Residue of e^(az)/(1+e^z)^2 at I Pi

[lunde]

Homework Statement

I need to find the residue of e^(az)/(1+e^z)^2 at I Pi. For some reason this is such much harder than I thought it was going to be. Mathematica is not even helping :(.

Homework Equations

Cauchy's kth Integral formula.

The Attempt at a Solution

I made an attempt at doing a u substitution of u=e^z, but I ended up with a residue of 0, which was not what I was expecting. All this lead me to believe that you probably can't use u-substitution strategies with line integrals and expect them to work.

 

[vela]

Try expanding the function as a Laurent series. Use the fact that

$$e^{az} = e^{a(z-i\pi+i\pi)} = e^{ia\pi}e^{a(z-i\pi)}$$

Don't forget to expand both exponentials (in the original function) as series. Remember that all you're interested in is the coefficient of 1/(z-iπ), so just concentrate on the terms that will contribute to that.