Goursat's integral lemma: Difference between revisions

Latest revision as of 19:13, 18 May 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

For a triangle

Suppose $U$ is a domain in $\mathbb {C}$ , and $f:U\to \mathbb {C}$ is a holomorphic function. Suppose $\triangle$ is a triangle contained completely inside $U$ (i.e. the interior and boundary are contained inside $U$ ). Then, we have:

$\int _{\partial \Delta }f(z)\,dz=0$

For a region bounded by piecewise smooth curves

Suppose $U$ is a domain in $\mathbb {C}$ , and $f:U\to \mathbb {C}$ is a holomorphic function. Suppose $V$ is an open subset whose closure is a compact subset of $U$ , such that $\partial V$ is piecewise $C^{1}$ . Note that $V$ may have disconnected boundary; for instance, $V$ may be an annulus.

Then we have:

$\int _{\partial V}f(z)\,dz=0$

Note that this is a slight generalization of the previous case, where we restrict $V$ to the interior of a triangle.

Related facts

Goursat's integral lemma for complex-differentiable functions: This is a slightly more general version, giving the conclusion for triangles under the weaker assumption that $f$ is only complex-differentiable, rather than holomorphic. It is used to establish that in fact, complex-differentiable is equal to holomorphic.
Homotopy invariance formulation of Cauchy's theorem: This is a more general version of the Goursat's integral lemma which replaces the condition of being the boundary of an open subset, by the condition of being a zero-homologous cycle. More generally, it says that if two cycles are homotopic, they give the same integral for all holomorphic functions.

Facts used

@@ Line 1: / Line 1: @@
+{{basic fact}}
 ==Statement==
@@ Line 17: / Line 18: @@
 Note that this is a slight generalization of the previous case, where we restrict <math>V</math> to the interior of a triangle.
-==Statement in terms of cohomology and loops==
+==Related facts==
-Note that [[integral of a complex-valued function along a curve|integration along piecewise smooth curves]] gives a map as follows:
+* [[Goursat's integral lemma for complex-differentiable functions]]: This is a slightly more general version, giving the conclusion for triangles under the weaker assumption that <math>f</math> is only complex-differentiable, rather than holomorphic. It is used to establish that in fact, complex-differentiable is equal to holomorphic.
+* [[Homotopy invariance formulation of Cauchy's theorem]]: This is a more general version of the Goursat's integral lemma which replaces the condition of being the boundary of an open subset, by the condition of being a zero-homologous cycle. More generally, it says that if two cycles are homotopic, they give the same integral for all holomorphic functions.
-Piecewise smooth curves <math>\times</math> Continuous functions <math>\to \mathbb{C}</math>
-We can restrict attention to piecewise smooth loops at a point <math>z_0 \in U</math>. What this says is that if the loop is nullhomologous, or if we take a sum of loops whose homology classes add up to zero, then evaluating on any holomorphic function, gives the value zero.
-This helps us tells us that the map we have descends to a bilinear map:
-<math>H_1(U) \times H^1_{DR}(U) \to \mathbb{C}</math>
 ==Facts used==