Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I need help with the analytical solution of this integral:

$$\int_{0}^{2\pi}\frac{1}{\sqrt{a^2-b^2\cos^{2}2\phi}}\exp{\left(-\frac{(x-c\cos\phi)^2}{a+b\cos2\phi}-\frac{(y-d\sin\phi)^2}{a-b\cos2\phi}\right)}\mathrm d\phi$$

where $a^2-b^2=1$ and $c, d$ are constants. I know the solution for $a=1$, $b=0$, using the generating function of the modified Bessel functions of first kind, but I think it can't be used in this case. ┬┐Can it be solved using the stationary phase method?

Any help would be welcomed.

share|cite|improve this question
Have you tried asking a CAS for help? – CBenni Feb 2 '13 at 12:47
Perhaps differentiating with respect to $x$ and/or $y$ helps in understanding this function... Might try some trigonometric substitution, like $u = \tan \phi$, or something based on the expression under the root. – vonbrand Feb 2 '13 at 13:50
CBenni, I tried Mathematica but it doesn't generate a solution. – R2D2 Feb 2 '13 at 15:34

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.