Contour integral with exponential in the denominator

[jncarter]

Homework Statement

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∏.

Homework 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.

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.

 

[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.