I want to solve this integral $$ \int_{0}^{\infty}\int_{0}^{2\pi}\exp(-ar^2)\exp(r\,b(\cos\theta+\sin\theta))r^{m}\cos^{m}(2\theta)d\theta \,dr,$$ where $a$ and $b$ are constants. I know how to solve this integral when $b=0$. I need some suggestions from experts to solve this integral. Of course, I can expand the second exponential function containing sine and cosine as a power series then I can write the integral as a complicated summation over many indices containing gamma functions and beta functions. I need a very simple result, If somebody can connect this to another special functions, I will be happy.

  • $\begingroup$ first of all i would reduce argument in the exponential to $rb\mathrm{cos}\left(\theta -\frac{\pi}{4}\right)$ then change variables as $\theta\rightarrow\theta-\frac{\pi}{4}$. $\endgroup$ – Chinny84 Feb 23 '14 at 1:20
  • $\begingroup$ OK Thanks for the first step. $\endgroup$ – Sijo Joseph Feb 23 '14 at 2:26

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