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I'm searching for a way to evaluate the following integral: $$\int_0^a x^{v/2} e^{-\alpha x} J_v(2\beta\sqrt{x}) dx$$

where $J_v(x)$ are the Bessel-functions, and $v \in \mathbb{N}, (a,\beta) \in \mathbb{R}, \alpha \in \mathbb{C}$. The integral has a closed form when $a \to \infty$ (Gradshteyn/Ryzhik). However, is there also a way to solve the integral with a general upper interval-limit?

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  • $\begingroup$ If you could do this for general $a$, you'd have a closed-form antiderivative for the integrand. I don't think that's likely. $\endgroup$ – Robert Israel Dec 11 '14 at 0:09
  • $\begingroup$ What you get by expanding $J_\nu$ as a Taylor series and integrate it termwise against $x^{\nu/2}e^{-\alpha x}$ is a series of incomplete Gamma functions. $\endgroup$ – Jack D'Aurizio Dec 11 '14 at 1:13
  • $\begingroup$ Are there any constraints on the parameters? $\endgroup$ – David H Dec 12 '14 at 4:51
  • $\begingroup$ @DavidH I added the domains of the paramters. There is no obvious other constraint that I could give. $\endgroup$ – NicoDean Dec 12 '14 at 17:49
  • $\begingroup$ @JackD'Aurizio Sounds interesting, I will try to see how that could help. Thanks. $\endgroup$ – NicoDean Dec 12 '14 at 17:49

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