Find the contour integral of $$\int_C\frac{(z+a)(z+b)}{(z-a)(z-b)} \mbox{d}z,$$ where the modulus of $a$ and $b$ are less than $1$, and the integral path $C$ is the anticlockwise unit circle ($|z|= 1$).
I am a beginner in contour integration, and I cannot find how to parameterize the given function. Any hints/starting approach would be good.
Let $$f(z)=\frac{(z+a)(z+b)}{(z-a)(z-b)}$$ so $a$ and $b$ are simple poles for $f$ and then $$Res(f,a)=2a\frac{a+b}{a-b}\quad;\quad Res(f,b)=2b\frac{a+b}{b-a}$$ so $$\int_{C(0,1)} f(z)dz=2i\pi(Res(f,a)+Res(f,b))=4i\pi(a+b)$$