- #1
appelberry
- 23
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Hi,
I am trying to find an analytic solution to the following double integral:
<br /> \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
<br /> \frac{2\pi\exp\left(\frac{u^2}{a}\right)}{\gamma}\left(\exp\left(\gamma\right)-\exp\left(-\gamma\right)\right)<br />
where
<br /> \gamma = \frac{2u}{a}\sqrt{v_x^2+v_y^2+v_z^2}\right)<br />
but I cannot find the more general solution.
I am trying to find an analytic solution to the following double integral:
<br /> \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
<br /> \frac{2\pi\exp\left(\frac{u^2}{a}\right)}{\gamma}\left(\exp\left(\gamma\right)-\exp\left(-\gamma\right)\right)<br />
where
<br /> \gamma = \frac{2u}{a}\sqrt{v_x^2+v_y^2+v_z^2}\right)<br />
but I cannot find the more general solution.
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