Evaluating $\int \frac{z^d f(z)}{z-a}\, dz$ over a closed contour.

Question

We know that for $z \in \mathbb{C}$

$$ \int_C \frac{f(z)}{z-a}\ dz = f(a) $$

where $C$ is a closed contour.

Then how can we compute

$$ \int_C \frac{z^d f(z)}{z-a} dz $$

Answer 1

Let $g(z)=z^d f(z)$. Then $$\int_C \frac{g(z)}{z-a}dz = 2\pi i \cdot g(a) = 2\pi i \cdot a^d f(a)$$

assuming that $C$ is a closed, positively oriented, rectifiable curve with winding number $n=1$, enclosing the point $z=a$, and that $f(z)$ is holomorphic on the region enclosed by $C$.