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General expression for contour integral of a holomorphic function

Given a non-trivial complex function $f:\mathbb{C}\setminus\{a\} \to \mathbb{C}$, I would like to compute the contour integral
$$\int_C f(z)\ dz,$$
where the closed curve $C$ is the circle around $a$. Is there a concise general formula like the one I know for the complex Laplace integral? For example, if $f(z)=z^n$ for $n \in \mathbb{N}$ it holds that
$$\int_C z^n \ dz = \frac{z^{n+1}}{n+1} + C.$$

A:

If $f\colon D\setminus \{z_0\}\to\mathbb C$ is holomorphic and $D$ is a disc with center $z_0$ such that $f(z_0) eq 0$, then $\int_C f(z)\,\mathrm dz= \int_\gamma f(z)\,\mathrm dz$ where $\gamma\colon [0,2\pi]\to D\setminus \{z_0\}$, $\gamma(t)=e^{it}z_0$.

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