Consider the integral \(\oint_{C} \frac{2 d z}{z^{2}-1}\), where \(C\) is a circle of radius \(1 / 2\) and centered at 1 , and positively oriented.

(a) Using a partial fraction method, show that \(\frac{2}{z^{2}-1}=\frac{1}{z-1}-\frac{1}{z+1}\).

(b) Evaluate the integral using Cauchy's integral formula and Cauchy's theorem with these partial fractions, respectively.