Assume we have a real-analytic function $f(x, y)>0$ in some neighborhood of 0. When does there exist a complex-analytic function $w(z)$ such that $|w(z)|=f(x,y)$ for $z=x+iy$.
One necessary condition is that $\Delta\ln f=0$. Is there anything else?
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4$\begingroup$ Why aren't you done then? If $\ln f$ is harmonic, then you can add its harmonic conjugate, exponentiate et voila $\endgroup$– Anthony QuasJun 19, 2013 at 8:33
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$\begingroup$ This is true if your neighborhood is simply connected. $\endgroup$– Alexandre EremenkoJun 19, 2013 at 18:25
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$\begingroup$ @Alexandre, Yes, I meant small $\epsilon$-neighborhood $\endgroup$– Dmitri ScheglovJun 19, 2013 at 18:35
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$\begingroup$ Voting to close as it is answered in the comments. $\endgroup$– Willie WongSep 18, 2013 at 12:36
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1 Answer
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To formally complete the question: the answer is by Anthony Quas in the comment below.