Derive a closed-form expression for $\int\limits_0^\infty \frac{u \operatorname{J}_0(xu)}{\sqrt{1+u^2} \sqrt{1+\beta u^2}} \, \mathrm{d} u$

Asked Oct 16 '23 at 12:04

Active Oct 16 '23 at 15:19

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My objective is to derive analytically a closed-form expression for the following integral: $$ \int\limits_0^\infty \frac{u \operatorname{J}_0(xu)}{\sqrt{1+u^2} \sqrt{1+\beta u^2}} \, \mathrm{d} u, \tag{1} $$

Here, $\operatorname{J}_0$ represents the zeroth-order Bessel function of the first kind, and both $x$ and $\beta$ are positive real numbers.

It is worth noting that obtaining a closed-form expression for integral (1) is straightforward when $\beta$ takes on the values of either $0$ or $1$.

This question is inspired from this question in which Mr Zacky suggested using this question to obtain a solution of the problem. Any help with this is highly appreciated. Thank you!

edited Oct 16 '23 at 15:19

asked Oct 16 '23 at 12:04

Eulerian

1

It is a definite integral, not indefinite integral. – user10354138 Oct 16 '23 at 12:25
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"Indefinite" or "definite"? – David G. Stork Oct 16 '23 at 12:25
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Perhaps less straight forward, but $\beta=-1$ results in $$\frac{4}{\sqrt{\pi} x^2} G_{0,4}^{3,0}\left(-\frac{1}{16} \left(i x^2\right),\frac{1}{2}| \begin{array}{c} \frac{1}{2},\frac{1}{2},1,\frac{1}{2} \ \end{array} \right)$$ in terms of the Meijer G-function. – Steven Clark Oct 17 '23 at 03:10

Derive a closed-form expression for $\int\limits_0^\infty \frac{u \operatorname{J}_0(xu)}{\sqrt{1+u^2} \sqrt{1+\beta u^2}} \, \mathrm{d} u$

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