Integral Representation for Bessel's Functions of the First Kind and Neumann Series

摘要

A Fourier-type integral representation for Bessel's functions of the first kind and complex order is obtained by using the Gegenbauer extension of Poisson's integral representation for the Bessel function along with a suitable trigonometric integral representation of Gegenbauer's polynomials. By using this representation, expansions in series of Bessel's functions of various functions which are related to the incomplete gamma function can be obtained in a unified way. Neumann series are then considered and a new closed-form integral representation for this class of series is given. The density function of this representation is the generating function of the sequence of coefficients of the Neumann series on the unit circle. Examples of new closed-form integral representations of special functions are thus presented.