  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.
