08 - Cauchy's Theorem for Simply Connected Domains

Cauchy's Theorem for Simply Connected Domains

Suppose $U$ is a simply connected domain
Let $a,b\in U$

Let $f:U\to C$ be a holomorphic function on $U$ then

If ${\gamma }_1,{\gamma }_2$ are paths from $a$ to $b$ then

$${\int }_{{\gamma }_1}f(z)dz={\int }_{{\gamma }_2}f(z)dz$$

In particular if $\gamma$ is a closed oriented curve then ${\int }_{\gamma }f(z)dz=0$
so any holomorphic function on $U$ has a primitive

Proof

Since $U$ is simply connected then
Any two paths from $a$ to $b$ are homotopic so first part is by Homotopy Cauchy’s Theorem

In a simply connected domain, any closed path

$$\gamma :[0,1]\to U\text{ with }\gamma (0)=\gamma (1)=a$$

is homotopic to some constant path $c(t)=z_0$ hence

$${\int }_{\gamma }f(z)dz={\int }_{c}f(z)dz=0$$