Lemma from Goursat

The lemma Goursat , sometimes referred to as a set of Goursat, is a set of the function theory .

Goursat's lemma is a preliminary stage of Cauchy's integral theorem and is also often used for its proof. It plays an important role in the structure of function theory. It is noteworthy that the lemma only presupposes the complex differentiability , but not the continuous differentiability. The lemma was proven by Édouard Goursat ( 1858 - 1936 ) in the rectangular form and published in 1884 . The triangular shape common today comes from Alfred Pringsheim .

Goursat's lemma

Goursat's lemma for triangles

Be open and holomorphic (complex-differentiable). Then, for the path integral of the boundary curve along each in lying triangle : ${\displaystyle U\subset \mathbb {C} }$ ${\displaystyle f\colon U\rightarrow \mathbb {C} }$ ${\displaystyle \partial \triangle }$ ${\displaystyle U}$ ${\displaystyle \triangle }$

${\displaystyle \oint _{\partial \triangle }f(z)\mathrm {d} z=0}$

Goursat's lemma for rectangles

Sometimes Goursat's lemma is also formulated for rectangles:

Be open and holomorphic. Then for all edge curves one in nearby rectangle : ${\displaystyle U\subset \mathbb {C} }$ ${\displaystyle f\colon U\rightarrow \mathbb {C} }$ ${\displaystyle \partial \Box }$ ${\displaystyle U}$ ${\displaystyle \Box }$

${\displaystyle \oint _{\partial \Box }f(z)\mathrm {d} z=0}$

Goursat's lemma tightened

Goursat's lemma also applies under somewhat weaker conditions:

Be open, and continuous and holomorphic , then applies to all boundary curves of triangles located in with as corner point: ${\displaystyle U\subset \mathbb {C} }$ ${\displaystyle z_{0}\in U}$ ${\displaystyle f\colon U\rightarrow \mathbb {C} }$ ${\displaystyle U\setminus \{z_{0}\}}$ ${\displaystyle \partial \triangle }$ ${\displaystyle U}$ ${\displaystyle \triangle }$ ${\displaystyle z_{0}}$

${\displaystyle \oint _{\partial \triangle }f(z)\mathrm {d} z=0}$

literature

• Eberhard Freitag & Rolf Busam: Function theory 1 , Springer-Verlag, Berlin, ISBN 3-540-67641-4