# Evaluating Integrals using the Residue THM

1. Dec 13, 2009

### brianhawaiian

1. The problem statement, all variables and given/known data

integral |z|=1 of sinz/z2dz

2. Relevant equations
Rule #1 if f(z) has a simple pole at z0, then
Res[f(z),z0] = lim(as z goes to z0) (z - z0)*f(z)

Rule #2 if f(z) has a double pole at z0, then
Res[f(z),z0] = lim(as z goes to z0)d/dz (z - z0)2*f(z)

Rule #3 If f(z) and g(z) are analytic at z0, and if g(z) has a simple zero at z0 then,
Res[f(z)/g(z), z0] = f(z0)/g'(z0)

Rule #4 If g(z) is analytic and has a simple zero at z0 then,
Res[1/g(z), z0] = 1/g'(z0)

3. The attempt at a solution

Just confused on which one to use? Would I integrate first? And if so what would my z0 be in Rest[..., z0]

Thanks

2. Dec 13, 2009

### phsopher

What kind of of pole does the function sinz/z2 have and where is it? Although I think 1 and 2 will give the same result.

3. Dec 13, 2009

### brianhawaiian

double pole at i/2 and -i/2?

Edit: Actually simple pole at z = 0?

Last edited: Dec 13, 2009
4. Dec 13, 2009

Yep.