(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use the contour integral

[itex]\int_{C}\frac{e^{pz}}{1+e^z}dz[/itex]to evaluate the real integral

[itex]\int^{\infty}_{- \infty}\frac{e^{px}}{1+e^x}dx[/itex]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

[itex]\oint f(z)dz = 2 \pi i \sum Res[f(z)] [/itex]

[itex] Res[f(z=z_{0})] = (m-1)! \frac{d^{m-1}}{dz^{m-1}}(z-z_{0})^{m}f(z)|_{z=z_{0}}[/itex]

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.

[itex] Res[f(z=i \pi)] = (z-i \pi) \frac{e^{pz}}{1+e^z}|_{z=i \pi}[/itex]

In the past I've had to fiddle with the denominator to get the z-z_{0}terms to cancel out, but in those cases it involved something nice and simple. I have no idea what to do with this equation.

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# Contour integral with exponential in the denominator

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