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.

File:Morera's Theorem.png


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