Calculating Residue of cos(z)/z in Contour Integral

[HappyEuler2]

Homework Statement

So the problem at hand is to calculate the contour integral \oint cos(z)/z around the circle abs(z)=1.5 .

Homework Equations

The integral is going to follow from the Cauchy-Integral Formula and the Residue theorem. The problem I am having is figuring out what the residue is going to be.

The Attempt at a Solution

So I know the pole is at z = 0, which lies inside of the contour. So the integral reduces to I=2*pi*i*residue @ 0. What I can't figure out is how to determine the residue. If I use maple, I know that the residue is 1, but I want to figure out where it comes from it. Any help?

 

[Dick]

If f(z) has a simple pole at z=c then the residue is lim z->c of (z-c)*f(z), isn't it? What does that give you?

 

[HappyEuler2]

Ah, thank you. I just needed someone to write it out clearly for me.

So the formal answer =2*Pi*i