# Contour Integration used to solved real Integrals

1. Oct 11, 2013

### Jufro

1. The problem statement, all variables and given/known data

Essential Mathematical Methods for the Physics Sciences Problem 15.7

Show that if f(z) has a simple zero at z0 then 1/ f(z) has a residue of 1/f'(z0). Then use this information to evaluate:

∫ sinθ/(a- sinθ) dθ, where the integral goes from -∏ to ∏.

2. Relevant equations
The book has further hints:
The unit circle has only one pole at z= i*a-i(a2-1)^1/2 and therefore has a residue of -i/2 *(a2 -1)-1/2.

3. The attempt at a solution

The first part is:
1/f(z) has a simple pole at z0 leaving the residue to be limz→z0 (z-z0) * 1/f(z) = 1/f'(z0) by definition of the derivative.

But using this information to do the integral is escaping me. Can anyone just help me understand how to use the hint. I don't actually need the whole step by step guide to the integral.

2. Oct 11, 2013

### vela

Staff Emeritus
I haven't worked it out, but I'd try using the substitution $z=e^{i\theta}$ and expressing $\sin\theta$ as $\frac{z-1/z}{2i}$.

3. Oct 12, 2013

### jackmell

What is a? That matters. For example if $|a|<1$ and real, the integral becomes a principal-valued integral.

4. Oct 12, 2013

### Jufro

Sorry, a is real and a > 1. Rewriting sin as vela said came to the right answer, thanks.