Contour integral around circle centered at 2i with radius r

[strangequark]

Homework Statement

Let $$\gamma_{r}$$ be the circle centered at 2i with a radius r. Compute:

$$\oint \frac{f(z)}{z^{2}+1}dz$$

Homework Equations

$$2 \pi i f(w)=\oint \frac{f(z)}{z-w}dz$$

Cauchy's integral formula... maybe?

The Attempt at a Solution

I can see how to find solutions for two separate cases:

0<r<1
0<r<3
r>3

I have no idea how to find a general formula for this... nor am I sure what to do when $$\gamma$$ passes thru a singularity...

any help on how to get started would be MUCH appreciated... thanks in advance

 

[Dick]

If the curve passes through the singularity then it's really only defined in the principle value sense. I wouldn't worry about that case. But you are doing it right. You have to split the answer into cases, not write on big formula.