Problem 11.4.2 Arfken
Evaluate $\oint_C \frac{dz}{z^2-1}$, where $C$ is the circle $|z-1||=1$.
My approach to the solution is that to express the integral as partial fractions.
$\oint_C \frac{dz}{z^2-1}=\frac12 \oint_C ( \frac1{z-1} - \frac1{z+1} ) dz$. Since the point $z=1$ is within the contour, and $z=-1$ is outside the contour, using Cauchy's Integral Formula, one gets $\pi i$. But the answer in the instructor's manual is $0$. So where am I going wrong?