Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
For $U \subset \mathbb{C}$ an open subset of the complex plane $\mathbb{C}$ and for $C$ a Jordan curve in $U$, a holomorphic function $f$ of $U$ sends a point $\zeta$ enclosed by $C$ to the contour integral
Hence the contour integral picks out the enclosed residues.
More generally, this implies, by Taylor series expansion of $f$, that for $n \in \mathbb{N}$ the $n$th complex derivative $f^{(n)}$ is
This is also known as Cauchy’s differentiation formula.
Here is a proof written in terms of synthetic infinitesimals as in synthetic differential geometry:
Let $\epsilon$ be a nilpotent. Let $S_\epsilon$ denote the circle of radius $\epsilon$ centered at $\zeta$. By the holomorphicity of $f$, the differential 1-form $f(z)/(z-\zeta) \,\mathrm{d}z$ is closed in the region bounded by $C$ and $S_\epsilon$. By the Stokes theorem,
Parametrize $S_\epsilon$ by $(t\mapsto \zeta + \epsilon \,\exp(2\pi\mathrm{i}t))$ to transform the above integral to
By the infinitesimal Taylor formula and the holomorphicity of $f$,
Hence the above integral is equal to
Last revised on January 29, 2024 at 00:01:22. See the history of this page for a list of all contributions to it.