## Abstract

By an *integral equation* is meant an equation to determine an unknown function φ(*x*) (φ(*x*) is defined in *a* < *x* < *b*) such that in the equation appears an integral in which the integrand is dependent on the desired function φ(*x*). Naturally in such an equation there can occur other terms—not necessarily in the form of an integral—which depend directly on φ(*x*).

## Keywords

Integral Equation Characteristic Function Product Kernel Fredholm Integral Equation Volterra Integral Equation## Preview

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## Notes

- (1).This integral can be evaluated through the substitution
*x*=*y*+ (η −*y*)*u*. It follows that \(\int\limits_{0}^{1}\frac{du}{\sqrt{u-u^2}}=[\textup{arc}\;\textup{sin}\;(2u-1)]_{0}^{1}=\pi\) (cf. p. 432, formula (264)).Google Scholar - (1).In order to avoid confusion we replace
*y*by η.Google Scholar - (1).In this example the exceptionally small number
*n*= 3 suffices.Google Scholar - (1).From the process on p. 693 it follows that for \(\lambda =\frac{5}{2}\;[25\pm \sqrt{601}]\) the integral equation has the following two linearly independent characteristic solutions: \(\varphi _{1}=C_{1}[6x-119\sqrt{x}-5\sqrt{601}\sqrt{x}]\), \(\varphi _{2}=C_{2}[6x-119\sqrt{x} +5\sqrt{601}\sqrt{x}]\). Since an inhomogeneous integral equation in the case of its characteristic values is solvable if and only if the perturbation function
*f*(*x*) is orthogonal to all the characteristic solutions of the transposed integral equation, the conditions of solvability are, when one notices, that the integral equation coincides with its transpose: \(\int\limits_{0}^{1}f(x)\varphi _{1}(x)\;dx=0,\;\;\;\int\limits_{0}^{1}f(x)\varphi _{2}(x)\;dx=0\). One sees easily that in the case*f*(*x*) =*x*this condition will not be fulfilled.Google Scholar - (1).A zero
*x*_{0}of a function is said to be of order*n*if it is also a zero of the first*n*− 1 derivatives but not of the*n*-th. Accordingly, a simple zero is a zero for which the value of the first derivative is different from zero.Google Scholar - (1).The reader should compare this with the section on degenerate kernels.Google Scholar
- (1).If the function φ(
*x*) were only integrable (cf. the footnote on p. 454) then from the continuity of the function*K*(*x*,*y*) considered as a function of*x*it would follow that the integral \(\int\limits_{a}^{b}K(x,y)\varphi (y)dy\) is also a continuous function of*x*. On the left side of (1) would therefore stand the sum of two continuous functions of*x*, and therefore the function φ(*x*) would have to be continuous also.Google Scholar - (1).
- (1).Where (
*x*,*g*) denotes the scalar product of the function*x*with the function*g*(*x*); \((x,g)=\int\limits_{0}^{1}xg(x)dx=\int\limits_{0}^{1}yg(y)dy\).Google Scholar - (1).Where (
*g*, sin*x*) again is the scalar product \((g,\textup{sin}\;x)=\int\limits_{-\pi}^{+\pi}g(x)\;\textup{sin}\;x\;dx=\int\limits_{-\pi}^{+\pi}g(y)\;\textup{sin}\;y\;dy\) and analogously \((g,\textup{cos}\;x)=\int\limits_{-\pi}^{+\pi}g(x)\;\textup{cos}\;x\;dx=\int\limits_{-\pi}^{+\pi}g(y)\;\textup{cos}\;y\;dy\).Google Scholar - (1).Since \(\varphi _{1}=\tilde{\varphi _{1}}/||\tilde{\varphi} ||\) the conditions \((g,\tilde{\varphi _{1}})=0\) and (
*g*, φ_{1}) = 0 are equivalent.Google Scholar - (1).There can be only finitely many
*K*such that λ*K*= λ. Thus for each characteristic value there can exist only finitely many linearly independent characteristic functions.Google Scholar

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© Verlag Harri Deutsch, Zürich 1973