Evaluating Integrals using the Residue THM

[brianhawaiian]

Homework Statement

integral |z|=1 of sinz/z²dz

Homework 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)

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

 

[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.

 

[brianhawaiian]

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.

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

Edit: Actually simple pole at z = 0?

 

[phsopher]

Yep.