# proof of Cauchy residue theorem

Being $f$ holomorphic by Cauchy-Riemann equations the differential form $f(z)\,dz$ is closed. So by the lemma about closed differential forms on a simple connected domain we know that the integral $\int_{C}f(z)\,dz$ is equal to $\int_{C^{\prime}}f(z)\,dz$ if $C^{\prime}$ is any curve which is homotopic to $C$. In particular we can consider a curve $C^{\prime}$ which turns around the points $a_{j}$ along small circles and join these small circles with segments. Since the curve $C^{\prime}$ follows each segment two times with opposite orientation it is enough to sum the integrals of $f$ around the small circles.

So letting $z=a_{j}+\rho e^{i\theta}$ be a parameterization of the curve around the point $a_{j}$, we have $dz=\rho ie^{i\theta}\,d\theta$ and hence

 $\int_{C}f(z)\,dz=\int_{C^{\prime}}f(z)\,dz=\sum_{j}\eta(C,a_{j})\int_{\partial B% _{\rho}(a_{j})}f(z)\,dz$
 $=\sum_{j}\eta(C,a_{j})\int_{0}^{2\pi}f(a_{j}+\rho e^{i\theta})\rho ie^{i\theta% }\,d\theta$

where $\rho>0$ is chosen so small that the balls $B_{\rho}(a_{j})$ are all disjoint and all contained in the domain $U$. So by linearity, it is enough to prove that for all $j$

 $i\int_{0}^{2\pi}f(a_{j}+e^{i\theta})\rho e^{i\theta}\,d\theta=2\pi i\mathrm{% Res}(f,a_{j}).$

Let now $j$ be fixed and consider now the Laurent series for $f$ in $a_{j}$:

 $f(z)=\sum_{k\in\mathbb{Z}}c_{k}(z-a_{j})^{k}$

so that $\mathrm{Res}(f,a_{j})=c_{-1}$. We have

 $\int_{0}^{2\pi}f(a_{j}+e^{i\theta})\rho e^{i\theta}\,d\theta=\sum_{k}\int_{0}^% {2\pi}c_{k}(\rho e^{i\theta})^{k}\rho e^{i\theta}\,d\theta=\rho^{k+1}\sum_{k}c% _{k}\int_{0}^{2\pi}e^{i(k+1)\theta}\,d\theta.$

Notice now that if $k=-1$ we have

 $\rho^{k+1}c_{k}\int_{0}^{2\pi}e^{i(k+1)\theta}\,d\theta=c_{-1}\int_{0}^{2\pi}d% \theta=2\pi c_{-1}=2\pi\,\mathrm{Res}(f,a_{j})$

while for $k\neq-1$ we have

 $\int_{0}^{2\pi}e^{i(k+1)\theta}\,d\theta=\left[\frac{e^{i(k+1)\theta}}{i(k+1)}% \right]_{0}^{2\pi}=0.$

Hence the result follows.

Title proof of Cauchy residue theorem ProofOfCauchyResidueTheorem 2013-03-22 13:42:04 2013-03-22 13:42:04 paolini (1187) paolini (1187) 7 paolini (1187) Proof msc 30E20