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Can anyone help me on deriving the closed form expression for the following definite integral of the modified Bessel function of the first kind, zero order, involving power and exponential functions

$$I={\displaystyle\int\limits_0^{a } {{x^{b}} \cdot \exp \left( { - \frac{{{x^2}}}{2}} \right){I_0}\left( {c x} \right)dx} }$$

where $a$, $b$, and $c$ are constant.

Many thanks for your help!

Chill2Macht
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junxv penn
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  • No nice closed form for this one in the general case. Even with $a=\infty$ it's gonna be tough. But I suggest you try the infinite series for $I_0(cx)$ or the integral representation of $I_0(cx)$, which contains the exponent, and then reversing the order of integration – Yuriy S May 21 '16 at 22:38
  • Thank you for your comment, Yuriy – junxv penn May 22 '16 at 12:44

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