# Ngô Quốc Anh

## July 20, 2009

### A generalization of the Morera’s Theorem

In complex analysis, a branch of mathematics, Morera’s theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

Morera’s theorem states that if $f$ is a continuous, complex-valued function defined on an open set $D$ in the complex plane, satisfying

$\displaystyle\oint_C f(z) dz = 0$

for every triangle $C$ in $D$, then $f$ must be holomorphic on $D$.

The assumption of Morera’s theorem is equivalent to that $f$ has an anti-derivative on $D$. The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. For instance, Cauchy’s integral theorem states that the line integral of a holomorphic function along a closed curve is zero, provided that the domain of the function is simply connected.

Now we state and prove the following generalization of the Morera’s theorem: Suppose that $f$ is continuous on $\mathbb C$, and

$\displaystyle\int_C f(z) dz = 0$

for every circle $C$. Prove $f$ is holomorphic.