- #1

Gulli

- 96

- 0

## Homework Statement

I have to prove the following: [itex]\int_0^{2\pi} \frac{\mathrm{d}\theta}{(a + cos(\theta))^2} = \frac{2pia}{(a^{2}-1)^{3/2}}[/itex] for a > 1.

## Homework Equations

I have an example at hand for [itex]\int_0^{2\pi} \frac{\mathrm{d}\theta}{a + cos(\theta)}[/itex] from which I know I have to substitute [itex]K = e^{i\theta} \rightarrow \mathrm{d}\theta = \frac{\mathrm{d}K}{iK}[/itex] and [itex]cos(\theta) = \frac{K + K^{-1}}{2}[/itex], use a keyhole contour based on the unit circle and the location of the singularity, use the residue theorem and use the fact that the cosine of a complex angle can apparently be bigger than 1 (otherwise there would be no singularities and the integral would be zero).

## The Attempt at a Solution

Unlike in the example the denominator doesn't yield a nice quadratic function, instead it yields a cubic function and K^-1 hasn't been eliminated.

How do I solve this?