Value of a contour Integral

Question

If we have the contour integral $\frac{1}{2\pi i} \oint_y \frac{f(C)}{(C-z)^2}dC$, where $f: \mathbb{C} \to \mathbb{C}$ is holomorphic and $y$ a closed circle with centre at $z\in \mathbb{C}$, which is traced anticlockwise. Am I correct in saying that the value of this contour integral is $f'(z)$?

Answer 1

Yes, it is a particular case of Cauchy integral formula. More generally, we have: $$ f^{(n)}(z)= \frac{n!}{2 \pi i} \int_{\gamma} \frac{f(\zeta)}{(\zeta - z)^{n+1}} d \zeta $$ where $f: U \rightarrow \mathbb{C}$ is holomorphic. $\gamma$ is some contour enclosing $z$, not necessarily a circle. This result can be generalised in various ways: just to cite a few, one is the so-called Cauchy-Pompeiu formula, while another one is the extension to several variables.