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Cauchy residue theorem (Theorem)

Let $ U \subset \mathbb{C}$ be a simply connected domain, and suppose $ f$ is a complex valued function which is defined and analytic on all but finitely many points $ a_1, \dots, a_m$ of $ U$. Let $ C$ be a closed curve in $ U$ which does not intersect any of the $ a_i$. Then

$\displaystyle \int_C f(z)\ dz = 2 \pi i \sum_{i=1}^m \eta(C,a_i) \operatorname{Res}(f;a_i), $
where
$\displaystyle \eta(C,a_i) := \frac{1}{2 \pi i} \int_C \frac{dz}{z-a_i} $
is the winding number of $ C$ about $ a_i$, and $ \operatorname{Res}(f;a_i)$ denotes the residue of $ f$ at $ a_i$.

The Cauchy residue theorem generalizes both the Cauchy integral theorem (because analytic functions have no poles) and the Cauchy integral formula (because $ f(x)/(x-a)^n$ for analytic $ f$ has exactly one pole at $ x=a$ with residue $ \operatorname{Res}(f(x)/(x-a)^n,a) = f^{(n)}(a)/n!)$.



"Cauchy residue theorem" is owned by djao. [ full author list (2) | owner history (1) ]
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See Also: residue, Cauchy integral formula, Cauchy integral theorem

Other names:  Cauchy residue formula, residue theorem

Attachments:
proof of Cauchy residue theorem (Proof) by paolini
example of using residue theorem (Example) by pahio
zero as contour integral (Corollary) by rspuzio
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Cross-references: Cauchy integral formula, poles, Cauchy integral theorem, residue, winding number, intersect, closed curve, points, analytic, function, complex, domain, simply connected
There are 10 references to this entry.

This is version 6 of Cauchy residue theorem, born on 2001-12-28, modified 2007-03-16.
Object id is 1154, canonical name is CauchyResidueTheorem.
Accessed 15557 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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