Integration of combination of Bessel Function and Exponential Function I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online literature. But I unable to solve a integration of combination of Bessel and Exponential function. The function is $$ \int_{-\frac{1}{2}}^{\frac{1}{2}}e^{-\iota \omega t} I_{0}\left [ b\sqrt{1-t^{2}} \right ]dt\ $$ I am using Walform Mathematica 9.1 for Mathematical analysis. Please Suggest me particular solution, Method or formulas to integrate this combined function. Also suggest any other Software for this type of integration.

This function is derived by Slepian and Pollak in 1961 and this is also know by Prolate-spheroidal wave function. For more detail read these articles

Prolate spheroidal wave functions, Fourier analysis, and uncertainty-I

Prolate spheroidal wave functions, Fourier analysis, and uncertainty-II

In 1964 J.F. kaiser introduce this function with simple approximate to family of window function nearly ideal properties

Everybody thanks for suggestions