## Abstract

Integral equations in which the unknown function appears in a convolution occur in some important situations. The equation , where f(x) and k(x) are given functions and λ a given constant, is an example of a Fredholm integral equation of the second kind. (An equation of the first kind is one in which the unknown function g does not appear outside the integral.) If the upper limit of integration b is replaced by the variable x, then (1) is said to be of Volterra, rather than Fredholm, type. By the change of variables x′ = x − a, y′ = y − a, (1) may then he written .

$$g(x) = f(x) + \lambda \int_a^b {k(x - y)\;g(y)\;dy} $$

(1)

$$G(p) = F(p) + \lambda \;K(p)\;G(p)$$

(2)

## Keywords

Integral Equation LAPLACE Transform Fredholm Integral Equation Pair Distribution Function Convolution Equation
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## Footnotes

- 1.See J. H. Giese, SIAM Review (1963), 5, 1.MathSciNetCrossRefGoogle Scholar
- 2.Some numerical values for the case a = -2 computed by Padé approximation may be found in L. Fox and E. J. Goodwin, Phil. Trans. Roy. Soc. Lond. (1953), A245 501.Google Scholar
- 3.N. Mullineux and J. R. Reed, Q. Appl. Math. (1967), 25, 327.MATHGoogle Scholar
- 4.Equations of this type may be solved by the Weiner-Hopf technique (see Section 18). However, we are interested here in a class of problems which can be solved by more elementary methods.Google Scholar
- 5.We must first take Re(p) a, and then use analytic continuation on the final result to extend it to Re(p) a.Google Scholar
- 6.As with (29), a process of analytic continuation may be involved.Google Scholar
- 7.This is the probability of finding two particles at the stated positions. For an infinite uniform system it is a function only of the relative positions of the two.Google Scholar
- 8.This identification is only valid in the Percus-Yevick approximation.Google Scholar
- 9.M. S. Wertheim, J. Math. Phys. (1964), 5, 643. The more general case where V(x) ≠ 0 for a ≤ |x| ≤ ℓ R is also analyzed using Laplace transforms.Google Scholar
- 10.The ensuing procedure is a simple example of the type of argument which is used in the Wiener-Hopf technique (Section 18).Google Scholar
- 11.Problems 8–13 and some related material may be found in D. O. Reudink, SIAM Review (1967), 9, 4.Google Scholar

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