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This is a doubt of mine on the basics of complex analysis.

I encountered a certain statement involving integrating a harmonic function, which would be nice for my research attempts if proved. When I strengthened the assumption to that the function is holomorphic, I could very easily do it using Cauchy's theorem. Is it always possible to treat a harmonic function as the real or imaginary part of a holomorphic function, and draw consequences from Cauchy's theorem?

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    It should be be said that you only talk about real valued harmonic functions. Linear combinations over $\mathbb{C}$ of harmonic functions are always harmonic.2011-09-16

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If $f$ is a harmonic function on a simply connected domain then it is the real or imaginary part of a holomorphic function.

If the domain is not simply connected then the above may not be true. Consider $f(x,y)=\log(\sqrt{x^2 +y^2})$ in the punctured unit disc.

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    Ah, ok, thanks everyone. This was something basic and I should have always known, but didn't. Your comments helped in understanding.2011-09-20