Integral representation for Bessel's functions of the first kind and Neumann series

TL;DR

This paper develops new integral representations for Bessel functions of the first kind and Neumann series, enabling series expansions of related functions and providing new closed-form integral formulas for special functions.

Contribution

It introduces a Fourier-type integral representation for Bessel functions of complex order using Gegenbauer extensions, and derives a new closed-form integral for Neumann series.

Findings

New integral representation for Bessel functions of complex order

Series expansion of functions related to the incomplete gamma function

Closed-form integral representations for Neumann series and special functions

Abstract

A Fourier-type integral representation for Bessel's function of the first kind and complex order is obtained by using the Gegenbuaer extension of Poisson's integral representation for the Bessel function along with a trigonometric integral representation of Gegenbauer's polynomials. This representation lets us express various functions related to the incomplete gamma function in series of Bessel's functions. Neumann series of Bessel functions are also considered and a new closed-form integral representation for this class of series is given. The density function of this representation is simply the analytic function on the unit circle associated with the sequence of coefficients of the Neumann series. Examples of new closed-form integral representations of special functions are also presented.