Complex Analysis - Contour Intergral

[castusalbuscor]

Homework Statement

The problem is to integrate:

$$\oint_{C}\frac{dz}{z^{2}-1}$$

C is a C.C.W circle |z| = 2.

Homework Equations

The Attempt at a Solution

I used the Cauchy integral formula:

$$\oint_{C}\frac{f(z)}{(z-z_{0})^{n+1}}dz = \frac{2 \pi i}{n!}f^{n}(z_{0})$$

Which gives an answer of $$2 \pi i$$ since there is a singularity inside of the contour C...

Does this look right?

 

[Dick]

That's not right at all. z^2-1 is not the same as (z-1)^2. And even if the problem were dz/(z-1)^2 the integral of that around |z|=2 would be zero (since f(z) in your formula is 1). Use z^2-1=(z-1)(z+1) and then look up the residue theorem. Or write 1/((z-1)(z+1))=A/(z-1)+B/(z+1) (figure out A and B) and then you can use the Cauchy integral formula on each part.