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I have an integral which arises while quantizing QED near a graphene sheet. It is

$$\int_{0}^{\infty}\mathrm{d}u\int_{0}^{2\pi}\mathrm{d}\theta\,u\frac{e^{iqu\cos{\theta}}}{\sqrt{u^2+z^2}}$$

Mathematica tells me the answer is $2\pi\,e^{-q|z|}/q$, but it invokes Bessel functions to perform the integration. My question is whether there is a way to peform the integral without appealing to Bessel functions (perhaps by performing the $\mathrm{d}u$ integral first).

[Asking on Physics because formally the $\mathrm{d}u$ integral diverges if you do it first, but physicists do this all the time.]

Bob Knighton
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  • I think the QED and graphene tags are not appropriate. This is pure maths, even if is does occur in the course of a physics calculation. No physics involved in the solution. If there are any tricks for handling the integral, the Mathematicians will have a better chance of knowing them. – sammy gerbil Aug 13 '16 at 01:58
  • I did ask the mathematicians first, but got nothing but low quality, unhelpful responses that didn't adress any of the questions I asked. – Bob Knighton Aug 14 '16 at 02:19
  • In physics we mostly leave the exact solution of difficult integrals or differential equations to the mathematicians, and consult tables of solutions. – sammy gerbil Aug 14 '16 at 15:47
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    So far I have been told that mathematicians never integrate things, so I should go ask physicists (because it's a physics problem for physics research) and that physicists don't know how to do integrals so I should ask mathematicians. Meanwhile, nobody has actually tried to answer the question. – Bob Knighton Aug 14 '16 at 18:04
  • Just to point out why the Bessel function approach shows up if you go through $\theta$ first: From the Jacobi-Anger expansion, the Fourier components of the periodic function $e^{i q u\cos\theta}$ are Bessel functions. In particular, the zeroth component will just be $J_0(u|q|)$, and that's that what the $\theta$-integral computes. – Semiclassical Aug 17 '16 at 15:45
  • Also, it looks like while Mathematica can do the $u$-integral (under proper Assumptions regarding $q\cos\theta$ and $z$), the result is a combination of modified Bessel functions and a Struve function. So that approach seems hardly better... – Semiclassical Aug 17 '16 at 15:55

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