Solution to an exponential integral

[appelberry]

Hi,

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

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

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 $$b=a$$ that the solution is


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

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

but I cannot find the more general solution.

 

[darkside00]

wont let you solve e^(sinx)