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.

  • $\begingroup$ As it stands, I don't see any particular reason to expect this to have a generic closed-form solution. Can you describe the motivation behind this integral? $\endgroup$ Aug 17, 2014 at 13:51
  • $\begingroup$ I have to develop Fractional Fourier Transform of this Bessel Function. I read somewhere its explanation of integration in Book Review: George A. Campbell and Ronald M. Foster, Fourier Integrals for Practical Applications by J. K. Lamond. In history it was solve by Ben Lagon. But I have no access to these literature. Please help. $\endgroup$ Aug 17, 2014 at 14:06
  • $\begingroup$ When you say 'solved', in what sense do you mean? I could well believe that there's a series expansion (in powers of $b$, say). It's harder for me to be confident that there's a closed-form solution, though it's not impossible. $\endgroup$ Aug 17, 2014 at 14:17
  • $\begingroup$ Two observations: The imaginary part vanishes. By changing the integration limits to $\pm1$, and letting $\omega=b=1$, the value becomes $2$. $\endgroup$
    – Lucian
    Aug 17, 2014 at 14:28
  • 1
    $\begingroup$ I am basically from engineering background. $\endgroup$ Aug 18, 2014 at 5:22

1 Answer 1


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


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