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In Stein and Shakarchi, Complex Analysis, Princeton lectures in Analysis, Chapter 2, Problem 2 an interesting question is posed. The problem section in each chapter contains more complicated problems, with a research taste.

Morera's theorem simply states that if a function $f$ is continuous on $\Bbb{C}$ and $\int_D f(z)dz=0$ for any triangle(rectangle) $D$, then $f$ is holomorphic in $\Bbb{C}$. (the theorem is still valid if we replace $\Bbb{C}$ by a disk).

The problem presented above, states that

Morera's theorem is still valid if we replace the contours of integration from triangles/rectangles to circles, and more generally, to any contour which is a translate and dilate of a toy contour $\Gamma$.

Is there a simple proof for this problem, or maybe a reference to an article in which I can find the proofs?

[I will post the hint given after the problem in the book but is quite long and I don't have the necessary time right now.]

share|cite|improve this question
a) AoPS/College Playground is the appropriate place for study questions. b) convolve with a mollifier, apply Green, conclude that $\bar \partial$ of the convolution is $0$, recall that the uniform limit of analytic functions is analytic. – fedja May 25 '11 at 11:07

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