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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.

3  
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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2  
You need to "accept" this answer to formally complete the question. – S. Carnahan Sep 17 '13 at 19:20

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