[Math] An extension of Morera’s Theorem

analytic-functionscv.complex-variables

Morera's Theorem states that

If $f$ is continuous in a region $D$ and satisfies $\oint_{\gamma} f = 0$ for
any closed curve $\gamma$ in $D$, then $f$ is analytic in $D$.

I have two questions:

  1. If $f$ is continuous in $D$ and $\oint_C f = 0$ for any circle $C$ in $D$,
    can we deduce that $\oint_{\gamma} f = 0$ for any closed curve $\gamma$ in $D$?

  2. (more ambitiously) If $f$ is continuous and $\oint_C f = 0$ for any circle $C$ in $D$, is $f$ analytic in $D$ ?

Partial ansers for question 2 seem to be here, but I doubt their argument, specificly, the construction of the original function.

Best Answer

The answer is yes, and a proof can be found for example on this webpage: http://anhngq.wordpress.com/2009/07/20/a-generalization-of-the-morera%E2%80%99s-theorem/

A brief summary: Suppose $f$ is continuous and $\int_C f = 0$ for every circle $C$, but $\int_\gamma f \neq 0$ for some closed curve $\gamma$. By convolving $f$ with a smooth approximation to the identity, we may assume $f$ is smooth. But then by applying Green's formula to $\int_C f = 0$ for small circles $C$, we see that $f$ must satisfy the Cauchy-Riemann equations, so $\int_\gamma f = 0$, a contradiction.

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