Complex Variables

**vincisonfire** · January 23rd 2009, 03:20 PM

Let $f (z)$ be an entire function such that for all x, y,
$|f (x +iy)| \neq 0$ and $arg(f (x + iy)) = xy$ . Find f .

First I thought that working using Euler's notation would make the thing simpler. Let $f (r,\theta)=\rho e^{i\phi}$
The condition are thus $|f(r,\theta)|=\rho \neq 0$
and $arg(f (x + iy)) = \phi = r^2cos(\theta)sin(\theta)$ .
Then $f(r,\theta) = \rho e^{ir^2cos(\theta)sin(\theta)}$ where
$\rho \neq 0$ . 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 $f (z)$ but I [think I] found many. Does what is above makes any sense?

**manjohn12** · January 23rd 2009, 07:17 PM

Originally Posted by vincisonfire

Let $f (z)$ be an entire function such that for all x, y,
$|f (x +iy)| \neq 0$ and $arg(f (x + iy)) = xy$ . Find f .

First I thought that working using Euler's notation would make the thing simpler. Let $f (r,\theta)=\rho e^{i\phi}$
The condition are thus $|f(r,\theta)|=\rho \neq 0$
and $arg(f (x + iy)) = \phi = r^2cos(\theta)sin(\theta)$ .
Then $f(r,\theta) = \rho e^{ir^2cos(\theta)sin(\theta)}$ where
$\rho \neq 0$ . 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 $f (z)$ but I [think I] found many. Does what is above makes any sense?

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

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