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