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Harmonic function

Consider the points O(0, 0) and  A(1, 0). Let \Gamma(x, y) be a point of the plane such that y>0. Set \varphi(x, y) to be the angle that is defined by O\Gamma and A \Gamma. ( the one that is less than \pi.) Prove that the function \varphi(x, y) is harmonic.

Solution [by Demetres Skouteris]

The complex function \log \left( 1 - \frac{1}{z} \right) has a holomorphic branch in the half plane \mathfrak{Im}(z)>0 and its imaginary part is the desired angle. Hence, the function of the angle is harmonic.

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