Closed form solution of an exponential kernel integral equation

Abstract

In this note a Fredholm integral equation of the first kind with exponential expressions for the kernel and right hand side is considered. The task of finding a practically usable solution to such an equation may need more effort than following a standard procedure, even when such a procedure yields a formal solution. An apparently elegant solution as an orthogonal polynomial expansion is obtained using the standard method based on transformation to a form where the kernel is an orthogonal polynomial generating function, but this is of limited use due to slow convergence. It is shown that this can nevertheless be transformed into a closed form solution that is computationally efficient.