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I am trying to find an analytic solution to the following double integral:

[tex]

\int_{0}^{2\pi}\int_{0}^{\pi}\sin\theta\exp\left(\frac{u^2}{a}\sin^{2}\theta+\frac{u^2}{b}\cos^{2}\theta-2u\left(\frac{v_x}{a}\sin\theta\cos\phi+\frac{v_y}{a}\sin\theta\sin\phi+\frac{v_z}{b}\cos\theta\right)\right)d\theta d\phi[/tex]

I have tried using Mathematica and various substitutions but with no success. Using the result given in a previous post

https://www.physicsforums.com/showthread.php?t=376233" , I know that in the limit of [tex]b=a[/tex] that the solution is

[tex]

\frac{2\pi\exp\left(\frac{u^2}{a}\right)}{\gamma}\left(\exp\left(\gamma\right)-\exp\left(-\gamma\right)\right)

[/tex]

where

[tex]

\gamma = \frac{2u}{a}\sqrt{v_x^2+v_y^2+v_z^2}\right)

[/tex]

but I cannot find the more general solution.

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# Solution to an exponential integral

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