Today I Learned

Some of the things I've learned every day since Oct 10, 2016

154: Morera’s Theorem

In complex analysis, Morera’s Theorem states that if D \subset \mathbb{C} is a region (open, path-connected) and f: D \rightarrow \mathbb{C} is continuous and has the property that

\int _\gamma f(z) dz = 0

for any closed curve \gamma \subset D, then f is holomorphic.

This is an immediate corollary to the observation in [153], since if F is a primitive of f on D then f itself is holomorphic on D.

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