Finding analytic function with given condition

Question

I have this task from complex analysis: Find analytic function $ f(z) $ such that $ |f(z)|=e^{{\rho}^2\cos(2\theta)} $ where $ z=\rho e^{i\theta}. $ I'm guessing I should use Cauchy-Riemann conditions, but I'm not sure how to do it. Could you help me, please?

Answer 1

Since $(\rho e^{i\theta})^2=\rho^2e^{2i\theta}$, you have $\operatorname{Re}\bigl((\rho e^{i\theta})^2\bigr)=\rho^2\cos(2\theta)$. Besides,$$e^{\operatorname{Re}\bigl((\rho e^{i\theta})^2\bigr)}=\left|e^{(\rho e^{i\theta})^2}\right|.$$So, take $f(z)=e^{z^2}.$

Answer 2

Hints: $|e^w|= e^{\text{Re}(w)}$. Now $\rho^2 \cos(2\theta) = \text{Re}(\rho^2 e^{2i\theta}) = \text{Re}(z^2)$.