nLab
Cauchy integral theorem

Idea

Cauchy’s integral theorem states that contour integrals of holomorphic functions, in any simply connected subspace of the complex plane, are invariant under homotopy of paths.

Statement

Let ${\gamma }_1,{\gamma }_2$ be two homotopic loops in a simply connected open subset $D\subseteq ℂ$. Let $f$ be a holomorphism on $D$. Then we have

$${\int }_{{\gamma }_1}f(z)\mathrm{d}z={\int }_{{\gamma }_2}f(z)\mathrm{d}z$$\int_{\gamma_1} f(z) \,\mathrm{d}z = \int_{\gamma_2} f(z) \,\mathrm{d}z

In particular we have

$${\int }_{{\gamma }_1}f(z)\mathrm{d}z=0$$\int_{\gamma_1} f(z) \,\mathrm{d}z = 0

for ${\gamma }_2≔0$.

category: analysis