1. ## Complex Variables

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?

2. 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?

I think you are correct. It says for all $\displaystyle x,y$, $\displaystyle |f(x+iy)| \neq 0$ and $\displaystyle \text{arg}(f(x+iy)) = xy$. So the argument is not constant. Nor is the modulus. Also polynomials of the form $\displaystyle f(x+yi)$ are analytic/entire.