general term converges, since
a quantity which obviously approaches zero as its limit when n = oc .
Consequently the series (7) converges absolutely.
Similarly the general term of (8) does not exceed in absolute value
Jnn Mn (h — a}n~1
O-l)!
1 JJn + -l V»+ 1 //^ 1\" ,r// x
and -^j~- = - - v / 1 + - M (u —a).
Bn n V \ ?2/
* Strictly speaking, the determinant and adjoint toj'tfe regard to the region S.
FREDllOLM'S SOLUTION
33
Since this last quantity approaches zero as its limit, the series of which
Bn is the general term converges ; and, this being a series of constant
positive terms, the series (8) converges absolutely and uniformly
in S.
It will, of course, be essential for us to discuss the question of the
continuity or discontinuity of the adjoint function D (x, y)*. For
this purpose let us write the general term of (8) in the form
'b
t
(x,y} ...
Ja Ja
(9),
where
rb r
L(4jr)«f .../
Ja Ja
(*, 60
(10).
Since the first term in (9) is the product of the general term of (7) by
K (x, y}, we may write
00 (— l)*
D (x, y) = D . K (x, y) + 2 - — ^- Qn (x, y) (11).
If we expand the determinant in (10) according to the elements of
its first row, we get
where
Let us change the notation for the variables of integration by
introducing in the /th integral in (12) £ in place of 6, f, in place of
6-+i, li+i in place of gu.2, etc., but leaving the variables 6, ••• 6-1
unchanged in notation. If then in this /th integral we bring the ith
* If we wished to assume that K (x, y) is continuous throughout S, we could
easily infer from the continuity of the terms in (8), and from the uniform
convergence of this series, that D (x, y) is continuous in S.
B. 3
FREDHOLMS SOLUTION
row of the determinant into the first place, all the terms of (12) are
seen to be equal, and we may write
n(x,y}=-nt\.. f
J a J
(13),
where
p =
•*- n
K (**-» ?)*-(&_„ £,) ... K(tn-lt 4-0
We are now in a position to examine the question of the continuity
of D (x, y}. For this purpose let us first show that the functions
Qn (%, y) as defined by (10) are continuous throughout 8. From either
(10) or (13) it is clear by a reference to Theorem 1, § 1, that $x (x, y} is
continuous throughout S. Let us assume that Q^, ... Qre_x are con-
tinuous there. If from this assumption we can infer that Qn is
continuous there, we shall have completed the proof that all Q's are
continuous by the method of mathematical induction. We may write
the second member of (13) in the form
rb rb
-n ... f K(x,t)
Ja Ja
The first of these integrals is merely a constant multiple of
Ja
and is therefore continuous by Theorem 1, § 1. The second may be
written
-n
and is therefore continuous for the same reason.
Having thus seen that all the $'s are continuous, we infer that the
function represented by the series in (11) is also continuous, since this
8] FREDHOLM'S SOLUTION 35
series is uniformly convergent in S, being the difference between the
uniformly convergent series (8) and the product of the series (7) by the
finite function K (x, y). We can now read off from (11) the finiteness
and the nature of the discontinuities of D (x, y).
The fundamental identity which connects K(x, y), D(x, y), and D
can now be established. If we substitute the value of Qn from (13)
in the series in (11), we get the same series we should obtain by
multiplying the series for D (£, y) by K (x, £) and integrating term
by term. Since this integration is permissible on account of the
uniform convergence of (8) and of the integrability of the value of this
series and of its terms, we have thus established the formula
In precisely the same way, if we expand the determinant in (10)
according to the elements of its first column, we find
n=l
2 Qn (*, y)=D (*, 0 K(t, y) d( (15).
-
By combining (14) and (15) with (11), we thus get the fundamental
formula
D (x, y}-D.K(x,y) = ! * K(x, fl D (*, y) d£
Ja
(16).
This is the same as formulae (4) and (5) of the preceding section,
which were there deduced without any attempt at accuracy.
We have thus proved
THEOREM 1. Every function K (x, y) which is finite in S and whose
fKaomfsfMttfMB are regularly distributed and for which K (x, x) is
integrable in I has a determinant D given by the absolutely convergent
series (7) and an adjoint D (x, y) given by the absolutely and uniformly
convergent series (8). This adjoint function is finite in S and is
continuous at every jmnt where K is continuous*. The function K, its
determinant and its adjoint satisfy the identity (16).
* If D = 0, D (x, y) is continuous throughout S. If D=4=0, D (x, y) is discon-
tinuous wherever A'(x, y) is discontinuous, and is discontinuous in such a way that
D (.T. y) - D . K (x, y) is continuous.
3—2
36 FREDHOLM'S SOLUTION [8
If D 4= 0, we may divide (16) by - D, and if we let
*<*») = -^ (17),
the identity (16) reduces to formula (9), § 6. Hence
THEOREM 2. Every function K (x, y) finite in S, whose discon-
tinuities are regularly distributed, for which K (x, x) is integrable in
I, and whose determinant is different from zero has a reciprocal, which
is given by formula (17).
By a reference to Theorem 1, § 6, we get at once the further result
THEOREM 3. If the kernel K (x, y} of the equation (6) is finite in
S, and any discontinuities it may have are regularly distributed, and
K (x, x) is integrable in I, and if its determinant D is not zero, and
f(x) is continuous in I, the equation (6) has one and only one continuous
solution, and this solution is given by the formula
In order to secure the existence of a determinant and an adjoint
we have been obliged to assume that K (x, x) is integrable, since
otherwise even the terms of the series (7), (8) would be meaningless.
Such a restriction on the kernel of equation (6) is, however, obviously
immaterial since a change of definition of K on the line x = y will
clearly have no effect on the solutions of this equation. In order to
cover all cases conveniently, we will lay down the following
DEFINITION*. Let K (x, y) be any function finite in S and whose
discontinuities are regularly distributed, and let
K (x ?/) = '
- o V , J) | Q when x-y ;
then the determinant D0 and the adjoint D0 (x, y) of K0 are called the
modified determinant and the modi/led adjoint of K respectively.
Using this definition, we obtain at once
THEOREM 4. If we drop the requirement that K (x, x) be integrable
in 7, Theorem 3 remains true provided we use in place of D and
D (x, y) the modified determinant and the modified adjoint of K
respectively.
It is to be noticed that the modified determinant and the modified
adjoint may exist when the determinant and adjoint do not. If,
however, the conditions of Theorem 1 are fulfilled, so that all four of
these quantities exist, it is a matter of some interest to know the
* The idea involved in this definition was used by Hilbert (Gottinger Nachrich-
ten, 1904, p. 82) for the purpose of treating a case in which K is not necessarily
finite.
8] FREDHOLM'S SOLUTION 37
relation between the determinant and the modified determinant, and
also between the adjoint and the modified adjoint. These relations
are given by the formulae
(18),
n / ,. ,A _ > when x *y
0 I*' y) - V [D (x, y) - DK (x, y)] when x = y
where D and D (x, y) are the determinant and adjoint of K, and Z),,
and D0 (x, y) the modified determinant and modified adjoint, and
(20).
The proof of (18) and the first half of (19) consists simply in
multiplying the series for D0 or Dn (x, y) by the series
while the second half of (19) may readily be deduced either in the
same way or from (16) in combination with (18) and the first half of (19).
The details of these proofs we leave to the reader*.
The case considered by Liouville and Volterra in which the upper
limits of integration are variable may readily be seen to come under
the last theorem. We get this case, as we have seen, by supposing
that K (x, y) - 0 when y > x. It is not hard to show that D0 = 1, and
that the series (8) for D0 (x, y) reduces to the series (5) of § 6, so that
the solution just given reduces in this case precisely to Volterra's
solution.
In conclusion, we turn very briefly to the case D = Q. If (6) has
a continuous solution u (x), we shall have
Multiply this equation by D (x, £) and integrate with regard to £ from
a to b. By reducing the result obtained by means of (16) we deduce
* The reader may also show that if K (x, y) is finite in S, and any discon-
tinuities it may have are regularly distributed, and its modified determinant D0*0,
it has a reciprocal which is given by the formula
k(x,y) =
38 THE INTEGRAL EQUATION [8, 9
THEOREM 5*. If the determinant of the integral equation (6)
vanishes, while the other conditions of Theorem 3 are fulfilled, a necessary
condition for the equation to have a solution continuous throughout
I is
/:
Apart from the special case in which D (x, y] = 0, it will be seen
that when D = 0 it is only for special functions f (x) that the equation
{6) can have a continuous solution.
Filially we note that if we drop the restriction that K (x, x) be
integrable, Theorem 5 remains true if we replace the determinant and
the adjoint by the modified determinant and the modified adjoint.
9. The Integral Equation with a Parameter. For many
purposes it is important to consider integral equations of the second
kind whose kernel contains a parameter A. In particular, the case in
which this parameter comes in only as a factor is of prime importance.
In this case the equation may be written
tb
Ja
(1).
It is customary to speak of K as the kernel of this equation. It
seems more consistent, however, to call \K the kernel, and this we
shall do.
It will be assumed throughout this section that / is continuous
throughout /, that K is finite in $, that any discontinuities K may
have are regularly distributed, and that K (x, x) is integrable through-
out /+. While we still suppose the functions /and K to be real§, we
shall allow the parameter A. to take on complex as well as real values.
This makes the kernel ^K no longer necessarily real, but this, as was
remarked in the second footnote to § 5 will not in any way affect the
* This theorem obviously corresponds to the fact that if the determinant
of a system of linear equations vanishes, the equations have, in general, no
solution.
t It may be noticed that if we let \ = l/p, Liouville's original equation (5), § 4,
has precisely this form.
* This last restriction need not be made for Theorems 1 and 2 below, while
in the later theorems it might easily be avoided by using the modified determinant
and adjoint.
§ It may readily be seen that the reality of K is not necessary for the proofs of
Theorems 1 and 2. That it is not necessary in the case of the later theorems
becomes evident if we make use of the more general form of Hadamard's Theorem
which refers to determinants with complex elements.
9] WITH A PARAMETER 39
developments of that section and the next, the main results of which
read as follows when we replace Kby ^K:
THEOREM 1. If K^ K», ... are the functions obtained by iteration
from K, then for any value of A for which the series
*(*,y, *) = -lKi(*,y)-KK*(*,y)-- (2)
converges uniformly in (x, y) throughout S, the function k (x, y ; A) is the
i -K-iprocal of \K (x, y}; and, for any such value of\, the equation (1) has
one and only one continuous solution, namely
u (x) =f(x) - bk (x, i • A)/(£) dt (3),
a solution which is obviously real if\ is real.
THEOREM 2. If M denotes the upper limit of | K(x,y) \ in S, the
hypothesis of Theorem 1 <"•/// 1 >t' fulfilled when
This hypothesis will be fulfilled for all values of \ if
K(x, y) = 0 when y > x (5).
Turning now to § 8, we first establish
THEOREM 3. The two power-series
brb
+
[I
*>-*;.
*(*,y)
-. ..(7),
r»n ft- rye for all values <>f A, and- the second converges uniformly in
(x, y ; X) for (x, y) in S and X restricted to any finite range of values.
For this range of values of(.r,y; A), the function D (x,y; A) represented
by the second series is finite and has no discontinuities except where K
is discontinuous.
These functions satisfy the identity
(8).
40 THE INTEGRAL EQUATION [9
If we restricted ourselves to real values of X, this theorem would be
hardly more than a special case of the results of § 8 obtained by
replacing K by A7T, the one new point being that the convergence of
(7) is uniform in A as well as in (x, y). This point, however, becomes
at once obvious when we replace A. by a positive constant p. such that
for the range of values of A considered | A. j ^ p..
In the case where A is complex* the theorem cannot be obtained as
a special case of § 8, since in that section we assumed that K was real,
and. \K, which now takes the place of K, is not real. The proof of the
theorem is, however, immediate if we run through the developments of
§ 8 again replacing K by ^K at every point. We leave it for the
reader to do this.
We add, also without proof, the following theorem which the
reader may establish by reasoning closely analogous to that used
in §8:
THEOREM 4. The series obtained by differentiating (7) p times term
by term with regard to A converges uniformly in (x, y ; A) for (x, y) in
S and A restricted to any finite range of values. The function
represented by this series is finite for this range of values and is discon-
tinuous only where K is discontinuous.
We next establish a lemma concerning power-series which we shall
find useful.
LEMMA t. If in the power-series
P(x,y\ X) = a«(x,y} + al(x,y}\ + az(x,y)X2+ ... (9)
each of the coefficients a,- (x, y) is finite in S, and if there exists a positive
constant R such that when A | < R the series (9) converges throughout S ;
and: if the function P represented by the series is a finite function of
(x, y ; A) when (x, y) varies throughout S and A varies on the circum-
ference \\\=-r, where r is a positive constant less than R; tk#n, if p is a
positive constant less than r, the series (9) converges uniformly in (x,y ; A)
for (x, y) in S and A ^ p.
For denote by M the upper limit of P when (x, y) is in S nnd
| A | = r. By a well-known theorem on power-series (cf. Forsyth, Theory
of Functions, 2nd ed., p. 34)
M^\an\ rn.
* No special consideration of this case would be necessary if Hadamard's
Theorem had been established for determinants whose elements are complex.
t This lemma may be extended at once to the case in which the a's are
functions of any number of variables, and S is replaced by any finite or infinite
region.
9] WITH A PARAMETER 41
Consequently, when (a; y) is in S and A | ^ p, the terms of (9) do not
exceed in absolute value the corresponding terms of the series
M+M?
r
and, this being a convergent series of positive constant terms, the
uniform convergence of (9) is established.
The exceptional character of the case in which D = 0 was evident in
§ 8. We lay down the following
DEFINITION. The roots of the Integral analytic function D (A) shall
be called simply the roots for the function K (x, y)*.
We are already familiar with one case in which D (A) has no roots,
namely the case in which K(x, y) is zero when y>x. A simple case
in which D (A) does have a root is when K(x, y) = l. Here
D (A) = 1 - A (b - a),
so that D (A) has one and only one root, namely A = l/(b - a). Other
important cases in which D (A) has at least one root will be considered
in§ 11.
By the proof used for Theorem 2, § 8, we obtain
THEOREM 5. Except when A is a root for K (x, y), the function
\K(x, y) has a reciprocal k(x, y ; A) given by the formula
This reciprocal is therefore an analytic function of A which has no
singularities in the finite part of the \-plane except perhaps poles at the
roots of D (A). We will now show that at any such point £ does have a
pole, at least for some points (.r, y) in 8.
For this purpose we first note the formula
whose truth follows at once from formulae (6), (7)t.
Let Aj be the root we wish to consider, and suppose that it is a root
of the mih order of D (A), so that we may write
#(A) = cm(A-A1)'» + Cm+1(A-A1)m+1+.... (w>0,<;m*0) (12),
this equation being valid for all values of A.
* We note that the roots of the modified determinant Z>0 (\) are the same as
those of D (X), cf. formula (18), § 8. We may therefore compute the roots for
K(x, y) as the roots of D0(X). We should so define them if K(x, x) were not
integrable.
t That D (£, £ ; A) is integrable in I is clear, since by (11), § 8, it differs by a
continuous function from a constant multiple of K (f , £). This same formula shows
us that the series for D (|, £ ; X) may be integrated term by term.
42 EQUATION WITH PARAMETER [9
On the other hand D(x,y; A), being analytic in A. for all finite points
in the A-plane, may be developed about A. = Aj in a power-series whose
coefficients are functions of (x, y}. Some of the first of these co-
efficients may be identically zero throughout S. They cannot, however,
all be identically zero, for then, as we see from (7), we should have
K(x, y) = 0, and consequently Z>(A) = 1, and this is inconsistent with
the hypothesis that D (A) has a root Aj . We may therefore write
D(x,y,*)= gk (m, y} (A. - ^)k + gk+l (a, y) (A - A,)**1 + . . . (k ^0, gh$0)
(13).
The coefficients gk, gk+1, ... differ merely by constant factors from
successive derivatives of D (x, y • A) with regard to A when A = Ax .
From Theorem 4 we therefore see that all these g's are finite through-
out S and are discontinuous only where K is discontinuous.
A reference to our Lemma now shows us that the series (13) con-
verges uniformly in (x, y ; A) when (x, y) varies throughout S and
A varies over any finite region in the A-plane. We may then substitute
the series (12) and (13) in (11), and integrate term by term on the left,
thus getting
S (A - A,)* (bgt (£ t) dt = A 5 ict (A - A,)'-'.
i=k Ja i=m
If here we write A = Ax + (A - Aj), we see that the second member of this
equation may be developed in a series proceeding according to positive
integral powers of A — A1; and beginuing with the term in (A — Aj)'"
whose coefficient is not zero, since Aj ^ 0, cm * 0. Since the left-hand
side is also a power-series in A- A1; it follows that it must also begin
with the same term, so that k ^ m — 1, that is
k<m (14).
Since /; (x, y; A) is the negative of the ratio of (13) to (12), we thus get
THEOREM 6. If Ax is a root for K(x, y), and k (x, y ; A) the recipro-
cal of \K(x, y), then, at least for some points in S, k(x, y ; A) has a
pole at the point A = Aj .
Instead of considering the development of k about a root of D (A),
let us now consider its development about the point A - 0. This is
certainly not a root of D (A), as we see from the form of the series for
D (A). Consequently the function k (x, y ; A) is analytic in A at the
point A = 0, and can be developed about this point in a power-series
which converges in a circle described about the origin as centre and
reaching out to the point in the A-plane nearest to the origin which
represents a root of D(\). It is possible that for some points (,r, //)
this series may converge outside of this circle, but it cannot do so for
9, 10] HOMOGENEOUS INTEGRAL EQUATIONS 43
all points of S. This development must be precisely the series (2),
since we have seen that that series represents /• (.r, y\ A) when A. is
sufficiently small.
By means of our Lemma we can go a step further. For, if we
denote by /• a positive constant smaller than the absolute value of
any of the roots of D (A), the function D (A) is continuous and different
from zero when A -r. We know also from Theorem 3 that D(x,y\ A)
is finite when i A - r and (x, y) is in S. Consequently, by (10), the
same will be true of /• (.r, y ; A). We may then infer by means of our
Lemma the uniform convergence of (2), and we thus get
THEOREM 7. 7/Aj is the nx/tfor K(.r,y) of smallest absolute calue,
the series (2) converges uniformly in (x, y; A) when A = p < Aj , and
(.r, y) is in S, to the function i- (,r, y; A) reciprocal to \K(x, y).
From Theorem 5 and Theorem 1 , § 6, we infer at once
THEOREM 8. If A is not one of the roots for K, equation (1) has one
and only one solution continuous in j", namely
u(x, A) =/(,)
It will readily be seen that u (x, A) is a function analytic at all finite
points of the A-plane except at the roots of D (A) where it can have no
other singularities than poles.
10. The Fundamental Theorem concerning Homo-
geneous Integral Equations, with some Applications. We
consider in this section the equation
where, as before, we assume that K is finite in S, that any discon-
tinuities it may have are regularly distributed, and that K(x, x) is
integrable in /*.
From Theorem 8, § 9, we see that, when A is not a root for K, the only
continuous solution of (1) is the obvious solution u = 0. It remains
merely to consider the case in which A is a root of D(\). We will
prove that in this case (1) always has a continuous solution which does
not vanish identically T.
* As in § 9, this last restriction might be avoided by using the modified
determinant and adjoint.
t This fundamental theorem was first established by Fredholm. We follow
Kneser, Rendiconti of Palermo, vol. 22 (1906), p. 233.
44 HOMOGENEOUS INTEGRAL EQUATIONS [10
Let Aj be the root in question, and develop the determinant and the
adjoint of \K into the series (12) and (13) of § 9. These series we
substitute in the identity (8), § 9, and, since the second converges
uniformly, we obtain, after dividing by (A - A^*, the formula
= x2(x-A1y-fc Jr<«,0fl($jr)<« (2).
i=fc Jo.
If here we let A = A, , we see, on referring to the inequality k < m
(cf. (14) § 9), that all the terms reduce to zero except the first term in
the first series and the first term in the last series. We thus get
9* (#, y) = *i ( f(*, £) g* (6 y) dt (3).
Ja
Thus we see that whatever constant value in the interval / we may
assign to y, the function <fe(#» y) is a solution of (1) when X = Xa.
Moreover, since we saw in § 9 that gk is finite in 8 and that any
discontinuities it has are regularly distributed, we see, by referring to
Theorem 1, § 1, that the second, member of (2) is a continuous function
of (x, y) throughout S. Hence gk is a continuous solution of (1).
Finally, we know that gk does not vanish identically. Thus we have
proved
THEOREM 1. A necessary and sufficient condition that equation (1)
have a continuous solution which is not identically zero is that X 60 « root
forK.
DEFINITION 1. By a principal solution J "or K(x, if) is understood a
continuous solution of t/ie homogeneous equation (1) which is not
identically zero*.
The last theorem established shows that to every root of D (X)
correspond one or more principal solutions. In fact it is easily seen
that to a root of D (A) will always correspond an infinite number of such
solutions. For if u^ (x) is a first principal solution corresponding to
the root X15 then clu1(x), where cl is any constant different from zero,
will clearly also be a principal solution corresponding to Xjt.
* A more complete term here would be principal solution in x (Ntilllosung
in x) ; by a principal solution in y being understood a continuous solution of
fb
u(y) = \ I K(x, y)u (.>•),!.,•
J «
which does not vanish identically.
+ Notice that the principal solution gk (x, y) obtained in the proof of Theorem 1
also contains a parameter, y.
10] HOMOGENEOUS INTEGRAL EQUATIONS 45
If, besides the solutions c1ul (.r), the equation (1) has, when A - X1}
other continuous solutions, let M, (x) be such a one. Then
will clearly be a continuous solution whatever the values of the constant
Ci and c» may be. Proceeding in this way, we see that unless the
equation (1) has when X = X1 an infinite number of linearly independent
continuous solutions the general continuous solution of (1) for X = AX
may be written
Ci K! (a?) + Ca M2 (a-) + . . . + c» «» (a-)
where uly ... «„ are linearly independent continuous solutions of (1) for
X = XX and c1} ... cn are arbitrary constants. The functions «1? ... um
may then be called a fundamental system of solutions of (1) when
X = Xj* ; and the number 72 may be called the index of the root Xj , a
term not to be confounded with the multiplicity of this root. We give
the formal definition of both of these terms :
DEFINITION •>. The number of linearly independent principal
solutions for K(x, y) which correspond to a root Xx for K is called
the index <//'X1 ; the number of times ^ occurs as a root of the analytic
function Z>(X) (the determinant of\K) is called the multiplicity o/Xa.
From analogy with the theory of linear algebraic equations we
should expect that the determination of the index of a root for K
would require the introduction of series which correspond to the
second minors, third minors, etc., of the determinant of the system
of linear equations just as the series for D(x,y\ X) corresponds to the
first minors of this determinant. Such series were, in fact, introduced
by Fredholm, and form an essential part of the article already cited.
By means of them he was able to determine completely the index of any
root of D (X), and among other things he established the interesting
fact that the index of a root of D(\) can never exceed its multiplicity.
In particular, since the multiplicity is necessarily finite, it follows that
the index is also finite. For the discussion of these questions, we refer
the reader to Fredholm's article. In § 12 we shall give a different
proof, due to E. Schmidt, that the index of a real root of D(X) is
finite.
We conclude this section with two fairly obvious applications of the
theory of the homogeneous equation, first to the theory of reciprocal
* AH this is in perfect analogy with the theory of homogeneous linear algebraic
equations. Cf., for instance, the author's Introduction to Higher Algebra (Mac-
millan, 1907), p. 49.
46 SYMMETRIC KERNELS [10, 11
functions, and secondly to the theory of the non-homogeneous
equation.
THEOREM 2. If K(x, y) is finite in 8 and any discontinuities it
may have are regularly distributed, and K(x, x) is integrable in /, a
necessary and sufficient condition that K have a reciprocal is that t/w
determinant of K do not vanish.
That this is a sufficient condition was already proved in Theorem 2,
§ 8. To prove it necessary, suppose the determinant were zero. Then
the equation (1) has an infinite number of continuous solutions, and
this is seen by Theorem 1, § 6, to be impossible if Kha,s a reciprocal.
THEOREM 3. If A. is a root f 01- K(x, y), the equation*
u (x} =f(x) + A (bK(x, i) u (() d€ (4)
Ja
has either no continuous solution or an infinite number of continuous
solutions.
For it is clear that by adding to a first continuous solution of (4)
any continuous solution of the homogeneous equation (1), we get
another continuous solution of (4).
In fact it is readily seen that the general solution of (4) will be
obtained by adding to a particular solution of (4) the general solution
of (1). Cf. the corresponding well-known fact for non-homogeneous
linear differential equations.
11. Symmetric Kernels. A function K(x, y) is said to be
symmetric if K(x, y) = K(y, x). Integral equations whose kernels are
symmetric not only play, as has been shown by Hilbert, a very
important part in the applications, but their theory may be used
(cf. the papers of Hilbert and Schmidt t) as a foundation for the theory
of integral equations whose kernel is not symmetric. We assume
throughout this section that K is finite in 8, and that its discon-
tinuities are regularly distributed.
* We assume, of course, that f(x) is continuous in 7, and that K satisfies the
conditions stated at the beginning of this section.
t A series of papers by Hilbert in the Gottinger Nachrichten, beginning in 1904,
entitled "Gruiidziige einer allgemeinen Theorie der linearen Integralgleichungen " ;
and E. Schmidt, Math. Ann. vol. 63 (1907), p. 433. The greater part of this last
paper originally appeared in 1905 as a Dr. dissertation. A continuation, referring
to non-linear integral equations, appeared in Math. Ann. vol. 05 (1908), p. 370.
11] SYMMETRIC KERNELS 47
THEOREM 1. If K(x, y) is symmetric and does not vanish at nil
point* <>f .s' whrre it is continuous, then all of the iterated functions
K.2 (.r, y), K-A(x, y), ... are symmetric and none of them is identically
ten.
For let Kn (x, y) be the first of these functions which is not
symmetric. Then
Kn (x, y} = i V.-! (x, *) JCi (£, y) ft (I).
.'a
This, by the symmetry of Kl and Kn_i, reduces to
which is precisely Kn (y, a:}. Thus Kn is symmetric.
On the other hand, if some of the iterated functions vanish identically,
let KH-i (x, y) be the first one to do so. We see by (1) that Kn(x, y)
also vanishes identically. One of the two integers n — 1 and n is even.
Calling this even integer 2m, we have
0 = K,m (x, y) = \bKm (x, *) Km ((, y} d(.
Ja
Accordingly, owing to the symmetry of Km,
This, however, is possible only if Km vanishes at all points of S where
it is continuous. Since, as we saw in § 6, K2, K3, ... are continuous
throughout S, we are thus led to a contradiction, and our theorem is
proved.
"We now come to a fundamental existence theorem first established by
Schmidt, and which we shall prove by a method due to Kneser*.
THEOREM 2. For every symmetric function K(x, y} which does not
vanish at t eery point irkere it is continuous, there is at least one root.
This theorem will be established (cf. Theorem 7, § 9) if we can show
that the series (2) of § 9 is not uniformly convergent in (x, y) for every
value of A. Let us then assume that it is uniformly convergent
throughout S for every value of A. The series
is then, for every value of A, uniformly convergent throughout /. It
* Loc. cit. p. 236.
48 SYMMETRIC KERNELS [11
can therefore, since its terms are continuous, be integrated term by
term, and, if we let
rb
Un= \ Kn (x, x} dx (2),
Ja
the series Z72A2 + £73A3+ ...
is convergent for all values of A. Since this is a power-series, it is
necessarily absolutely convergent, and hence a series formed from part
of its terms is convergent. Thus
Z72A2+ U4\4+ £76A6+... (3)
converges for all values of A.
If, now, we remember that K is symmetric, we get, by formula
Un+m = f IK* (x, £) Km (x, I) d£dx,
Ja Ja
and in particular
Ja Ja
Hence by expanding the obvious inequality
r*> rb
I I [pKn+l (x, £) + qKn-i (•?, £)]2 d^dx = 0,
Ja Ja
where p and q are arbitrary real parameters, we get
that is, the first member of this inequality is a positive definite quadratic
form in (p, q). Consequently
or, since by (4), in combination with Theorem 1, the V's with even
subscripts are positive (not zero),
2n+2 > 2n /c\
TT ~ TT W-
Now in the series (3), the ratio of two successive terms is
^at+2 >,2
~TT~ '
i^2n
which, by a successive application of (5), we see is not less than
U.'
If then we let A= \ff72/l74, we see that the terms of (2) do not
decrease as we go out in the series, and consequently the series cannot
converge. Since this is a contradiction, our theorem is proved.
11] SYMMETRIC KERNELS 49
Our proof establishes at once the
COROLLARY. Under the hypothesis of Theorem 2, there is at least
one root for K(x, y) which in absolute value does not exceed >JU^Ut)
where the U's are defined by (2).
Let us now suppose that for the symmetric function K(x, y) there
are two roots, Ax and A^, and let us denote by Ui (x) and u2 (x) principal
solutions for K which correspond to \ and A^ respectively. Then
Let us multiply the first of these equations by ^u^(x), the second by
AjMj (x) and subtract. This gives
rb
(\2 - A.J) Ul (x) u2 (x) = A! A, K (x, £) [ttj (£) u» (x) - w2 (£) ^ (x)] d£,
Ja
and from this follows
rb
(A.J - Aj) / MJ (x) uz (x) dx
Ja
= A, A, f & /*[JT (ar, 0 u, (*) w2 (ar) - ^(f , a?) a. (ar) «, (0] dtdx.
Ja Ja
By interchanging the variables of integration in the second half of this
double integral, we see that the second member of this equation reduces
to zero, and, since by hypothesis A2 4= Aj , we get the result
THEOREM 3. If ^(x) and u^(x) are principal solutions for a
symmetric function K (x, y) which correspond to two distinct roots,
then
.'a
= 0.
The method we have just used is essentially that invented by
Poisson in the analogous case of linear differential equations of the
second order, and we will follow out the analogy still further by
proving, also by Poisson's method, as Schmidt has done,
THEOREM 4*. For the symmetric function K (x, y), there can be
only real roots.
Since the coefficients of the power-series D (A) are real, imaginary
* This theorem was first proved by a wholly different method by Hilbert as
an extension of a well-known theorem concerning symmetrical determinants.
Cf. Weber, Algebra, 2nd Ed. vol. 1, p. 309.
B. 4
50 SYMMETRIC KERNELS [11
roots necessarily occur in conjugate imaginary pairs. If possible let
/x. + vi and /A - vi be two imaginary roots. Let v (x) + iw (x} be a
principal solution corresponding to /A + vi, so that
rb
v (x) + iw (x} = (//, + vi) I K(x, £) [v (£) + iw (£)] dg.
Ja
If in this equation we separate real and imaginary parts, and then
recombine the resulting equations, we see that v (x} - iw (x} is a
principal solution corresponding to /A - vi. Consequently by Theorem 3
/'
Ja
This, however, is impossible, since v and w cannot both vanish
identically, as otherwise v + iw would not be a principal solution.
Thus our theorem is proved.
We saw in § 9 that the roots for K(x, y) are the poles of the
function Jc(x, y \ X) reciprocal to XK(x, y). In the case we are now
considering, in which K(x, y) is symmetric, we will show that these
poles are of the first order ; or, to use the notation of § 9, that
for every root X1} m = k+\. Suppose this were not the case, so that
m>k+l. Then from formula (2) of § 10 we should not only get,
by letting X = Xx ,
ff* & y} = AI f V(ar, £) gk (t, y} <ti (6),
Ja
as in § 10, but also, by first differentiating with regard to X and then
letting X = Xj,
rb rb
gk+i (*?, y) = I K(%, I) 0t (£, y) dt + *il K(x, |) gk+1 (£, y) d£.
Ja Ja
This may be reduced by (6) to the form
(x, () gk+l (t, y) dt (7).
We now multiply (7) by gk(x, y), (6) by gk+l (x, y\ subtract (6) from
(7), and integrate the resulting equation with regard to x from a to b.
This gives
rb rb
+ Xj IK (x, £) [gk (x, y) gk+l ((, y) - gk+l (x, y} gk (& y}} d£dx.
Ja Ja
The double integral is seen, precisely as in a similar case in the proof
11] SYMMETRIC KERNELS 51
of Theorem 3 above, to have the value zero, and we thus get the
formula
.'a
this equality holding for all values of y. This, however, is impossible,
since gk (x, y) is not identically zero. Thus we have proved
THEOREM 5*. If K(x,y) is symmetric, and k(x,y; X) is the
reciprocal of \K(x, y), then k has no singularities in the finite part of
the \-plane other than poles of the first order.
It should be noticed that this theorem does not by any means assert
that D (A) has only simple roots.
The results we have obtained in this section for symmetric kernels
may be extended to certain more general cases by the following device
due to Goursat t :
Form the function
K' (x, y) = >Jp(x)p(y)q(x)q(y) * (x, y)
where K is symmetric and does not vanish at every point where it is
continuous, while p and q are continuous and positive (not zero)
throughout /. The function K' then satisfies all the conditions im-
posed on K in Theorems 2, 4, 5. Now let
q(x)
and form the function
K(x, y} = K' (x, y)=p (x} q (y) * (x, y) .
Let us denote by k' (x,y; X) the reciprocal of \K' (x, y). Then the
reciprocal of \K(x, y) will, by the footruote to the definition of reciprocal
functions in § 6, be
r (x) , , . . .
~r k (x, y ; X).
r(y)
* Cf. the proof of this theorem in a somewhat more general case by Boggio, Paris
C. B. Oct. 14, 1907. The case considered by Boggio is less general than the result
of Goursat to be given in a moment. We mention in passing that the present
theorem is the extension to the case of an infinite number of variables of part of the
well-known algebraic theorem which says that the elementary divisors of the deter-
minant Dn of § 7, when this determinant is symmetric and K is replaced by \J£,
are all of the first degree. Cf., for instance, the writer's Introduction to Higher
Algebra, p. 305.
t Paris C. E. Feb. 17, 1908.
4—2
52 ORTHOGONAL FUNCTIONS [11, 12
Now the roots for a function are the poles of the reciprocal of X times
this function. Hence the roots for K(x, y) are the poles of the function
last written, and since these are the same as the poles of k' (x, y ; A),
which in turn are the roots for K ' (a, y), we see, by Theorem 2, that
there is necessarily at least one root for K(x, y\ and, by Theorem 4,
that these roots are necessarily real. Finally, by Theorem 5, the
reciprocal of \K(%, y} has no finite poles of order higher than the first.
Thus we get the result, which includes Theorems 2, 4, 5 as special
cases,
THEOREM 6. If K (x, y) is symmetric and does not vanish at all
points where it is continuous, and if p (x) and q (x) are continuous
throughout I and do not vanish there*, then for the function
K(x,y)=p(x}q(y)K(x,y)
there is at least one root, all such roots are real, and the function
reciprocal to h.K(x, y) has no finite poles of order higher than the first.
12. Orthogonal Functions. We begin by laying down the
following
DEFINITION 1. The functions ^(x), u2(x), ..., finite or infinite in
number, are said to be orthogonal to each other in the interval I if
(a) they are real and continuous in I;
(b) no one of them is identically zero in I;
(c) every pair of them, Ui (x) and Uj (x), satisfy the relation
'
.la
The connection between the subject of orthogonal functions and the
theory of homogeneous integral equations with a symmetric kernel is
evident from Theorem 3, § 11. On the other hand we recall that in
other branches of mathematics, in particular in the developments of
arbitrary functions which occur in mathematical physics, systems of
orthogonal functions play an important part. Thus, for instance,
in the subject of Fourier's Series we have to deal with the system
of functions
fl cos# cos 2x cos ^x ...
1 sin x sin 2% sin 3^ ...
orthogonal in the interval 0 ^ x ^ 2tr. Or again, the Legendre's
Polynomials
P«(x], P,(x\ P,(x), ...,
* We are clearly justified in dropping the restriction that p and q be positive
since a change of sign of K does not affect our result.
12] ORTHOGONAL FUNCTIONS 53
in terms of which arbitrary functions may be developed in the interval
— 1 ^ x ^ + 1, are orthogonal in this interval. On the other hand the
Bessel's Functions
J0 (c4 x\ J0(<hX\ ...,
in terms of which arbitrary functions may be developed in the interval
0 = x ^ 1, where a1} a^, ... are the roots of the transcendental equation
J0 (a) = 0, are not orthogonal in this interval, but become so if they are
all multiplied by the factor >Jx.
This subject of orthogonal functions is therefore one which has long
been of importance for its own sake*. We shall first develop this
subject independently, and then make some applications of the results
obtained to the theory of integral equations t.
THEOREM 1. If iii(x), ... un(x} are orthogonal in /, they are
necessarily linearly independent there.
For, if possible, let there be a relation
C-L Ui (x) + . . . + CnUn (x) = 0
where at least one of the constants c, say cv, is not zero. Multiplying
this equation by uv(x) and integrating from a to b, we get
cv f [uv(x)Ydx = 0,
Ja
which is impossible.
THEOREM 2. If i^ (x), ... un (x) are real and continuous throughout
/, and are linearly independent, there exist n linear combinations with
constant real coefficients of these functions which are orthogonal in I.
Let us begin by assuming that this theorem is true in the case of
n — 1 functions, so that we can get linear combinations with real
constant coefficients ih(x\ ... vn-l(x) of the functions ^(x), ... un_^(x)
which are orthogonal in /. It remains merely to show that the real
constants c may be so determined that the function
V* (X) = C^\ (x)+ ...+ C^Vn-! (X) + lln (x)
is orthogonal to t\ (x), . . . v^ (x). Let k be any one of the integers
1, 2, ... n - 1. If we multiply the last equation by vk(x) and integrate
from a to b, we get
rb rb rb
\ v* (x) vt (x) dx = ck [vk (x)]2 dx + M, (x) vk (x) dx.
Ja Ja Ja,
* So far as the writer knows, the term orthogonal was first used in this sense
by F. Klein in a course of lectures on the differential equations of mathematical
physics, delivered in Gottingen in the summer of 1889.
t Cf. Gram, Crelle, vol. 94 (1883), p. 41 ; and E. Schmidt, Math. Ann. vol. 63
(1907), p. 443.
54 ORTHOGONAL FUNCTIONS [12
If, then, we let
rb
I un (x) vk (x} dx
the function vn(x) will be orthogonal to vl (x), ... vn^ (x).
In order that the proof of our theorem by mathematical induction
be complete, it is merely necessary to notice that the theorem is true
(though trivial) in the case of a single function.
We saw above that when we have an infinite set of orthogonal
functions u^(x), u2(x), ... the problem of developing an arbitrary
function / (x} in a series of the form
f (x) = c^u^x} + c.2u<i(x) ... (1)
frequently presents itself, the series to hold in the interval / in which
the functions are orthogonal. Without attempting to consider this
question, we merely give the familiar formal determination of the
coefficients, a determination which is completely justifiable if we know
that f(x) can be developed into a series of this form which is uniformly
convergent, or, more generally, which after multiplication by any con-
tinuous function can be integrated term by term.
This formal determination of the coefficients consists in multiplying
(1) by iii (x) and integrating term by term from a to b. All the terms of
the series but one then drop out on account of the orthogonality of the
functions, and we get
fb
I f (x) ut (x} dx
(2)-
Ja
The ordinary formulae for the coefficients of a Fourier's Series or of a
series in terms of Legendre's Polynomials or of Bessel's Functions are
of course only special cases of (2).
By multiplying the functions Ui, uz, ••• by suitable real constants
we can obviously make the denominators in (2) take on any positive
values we please. If, in particular, we make them all take on the
value 1, we say that the functions are normalized.
DEFINITION 2. A set of functions orthogonal in I are said to be
normalized if the integral of the square of each one extended over I is 1 .
A much more elementary problem than the problem of development
we have just touched upon is the problem of getting an approximate
representation of a function f(x) in the interval / by means of a given
12] ORTHOGONAL FUNCTIONS 55
finite set of orthogonal functions HI(#), ... uk(x). We will suppose
f(x) to be finite in / and to have at most a finite number of discon-
tinuities there. We wish to determine the constants c,, ... ct in such a
way that the function
F(x) = c^ul(x)+ ... + ckut(x)
gives the best approximate representation of f(x} in / in the sense of
the method of least squares. That is, we wish to determine the
constants cl} ... ck so that
shall be a minimum. We have
™ = - 2 ?- Ff(x) F(x) dx + j-
CC- ' OC-
Now since
[b » fb
we get
! fb
>»\ ji I yt\ // f* \_ O/» I I II I fl I /T
/ * \ / ^'*^ "~^^i § l"i \ / I
Jo
Equating this to zero, we get as the desired values of d, ... ct precisely
the quantities (2).
Since cy/?cr>0, &J/dCidCj = 0, we see that formulae (2) really
correspond to a minimum of J, and we have thus proved
THEOREM 3. If Ui(x), ... itk(j") are orthogonal in I and f(x)
is finite in I and has at most a finite number of discontinuities, a
necessary and sufficient condition that the finite series
give the best representation of f(x) in I in the sense of the method of
least squares is that the c's be determined by (2).
For the sake of getting simpler formulae, let us suppose that the
functions «,(.r) are normalized. Then the minimum value of the
integral J is seen, by making use of the values (2), to be
- f [/(£)? #-*( I
Ja i=l\Ja
56 ORTHOGONAL FUNCTIONS [12
Since the left-hand side of this equation is positive or zero, the same is
true of the right-hand side, and we thus get the important inequality
2 ( f /(*) «, (*) #)' ^ f [/(£)? ** (4).
i=l\ya / ya
As an application of this inequality, we will establish the result,
already stated without proof in § 10,
THEOREM 4. If K(x, y) is finite in S and its discontinuities are
regularly distributed, then every real root for K (x, y) has a finite
index.
For suppose that to the real root \ there correspond n linearly in-
dependent real principal solutions for K(x, #)t. From these by means
of Theorem 2 we can form n orthogonal principal solutions. Let us
call these functions, after they have been normalized, u± (x}, ... un (x).
We will take these for the functions u of (4), and for /(£) we will take
K(x, £), which, for any fixed value of x, is finite in / and has only a
finite number of discontinuities there. When we remember that
i
«* (*) = xi K(%> £) Ui (£) d£ (i =1,2,... n\
Ja
the inequality (4) reduces to
AI i=l
If we integrate this with regard to x from a to b, we get on the
left, when we remember that the w's are normalized, n/\iz ; and thus
finally
(5).
This inequah'ty gives us an upper limit for the number n of linearly
independent real principal solutions corresponding to Xlt It is easily
seen, however, that the number of linearly independent complex
principal solutions cannot be greater than the number of linearly in-
dependent real principal solutions. Thus our theorem is proved.
* Formula (3) is called by E. Schmidt Bessel's Identity, and (4) Bessel's
Inequality, because in the Astr. Nachr. vol. 6 (1828), p. 333, Bessel had in the
special case in which the w's are trigonometric functions considered a problem
analogous to the one here treated, in which, however, no integrals but only finite
sums present themselves. The first to consider precisely the problem here
treated both for trigonometric functions and for Legendre's polynomials seems to
have been Plarr, Paris C. R. vol. 44 (1857), p. 985.
t We do not here exclude the possibility of still other linearly independent
principal solutions.
12] ORTHOGONAL FUNCTIONS 57
The inequality (5) may also be regarded as giving us a lower limit
for the absolute value of the real root Xj. By letting n = 1, we thus get
a lower limit for the absolute value of any real root. Hence we see that
in the case of a symmetrical kernel, or in the more general case of
Theorem 6, § 11, no root can be in absolute value less than
In these cases we may therefore replace Theorem 2, § 6, by the
following more far-reaching result :
THEOREM 5. If the finite function K whose discontinuities are
regularly distributed satisfies the conditions of Theorem 6, §11, and
the series (5) of§ 6 converges absolutely and uniformly.
In the special case in which K(x, y) is symmetric, the roots and
the real principal solutions for K are also known as the characteristic
numbers and the characteristic functions* of K respectively. We saw
in§ 11 that these characteristic numbers are real, and that two charac-
teristic functions corresponding to different characteristic numbers are
orthogonal to each other. Moreover, by Theorem 4, we see that each
characteristic number has a finite index. For each characteristic
number whose index n is greater than 1, we can pick out a set of
n orthogonal characteristic functions on which every other character-
istic function corresponding to this characteristic number will then be
linearly dependent. We thus get characteristic functions UI(-T),
w2(.r), ..., finite or infinite in number as the case may be, ortho-
gonal to each other in /, and such that every characteristic function
of K is linearly dependent upon a finite number of them. Such a
system we speak of as a complete orthogonal system of characteristic
functions. We may, of course, normalize it if we choose.
A problem of fundamental importance is: under what conditions
can a function J (j-) be developed in a series whose terms are constant
multiples of the elements of this complete orthogonal system of
characteristic functions, the coefficients of the series being deter-
mined by formula (2) ? For such treatment as has, up to the
present time, been given of this problem we refer to the papers
* Eigemcerte and Eigenfunktionen. It is only when A' is symmetric that the
definition here given of these terms is correct. For the case of an unsymmetric
kernel K, cf. the definition given by E. Schmidt, loc. cit. p. 461.
58 ORTHOGONAL FUNCTIONS [12
already cited by Hilbert and Schmidt*. We confine ourselves to a
single important, though very special, case, namely, that in which
the function f(x) to be developed is the symmetric kernel K(x, y}
itself, and even here we do not attempt to treat the problem completely.
Let us suppose that the complete orthogonal system of character-
istic functions Ui (x), u2 (x), . . . has been normalized, and denote by
AU AS, ... the characteristic numbers corresponding to themt. For-
mula (2), when we let J (x} - K(x, y), now reduces to
(bir/ \ i \ j
= I K (x, y) uv (x) ax =
Ja
We are thus led, by this purely formal work, to inquire whether the
series M*)*»(y) + «.(g)m(y) + (6)
A! A2
really converges and represents the function K(x, y). We prove here,
following the method of Schmidt, the theorem due to Hilbert,
THEOREM 6. If the symmetric function K(x, y) is finite in S and
its discontinuities are regularly distributed, and if Ui (x\ u2(x), ... is a,
complete normalized orthogonal system of characteristic functions of K,
and A15 A2j ... are the corresponding characteristic numbers ; then if the
series (6) converges uniformly throughout S, it represents the function
K(x,y) at every point where this function is continuous.
Since the series (6) converges uniformly and its terms are continuous,
the function represented by (6) is continuous throughout S. It is also
obviously symmetric. Consequently the function
Q(.,Q.f(f, $-!*!<&!& (7)
is symmetric and finite in S, and its discontinuities are regularly
distributed. Let us assume that the theorem is false. Then Q does
not vanish at every point where it is continuous ; so that, by Theorem 2,
§ 11, it has at least one characteristic number c. Let </>• (x) be a
characteristic function of Q corresponding to c, so that
(8).
* Cf. also Kneser, Math. Ann. vol. 63 (1907), p. 477, where application is made
to the problem of developing an arbitrary function according to the solutions of
a linear differential equation of the second order. This application had already
been made by Hilbert, though much less successfully.
t It should be noticed that the X's are not necessarily all distinct.
12] ORTHOGONAL FUNCTIONS 59
If we multiply (7) by iti(£)d£ and integrate, we get, when we remember
that the w's are normalized,
= I
./a
(9).
From (8) and (9) we deduce
f V (*) «« (0 # - c f f Q (£, &) «, (*) ^ (6) #1 # = 0 (10).
./a ./a Jo,
Multiplying (7) by ^ (£)<££ and integrating, and reducing by means of
(10), we find
The first member of this equation is, by (8), simply iff (x)jc. Hence
Since i^(.r) is continuous and does not vanish identically in 7, this
shows that $ (.r) is a characteristic function of K, so that it must be
linearly dependent on a finite number of the u's. This, however, is
impossible, by Theorem 1, since, by (10), ^ is orthogonal to all the us.
Thus the theorem is proved, since the assumption that it is not true
has led us to a contradiction.
The condition of uniform convergence of (6) is, of course, fulfilled if
the series has only a finite number of terms, that is if K has only a
finite number of characteristic numbers. In this case our theorem tells
us that K is equal, wherever it is continuous, to the finite series (6),
which is continuous throughout £ Any discontinuities which K has
must therefore be removable discontinuities, that is, such that K
becomes continuous throughout S if its definition at its points q/
discontinuity be suitably changed.
Conversely, if we can find a finite sum of the form
where the fs and <£'s are continuous in /, such that this sum is
equal to K at all points of S where K is continuous, it is readily proved
that K can have only a finite number of characteristic numbers. For,
in the first place, in computing these characteristic numbers we may
obviously use (11) in our kernel in place of K(x, y). Now if we sub-
stitute (11) for K in the series for D(^) (formula (6), § 9), it will be
seen that all the determinants of order higher than n in this series have
the value zero. Consequently D (X) is a polynomial of at most the nth
60 EQUATION OF FIRST KIND [12, 13
degree in X, and cannot, therefore, have more than a finite number of
roots. Thus we have proved
THEOREM 7. I/ the symmetric function K(x, y} is finite in 8 and
its discontinuities are regularly distributed, a necessary and sufficient
condition that K have only a finite number of characteristic numbers is
that there exist a sum of the form (11), where thefs and (J>s are continuous
in I, which is equal to K at all points of S where K is continuous.
In particular, if the symmetric function K(x, y) is finite in $and
has discontinuities which are not all removable and which are regularly
distributed, it will surely have an infinite number of characteristic
numbers.
13. The Integral Equation of the First Kind whose Kernel
is Finite. In this section and the next we take up briefly the subject
of integral equations of the first kind. We shall be concerned, in the
main, with the equation
/(«)-• ife 0*(0# (i),
which has been so extensively treated by Volterra* that Picard has
proposed to call it Volterra's Equation.
Let us assume that K is continuous throughout the triangle T, and
that it has a derivative
finite in T and whose discontinuities are regularly distributed. In
order that (1) have a solution u (x) continuous in /, it is evidently
necessary that / (x) be continuous in / and that f (a) = 0. Differen-
tiating (1), we get
/' (x) = K (x, x} u (x) + [XK^ (x, £ ) u (£) dt (2).
From this we deduce at once, as a further necessary condition, that
/ (x) have a derivative /' (x) continuous throughout /. In order that
(2) be an equation of the second kind satisfying the restrictions we
have imposed in §§ 5, 6, it is now sufficient to impose on K the further
restriction that K (x, x} do not vanish at any point of /. Making this
assumption, we can infer at once that (1) cannot have more than one
continuous solution, and, if it has any such solution, this will be the
continuous solution of (2).
* See the papers cited in § 6. Cf. also Holmgren, Atti of the Turin Academy,
vol. 35 (1900), p. 570 ; and T. Labesco, Journal de MatMmatique, 6th eer. vol. 4
(1908), p. 125.
13] WHOSE KERNEL IS FINITE 61
We may also, if we impose the conditions just stated, work back
from (2) to (1). For equation (2) may be written
Accordingly the continuous solution of (2) also satisfies the equation
f(x) - \X K (x, £) u (£) d£ = const.,
and by taking the limit as x approaches a, we see that this constant
has the value zero. Thus we have proved
THEOREM 1 *. If K (x, £) is continuous in T and has a derivative
Ki — ^KI^x finite in T and ivhose discontinuities are regularly dis-
tributed, and if K (x, x} does not vanish at any point of I, a necessary
and sufficient condition that (1) have a continuous solution is that f(x)
and its derivative f'(x} be continuous in I and f(a) — Q. If these
conditions are fulfilled, (1) has only one continuous solution, namely the
continuous solution of the equation of the second kind (2)t.
Without considering exhaustively the case in which K (x, x)
vanishes at one or more points in /, we wish to examine it sufficiently
to show that it is really an exceptional case.
Let us first suppose that K (x, x) vanishes identically in /. In
this case (2) reduces to
(3).
Assuming that not only K but also KI is continuous in T, and that
KI (x, x) does not vanish at any point of /, we see that equation (3)
comes under the case covered by Theorem 1 provided that KI has a
finite derivative whose discontinuities are regularly distributed. More-
* Cf. Lincei I. See also Torino I, and for a slightly earlier but less complete
discussion, Le Roux, Annales de VEcole normale superieure, 3rd ser. vol. 12
(1895), p. 243.
t Volterra has also indicated a second method for reducing the equation (1) to
an equation of the second kind. This consists in performing an integration by
parts. If we let
K, (x, y)=
,
we get from (1) by an integration by parts
-/.
f(x)=K(x,x) V(x) - K%(x, |) U(Qd£ (2').
We leave it for the reader to formulate conditions under which we can get the
general continuous solution of (1) by differentiating the continuous solution
of (2').
62 EQUATION OF FIRST KIND [13
over we see as before that any continuous solution of (3) is also a
solution of (1); and we get
THEOREM 2. If K (x, £) and K± (x, £) = dKfix are continuous in
T, and Ku (x, £) = (PK/dx2 is finite in T and its discontinuities are
regularly distributed, and K (x, x) = 0 while K^ (x, x) does not vanish
at any point of /, then a necessary and sufficient condition that (1) have
a continuous solution is that f (x}, f (x), f" (x} be continuous in I and
that f(a)=f (a) = 0. If these conditions are fulfilled, the equation (1)
has only one continuous solution, which may be found by solving (3).
We leave it for the reader to push the results of this theorem
farther by replacing the condition that K^ (x, x} shall not vanish at
any point of / by the condition KI (x, x} = 0.
The case in which K (x, x) vanishes at a finite number of points in
/has been discussed at length by Volterra*. We content ourselves
with treating a simple example in which the results are fairly typical
of the general case.
Consider the equation
0 = fX(a£ + fix) u (£) d£ (a + (3 4= 0) (4).
Jo
Here the kernel a| + fix reduces when £ = x to (a + ft) x, and thus
vanishes when and only when x = 0. By differentiating (4) twice we
see that any continuous solution which it may have has, except
perhaps when x = 0, a continuous derivative u' (x} ; and we obtain for
u (x) in this way the differential equation
(a + (3) X U (x) + (a + 2/3) U (x) - 0 (5).
The general solution of this equation is
__£__!
u(x) = cx a+0 (6).
If c 4= 0, this is continuous when and only when ft /(a + /?) ^ - 1 ;
that is, when
or, more simply, when
-l>^-2 (7).
If this condition is satisfied, the function (6) is readily seen to be
a solution of (4), and, since (6) contains an arbitrary constant, we have
proved
* Torino III, IV. See also the papers by Holmgren and Labesco cited at the
beginning of this section.
13] WHOSE KERNEL IS FINITE 63
THEOREM 3. Equation (4) has one and only one continuous solution
{namely u = 0) except when condition (7) is fulfilled, in which case it
has an infinite number of such solutions.
We leave it for the reader to extend this result to the more general
equation
/(a?)=f(«* + #r)«(*)# (8),
Jo
where f(x), f (x), f" (x) are continuous in /, and/(0) =/' (0) = 0.
We may add that the method we have just used enables us to find
not merely the continuous solutions of (4) or (8) but also, in some cases
in which (7) is not fulfilled, discontinuous but integrable solutions.
Thus, when a//2 < — 2, the function (6) is integrable and is a solution
of (4).
We turn now to the more general integral equation of the first
kind
(9).
a
Volterra's equation is the special case of this in which K(x, y) vanishes
when y > x. If we look more closely, we see that in the case covered
by Theorem 1 the kernel K (x, y) has a finite jump at every point of
the line x = y, passing from the value K (x, x) to the value 0 as we
cross this line. The cases which presented more difficulty were those
in which there is either no discontinuity along this line, or those in
•which there are a finite number of points where there is no dis-
continuity.
This suggests that we consider in place of the line y = x the curve
y = 4> (x) (where we will assume <f> (x) to be continuous and to have
a continuous derivative <£' (x) in /), which we will suppose crosses the
square S from left to right dividing it into an upper and a lower
portion ; i.e. we assume that when
a^x^b, a^<f>(x)^b.
Along this curve we suppose the function K (x, y) to have a finite
jump ; that is, if (x0, y0) is any point of this curve at which
y0*a, y0*b,
we assume that the two limits
K (x<>, y0 - 0) and K (x0, y0 + 0)
exist. The difference of these limits is what we call the magnitude of
the jump at (x0, y0), and we write
J(x0) = K(x0, y0 - 0) - K(x0, y0 + 0).
64 KERNEL FINITE [13
We will suppose this function J (x) to be continuous in /; and also
that K is finite in S and continuous except along the curve y = <f> (x),
and that it has a derivative
finite in S and whose discontinuities are regularly distributed.
Let us now write (9) in the form
/(*) = /* (X) K (x, |) u (*) dt + !" K(x, |) u (£) #.
./a y*Ca;>)
By differentiating we get
/' (an) = J (a) # (x) u (as) + (V, («, £) w (£) ^ (10).
In order that this be an integral equation of the second kind of the
type we have considered, it is sufficient to make the further assumption
that neither J nor <f> vanishes in /. "We then have an integral
equation whose kernel is
If the determinant (or modified determinant) of this function is not
zero, equation (10) has a continuous solution provided that /' (x) be
continuous. We cannot, however, infer from this that (9) also has
a continuous solution. The continuous solution of (10) satisfies, as we
readily see, an equation of the form
(*, *)«(£)# (11),
and satisfies it, of course, only for one value of X. Thus we get the
result :
THEOREM 4. If the following conditions of continuity are satis/led
(1) K(x,y) is finite in S, and, except on the curve y = <t>(x), con-
tinuous there;
(2) <f> (x) satisfies the inequality
a^^(x)^b,
is continuous and has a continuous derivative in I, and $ (x) does not
vanish in I;
(3) At every point of I, except where
<^(x) = a or <f> (x) = b,
the two limits
K(x, $ (x) - 0) and K(x, <j> (x) + 0)
13, 14] KERNEL OR INTERVAL NOT FINITE 65
exist, and their difference
J(x) = K(x,<f>(x)-0)-K(x,<t>(x) + 0)
is continuous and does not vanish in I;
(4) K (x, y) has a derivative
Ki (of, y} = ZKfiz
finite in S and whose discontinuities are regularly distributed;
(5) f(x) is continuous and has a continuous derivative in I ;
then if the determinant, or modified determinant, of the function
K, (x, y}
J (*)+'(*}
is not zero, there will be one and only one value of the parameter A. for
which the equation (11) has a continuous solution, and this solution will
be the continuous solution of (10).
The reader will find no difficulty in extending this theorem so as to
cover cases in which J(x) = 0.
14. Equations of the First Kind whose Kernel or whose
Interval is not Finite. We pass now with Volterra* to the case in
which the kernel K of the equation
(1),
a
has the form
where G is continuous in T. Precisely as in the case of Abel's
equation, of which (1) is an immediate generalization, we see that (1)
cannot have a continuous solution unless f(x) is continuous throughout
7 and /(a) = 0.
Assuming that (1) has a continuous solution, we may obtain it as
follows : Multiply (1) by (z-xY~ldx, where
a ^ z ^ b,
and integrate the resulting equation, getting
r zw* = r I, i' e
Ja (Z-X)1 A Ja (Z-X)l-^Ja (X -
The second member of this formula reduces by Dirichlet's Extended
Formula (cf. § 1) to
* Cf. Torino H.
66 EQUATION OF FIRST KIND [14
Accordingly if we write
— u~ \d>x
"~"r ' (3),
equation (2) takes the form
Z (4).
We will now show that the kernel L of this equation is continuous
in T, except on the line z = £ where it is not yet denned. For this
purpose introduce y as variable of integration in place of x by means
of the formula
We thus get, when z >
Since this integral remains convergent when we replace G by the upper
limit of its absolute value, it follows that it is uniformly convergent in
T, and, since G is continuous, L is also continuous wherever it is
denned.
In order next to see whether L approaches a limit as the point
(z, £) approaches a point (c, c) on the hypotenuse of T, we apply the
law of the mean for integrals to the original definition of L, getting
I*
Jt
-, -- —
t (z- x)l~^ (x - £)A sm A.T
Consequently
Taking this limiting value as the definition of L (c, c), we see that
L is continuous throughout T, and that if we demand that G (xt x)
shall not vanish at any point of /, it follows that L (x, x~) does not
vanish at any point of /.
In order then that the equation (4) come under the case covered
by Theorem 1, § 13, it remains merely to impose on G restrictions
which will make L (z, £) have a partial derivative with regard to z
finite in 8 and whose discontinuities are regularly distributed. In
order to avoid all complications, we will assume that G (z, £) has
a partial derivative
14] KERNEL OR INTERVAL NOT FINITE 67
which is continuous in T. If, then, we differentiate (5) with regard to
z under the integral sign, we get
rl / y \l->
Since this integral is obviously uniformly convergent, we see, by the
ordinary test for differentiating an infinite integral under the sign of
integration, that it represents
except perhaps on the hypotenuse z = £ of T where formula (5) was not
valid. From the expression just obtained for L1 we see that LI is
finite in T, and from the uniform convergence of this integral, that Zx
is continuous in T except perhaps on the hypotenuse z = £.
It is now possible to apply Theorem 1 of § 13 to the equation (4).
In order that this equation have a continuous solution it is therefore
necessary and sufficient \h&\>F(z) be continuous and have a continuous
derivative in /, and that F (a) = 0. The first and last of these
conditions will be fulfilled if f (x) is continuous in /, as we see by
a reference to Theorem 2, § 1. "We have thus obtained' as an additional
necessary condition for (1) to have a continuous solution that F (z)
have a continuous derivative.
It remains to show that, if all the conditions we have mentioned
are fulfilled, the continuous solution of (4) satisfies (1). For this
purpose differentiate (4) with regard to z and then, after multiplying
by (x-zY^dz, where
a ^ x ^ b,
integrate from a to &, getting
I d * /(fl
--* •
A reference to Theorem 3, § 1, shows that the first member of this
equation is equal to
By an application of Dirichlet's Extended Formula, this reduces to
68 EQUATION OF FIRST KIND [14
The second member of (7) may be written, as we see again by
a reference to Theorem 3, § 1,
By a three-fold application of Dirichlet's Extended Formula this
becomes
d
!' !
jf p=
Sin XT
Substituting the values we have just found on the two sides of (7),
we see that this equation reduces to (1). We have thus proved the
THEOREM. If G (&, £) and its derivative
are continuous in T, and G (x, x) does not vanish in /, a necessary and
sufficient condition that the equation
f(x} = f P^
have a continuous solution is thatf(x) be continuous in I, that
and that the function
have a continuous derivative throughout I. If these conditions are
fulfilled, the equation has only one continuous solution, namely the
continuous solution of (4) where L is defined by (3).
By a reference to Theorem 3, § 1, it will be seen that a sufficient
though not a necessary condition that F have a continuous derivative
is that f be continuous and have a finite derivative with only a finite
number of discontinuities.
If G (x, £) = 1, equation (1) reduces to equation (1) of § 3. In this
case
L = 7r/sin XTT,
14] KERNEL OR INTERVAL NOT FINITE 69
so that (4) becomes
tZ f(x) T [Z
1 — \ — d-K = —. — r — I u (£) d£,
from which, by differentiation, we get, under the assumptions there
made, precisely the solutions (4) and (5) of § 3.
The theorem just proved may be readily modified so as to apply to
cases in which
We leave it for the reader to carry through the discussion here.
A case which may be made to depend on the theorem of this section
is that in which in equation (1), § 13, the function K (x, x) vanishes
identically while the derivative K^ (x, y) has the form
^ifoy) = Jl(flr'u (o<x<i),
where G is continuous in T. Here too we leave the discussion and
the precise formulation of the result to the reader. Theorem 4, § 3, is
a special case of this.
An equation of some importance, which bears a certain resemblance
to the equation of this section, is
f(x) = f {ctn «•(*-*)+(? (x, 0} u
Jo
treated by Hilbert and Kellogg. Here G (x, £) is supposed to be
finite in S, and, in order that the integral should have a meaning, it is
in general necessary that we interpret it to mean the principal value
according to Cauchy. For a treatment of this equation we refer to
pages 17 — 27 of Kellogg's dissertation: "Zur Theorie der Integral-
gleichungen und des Dirichletschen Princips." Gottingen, 1902.
We close by a brief treatment of what was perhaps the first integral
equation to be successfully treated.
One form of Fourier's Integral Theorem is contained in the
formula
(*> = - r
TTJO JO
This formula is valid when 0 = x provided the function f (x)
satisfies certain conditions ; for instance, it is sufficient to demand that,
when 0 =x, f(x) be continuous and do not have an infinite number of
maxima and minima in the neighbourhood of any point, and that
jfl/M
dx
70 EQUATION OF FIRST KIND [14
converge. Formula (8) shows us that, under the conditions just
stated, the integral equation of the first kind
^Jo cos (*£)«(£)# (9)
has as a continuous solution
/2 f°°
u(x}-\/ ~ I cos(«^)y (t)«t (10).
Conversely, we see in the same way that (10) is the only continuous
solution of (9) which can be expressed by means of Fourier's Integral
Theorem.
The equation (9) has as its kernel the continuous symmetric
function
*/-cos(#£) (11),
and the only peculiarity is that the interval I is now infinite. This
change from a finite to an infinite interval makes a very essential
difference in the whole theory of integral equations*, as we see by
considering the characteristic numbers of the kernel (11). If the
interval were finite, there would, as we know, be an infinite number of
these numbers, each with a finite index f. Here, however, (at least if
we restrict the conception of characteristic functions to functions for
which Fourier's Integral Theorem (8) holds) there are only two
characteristic numbers, namely ± 1, each with an infinite index. To
prove this, let us consider the homogeneous equation of the second
kind,
cos (x£) u (£) d£ (12).
If X is a characteristic number and u (x) a corresponding characteristic
function, not only will this equation (12) be fulfilled, but also, as we
see by taking in (9) and (10) for /(a?) the function u (#)/A,
(13).
" JO
By introducing new variables
equation (9) with an infinite interval and a finite kernel may be reduced to an
equation with a finite interval and an infinite kernel.
t We are here assuming the easily established fact that cos (z£ ) cannot be
expressed as the sum of a finite number of terms each of which is the product of a
function of x by a function of £.
14] KERNEL OR INTERVAL NOT FINITE 71
By combining (12) and (13), we see that
u (x) = \*u (#),
so that, since u (z) does not vanish identically, it follows that A = ± 1
are the only possible characteristic numbers.
To prove that these are really characteristic numbers and that they
have an infinite index, we need merely notice that, when c has any
positive value, the functions
are solutions of (12) for A= + 1 and X = - 1 respectively.
The kernel (11) is only one of an important class of kernels having
similar properties. For a theory of such kernels we refer to the
dissertation by H. Weyl : "Singulare Integralgleichungen mit besonderer
Beriicksichtigung des Fourierschen Integraltheorems," GiJttingen, 1908.
INDEX
The numbers refer to pages.
,, linear, 24
Bessel's functions, 53, 54
identity and inequality, 56
Boggio, 51
Characteristic functions, 57
numbers, 57
Complete orthogonal system, 57
Determinant, 26, 32
modified, 36
Differential equations, 1, 12, 46, 58
Dirichlet's formula, 4
Discontinuous solutions, 10, 17, 63
Du Bois-Keymond, 1
Eigenfunktionen, 57
Eigenwerte, 57
Fischer, 29
Fourier's integral, 1, 69-71
Fourier's series, 52, 54
Fredholm, 2, 24-29, 31, 43, 45
Goursat, 10, 21, 51
Gram, 53
Hadamard, 29-31
Hilbert, 2, 13, 25, 31, 36, 46, 49, 58, 69
Holmgren, 60, 62
Hurwitz, 4
1,2
Index of a root for K, 45
Infinite kernels, 7-11, 18, 31, 36, 65
solutions, 10, 17, 63
Integral equations, homogeneous, 14, 43
non-linear, 46
of first kind, 1, 13, 60-71
of second kind, 1, 13 ft.
of third kind, 13
singular, 71
with parameter, 38, 64
Inversion of definite integrals, 1
Iterated functions, 19
Kellogg, 69
Kernel, 13. (See also infinite.)
Klein, 53
Kneser, 43, 47, 58
Labesco, 60, 62
La Vallee Poussin, 4
Legendre's polynomials, 52, 54, 56
Le Boux, 61
Linear algebraic equations, 24
Liouville, 2, 8, 11, 14, 37, 38
Mason, 14
Minors, 27, 45
Modified determinant and adjoint, 36
Multiple integrals, 1
Multiplicity of a root for K, 45
Neumann, 14
Non-linear integral equations, 46
Normalized functions, 54
Nulllosung, 44
Orthogonal functions, 52
Osgood, 27
Parameter in integral equation, 38, 64
Picard, 60
Plarr, 56
Poisson, 49
Principal solution for K, 44
Beciprocal functions, 21
Begularly distributed, 3
Boot for A", 41
Bouche, 2
S 2
Schmidt, 31, 46, 47, 49, 53, 56, 57, 58
Solutions, discontinuous, 10, 17, 63
principal, 44
Successive substitutions, 14
Symmetric kernel, 46
Systems of integral equations, 1
T, 2
Volterra, 2, 19, 24, 37, 60-63, 65-69
Volterra's equation, 60
Weyl, 71
Wirtinger, 29
CAMBRIDGE I PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.
Cambridge Tracts in Mathematics
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THE DEFINITE INTEGRAL, ITS MEANING AND
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THE FUNDAMENTAL THEOREMS OF THE DIF-
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THE CIRCLE AT INFINITY, by J. H. GRACE, M.A.
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Bocher, Maxime
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