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In my work, an integral of the following type arose:

$$\int_0^\infty dx J_l(a x) J_l(b x) \frac{1}{x - c + i 0}.$$

Here $J$ is the Bessel function of the first kind. Assume that $a$, $b$, and $c$ are real numbers (could be $<0$ or $>0$), $l\in \{1/2, 3/2, \dots\}$. I wonder, is there any way to take it analytically?

Here (https://dlmf.nist.gov/10.22), I found an expression which resembles my integral (10.22.69), but there it's $\frac{x}{x^2-c^2}$ instead of $\frac{1}{x-c}$.

Ideally, I'd like to obtain an analytical expression for both the real and imaginary parts using special functions. My current alternative is to integrate numerically, but I don't want to deal with that...

It would be grateful for any help.

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    $\begingroup$ What is the point of $i 0$ in the denominator of the integrand? $\endgroup$ Commented Nov 23 at 3:17
  • $\begingroup$ @StevenClark One can interpret that as $\text{Im}C<0$, or that the pole is fixed in the lower half-plane. In fact, if we could just obtain the real part (I'm thinking about Sokhotsky's formula, but that's just fantasy), that would be great. I really don't want to perform this integration numerically... $\endgroup$ Commented Nov 23 at 10:38
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    $\begingroup$ Integral can be expressed by 2 variable FoxH function or series: $$\int_0^{\infty } \frac{J_l(a x) J_l(b x)}{x-c} \, dx=-\sum _{m=0}^{\infty } \frac{(-1)^m a^{l+2 m} b^{-1-l-2 m} G_{4,2}^{2,3}\left(\frac{4}{b^2 c^2}| \begin{array}{c} 0,\frac{1}{2},\frac{1}{2}-l-m,\frac{1}{2}-m \\ 0,\frac{1}{2} \\ \end{array} \right)}{c \pi m! \Gamma (1+l+m)}$$ for: $c<0$ where: $G$ is MeijerG function. $\endgroup$ Commented Nov 23 at 10:54
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    $\begingroup$ The special case $a=b$ appears in the book Integrals and Series Volume 2 Special Functions by A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (eq. 2.12.32.1, p. 212). It involves $J$, $Y$ and the generalised hypergeometric functions ${}_2 F_3$ and ${}_1 F_2$. $\endgroup$ Commented Nov 23 at 11:01
  • $\begingroup$ Thank you all a lot, that helped me, however I think it's worth to expand J in sin/cos series, and algorithmize the calculation of simple integrals $\endgroup$ Commented Nov 27 at 18:24

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