Cauchy's Theorem Consequences

Subjects: Complex Analysis
Links: Homotopy Cauchy's Theorem, Homology Cauchy's Theorem

Morera’s Theorem

Let $f$ be a continuous function on a region $A$, and suppose that ${\oint }_{\gamma }f=0$ for every closed curve in $A$. Then $f$ is holomorphic on $A$, and $f=F^{\prime }$ for some holomorphic function $F$ on $A$.

Cor: Let $f$ be a continuous function on $A$ and holomorphic on $A\setminus \{z_0\}$ for some point $z_0\in A$. Then $f$ is holomorphic on $A$.

Logarithms of Functions

Let $f$ be an holomorphic function that is never $0$ on a simply connected domain $A$. Then there is a function $g$ holomorphic on $A$ and unique up to addition of constant multiples of $2\pi i$ such that $\exp (g(z))=f(z)$ for all $z\in A$.