Finding a Complex function

**mohammadfawaz** · March 12th 2011, 01:57 AM

Hello,
I have the following question:
A complex function f is differentiable in a region $\Omega \subset \field{C}.$ and satisfies the equation $|f(z)|=5$ for all $z\in \Omega$ . Determine the function f.

I started by saying $f = u + iv$ . We know $u^2+v^2 = 25$ .
And we know also the Cauchy-Riemann equations since f is differentiable. I tried taking derivatives in $u^2+v^2 = 25$ but I was stuck. An intuitive function to have is $f = 5c$ where c is a complex number with magnitude 1.

Any help is appreciated.

Thanx

**FernandoRevilla** · March 12th 2011, 02:23 AM

According to the Maximum Modulus Principle, $f$ must be constant in $\Omega$ so $f(z)=5e^{i\theta_0}$ with $\theta_0\in [0,2\pi)$ .

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