The method of finite-product extraction and an application to Wiener-Hopf theory

Abstract

In this work we describe a simple method for f nding approximate representations for special functions which are entire transcendental functions that can be represented by inf nite products. This method replaces the inf nite product by a f nite polynomial and Gamma functions. This approximate representation is shown in the case of Bessel functions to be very accurate over a large range of parameter values. These approximate expressions can be useful for f nding the roots of a transcendental equation and the WienerHopf factorization of functions involving such Bessel functions.The method is shown to be potentially useful for other transcendental and Wiener-Hopf problems, which involve other entire functions that have inf nite product representations.