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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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closed as off-topic by Michael Renardy, Ramiro de la Vega, Ryan Budney, Todd Trimble, Willie Wong Sep 18 '13 at 12:36

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Michael Renardy, Ramiro de la Vega, Ryan Budney, Todd Trimble
If this question can be reworded to fit the rules in the help center, please edit the question.

Why aren't you done then? If $\ln f$ is harmonic, then you can add its harmonic conjugate, exponentiate et voila – Anthony Quas Jun 19 '13 at 8:33
Thank you, how could I miss such an easy remark! – Dmitri Scheglov Jun 19 '13 at 14:09
This is true if your neighborhood is simply connected. – Alexandre Eremenko Jun 19 '13 at 18:25
@Alexandre, Yes, I meant small $\epsilon$-neighborhood – Dmitri Scheglov Jun 19 '13 at 18:35
Voting to close as it is answered in the comments. – Willie Wong Sep 18 '13 at 12:36
up vote 1 down vote accepted

To formally complete the question: the answer is by Anthony Quas in the comment below.

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You need to "accept" this answer to formally complete the question. – S. Carnahan Sep 17 '13 at 19:20

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