Originally Posted by

**vincisonfire** Let $\displaystyle f (z) $ be an entire function such that for all x, y,

$\displaystyle |f (x +iy)| \neq 0 $ and $\displaystyle arg(f (x + iy)) = xy $. Find f .

First I thought that working using Euler's notation would make the thing simpler. Let $\displaystyle f (r,\theta)=\rho e^{i\phi} $

The condition are thus $\displaystyle |f(r,\theta)|=\rho \neq 0 $

and $\displaystyle arg(f (x + iy)) = \phi = r^2cos(\theta)sin(\theta) $.

Then $\displaystyle f(r,\theta) = \rho e^{ir^2cos(\theta)sin(\theta)} $ where

$\displaystyle \rho \neq 0 $. Since this is a composition of entire function it should be an entire function.

I'm not comfortable with complex variables so I ask for some help. The question suggest there might be a single $\displaystyle f (z) $ but I [think I] found many. Does what is above makes any sense?