Assume we have a realanalytic function $f(x, y)>0$ in some neighborhood of 0. When does there exist a complexanalytic 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?
Assume we have a realanalytic function $f(x, y)>0$ in some neighborhood of 0. When does there exist a complexanalytic 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? 

closed as offtopic by Michael Renardy, Ramiro de la Vega, Ryan Budney, Todd Trimble♦, Willie Wong Sep 18 '13 at 12:36This question appears to be offtopic. The users who voted to close gave this specific reason:



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

