The solution of an integral equation

Abstract:

Various methods are developed to solve the integral equation $f(x) = \int _0^\infty {g(t)} k(xt)dt$, when the Mellin transform $K(s)$ of the kernel function $k(x)$ is decomposable. Each method corresponds to the way $K(s)$ is decomposed: Namely (i) $K(s) = 1/L(1 - s)M(1 - s)$, (ii) $K(s) = H(s)/M(1 - s)$ and (iii) $K(s) = N(s)H(s)$, where $L,M,N$ and $H$ are arbitrary functions of the complex variable $s$. Numerous special cases and examples are given to illustrate the technique and the advantage of these methods.