Solve $\int_{|z|=2}\frac{e^{3z}}{(z-1)^3}dz$ using residue.

Question

I'm trying to evaluate $$\int_{|z|=2}\frac{e^{3z}}{(z-1)^3}dz$$ using the residue theorem. I get a pole of order $3$ at 1 with a residue of $\frac{9}{2}e^3$. But since the absolute value of the residue (which in this case is exactly the residue) is bigger than 2, does that mean the integral is equal to zero? I'm just really confused, since this is the first in a series of practice problems and I was expecting to have to use the residue theorem.

Answer 1

The absolute value of the residue is irrelevant here. Since$$\operatorname{res}\left(1,\frac{e^{3z}}{(z-1)^3}\right)=\frac92e^3,$$then$$\int_{|z|=2}\frac{e^{3z}}{(z-1)^3}\,\mathrm dz=9e^3\pi i.$$