Homotopy-invariance formulation of Cauchy's theorem: Difference between revisions

Revision as of 18:49, 26 April 2008

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose $U \subset C$ is an open subset and $γ_{1}, γ_{2}$ are two smooth paths in $U$ such that there is a smooth homotopy between $γ_{1}$ and $γ_{2}$ preserving endpoints. Then, for any holomorphic function $f : U \to C$ , we have:

$\oint_{γ_{1}} f (z) d z = \oint_{γ_{2}} f (z) d z$

In particular, if $c$ is a zero-homologous cycle, we have $\oint_{c} f (z) d z = 0$ .

Related facts

Goursat's integral lemma: It states something very similar, albeit in a very special case: where $c$ forms the smooth boundary of a region.

@@ Line 1: / Line 1: @@
+{{basic fact}}
 ==Statement==
-Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>c_1, c_2</math> are two cycles (cycle being a sum of smooth simple closed curves) that are smoothly homotopic. Then, for any [[holomorphic function]] <math>f: U \to \mathbb{C}</math>, we have:
+Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>\gamma_1,\gamma_2</math> are two smooth paths in <math>U</math> such that there is a smooth homotopy between <math>\gamma_1</math> and <math>\gamma_2</math> preserving endpoints. Then, for any [[holomorphic function]] <math>f: U \to \mathbb{C}</math>, we have:
-<math>\oint_{c_1} f(z) dz = \oint_{c_2} f(z) dz</math>
+<math>\oint_{\gamma_1} f(z) dz = \oint_{\gamma_2} f(z) dz</math>
 In particular, if <math>c</math> is a zero-homologous cycle, we have <math>\oint_c f(z) dz = 0</math>.