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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

$\displaystyle \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 $
$\displaystyle = \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$
$\displaystyle 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$:

$\displaystyle f(z)= \sum_{k\in \mathbb{Z}} c_k (z-a_j)^k $
so that $ \mathrm{Res}(f,a_j)=c_{-1}$. We have
$\displaystyle \int_0^{2\pi} f(a_j+e^{i\theta})\rho e^{i\theta}\, d\theta = \sum... ...ta}\, 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
$\displaystyle \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
$\displaystyle \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.



"proof of Cauchy residue theorem" is owned by paolini.
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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, 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 2840 times total.

Classification:
AMS MSC: 30E20 (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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