# Homework Help: Contour integral with exponential in the denominator

1. Nov 14, 2011

### jncarter

1. The problem statement, all variables and given/known data
Use the contour integral
$\int_{C}\frac{e^{pz}}{1+e^z}dz$​
to evaluate the real integral
$\int^{\infty}_{- \infty}\frac{e^{px}}{1+e^x}dx$​
0<p<1
The contour is attached.
It is a closed rectangle in the positive half of the complex plane. It height is 2i∏.

2. Relevant equations
$\oint f(z)dz = 2 \pi i \sum Res[f(z)]$
$Res[f(z=z_{0})] = (m-1)! \frac{d^{m-1}}{dz^{m-1}}(z-z_{0})^{m}f(z)|_{z=z_{0}}$
where m is the degree of the pole.

3. The attempt at a solution
I found that there was a simple pole at z = i∏, so I must use the residue theorem to find the value of the complex integral.
$Res[f(z=i \pi)] = (z-i \pi) \frac{e^{pz}}{1+e^z}|_{z=i \pi}$
In the past I've had to fiddle with the denominator to get the z-z0 terms to cancel out, but in those cases it involved something nice and simple. I have no idea what to do with this equation.

2. Nov 14, 2011

### Dick

The residue is defined as the LIMIT of your expression as z->i*pi, it's not the value of the expression at z=i*pi (unless you do your magic cancellation). Try using l'Hopital to find the limit.