analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
…
…
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
Cauchy’s integral theorem states that contour integrals of holomorphic functions in the complex plane $\mathbb{C}$ are invariant under homotopy of paths. In particular, if a function is holomorphic on a simply connected subspace of $\mathbb{C}$, then its contour integral on a path depends only on the beginning and ending points of the path, and indeed can be given by subtracting the values there of an antiderivative (in accordance with the second Fundamental Theorem of Calculus).
Let $D$ be an open subset of the complex plane $\mathbb{C}$, let $a$ and $b$ be two points in $D$, let $\gamma_1$ and $\gamma_2$ be two curves in $D$ from $a$ to $b$, let the region between them also lie entirely within $D$, and let $f$ be a holomorphic function on $D$. Then we have
In particular we have
if $a = b$ (because then $\gamma_2$ may be taken to be a constant); in other words, the contour integral of a holomorphic function is zero around any loop whose inside lies entirely within the function's domain.
Last revised on January 29, 2024 at 00:04:56. See the history of this page for a list of all contributions to it.