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[parent] proof of Cauchy residue theorem (Proof)

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 $\int_{C'} f(z)\, dz$ if $C'$ is any curve which is homotopic to $C$ In particular we can consider a curve $C'$ which turns around the points $a_j$ along small circles and join these small circles with segments. Since the curve $C'$ 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 $dz=\rho i e^{i\theta}\, d \theta$ and hence $$ \int_C f(z)\, dz = \int_{C'} f(z)\, dz = \sum_j \eta(C,a_j)\int_{\partial B_\rho(a_j)} f(z)\, dz $$ $$ = \sum_j \eta(C,a_j) \int_0^{2\pi} f(a_j+\rho e^{i\theta}) \rho i e^{i\theta}\, d\theta $$ 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.




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Cross-references: Laurent series, fixed, contained, disjoint, balls, sum, orientation, opposite, segments, join, circles, points, homotopic, curve, integral, domain, connected, simple, closed differential forms, closed, differential form, equations, Riemann, holomorphic

This is version 3 of proof of Cauchy residue theorem, born on 2003-06-18, modified 2006-07-25.
Object id is 4377, canonical name is ProofOfCauchyResidueTheorem.
Accessed 3638 times total.

Classification:
AMS MSC30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)

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