1. ## Computing Residue

I need to compute the residue for $\frac{e^{-z}}{(z-1)^2}$. I'm going to be using it for Cauchy's residue theorem to evaluate the integral around the circle |z| = 3, so there is obviously a singularity at z=1. I'm confused how to manipulate the fraction and bring in series so that I can see what the coefficient on the $\frac{1}{z-1}$ term. Thank you.

2. Originally Posted by azdang
I need to compute the residue for $\frac{e^{-z}}{(z-1)^2}$. I'm going to be using it for Cauchy's residue theorem to evaluate the integral around the circle |z| = 3, so there is obviously a singularity at z=1. I'm confused how to manipulate the fraction and bring in series so that I can see what the coefficient on the $\frac{1}{z-1}$ term. Thank you.
z = 1 is a double pole. You will therefore have a formula for calculating this somewhere in your notes.

3. Hm, yes, thank you for the hint. However, the formula using the poles is three chapters after the section we are currently working on. This definitely makes the most sense to use though, so hopefully, I won't be penalized for using a different method. Thank you again

4. Originally Posted by azdang
Hm, yes, thank you for the hint. However, the formula using the poles is three chapters after the section we are currently working on. This definitely makes the most sense to use though, so hopefully, I won't be penalized for using a different method. Thank you again
Alternatively, calculate and substitute the first few terms of the Taylor series of $e^{-z}$ around $z = 1$ and then expand. Because all you need is the coefficient of the $\frac{1}{z - 1}$ term ....