Find entire $f$ from $|f|$

Question

$f$ is a Complex function.

Can we find entire function $f$ when we know the absolute value of $f$, $|f|$?

I want to know what entire function has absolute value as $$|f| = e^{(x^3-3xy^2)}$$

I do not think $f$ is unique... but can I find them all?

Answer 1

Hint: Suppose that $g$ is an entire function such that $\operatorname{Re}g(x+yi)=x^3-3xy^2$. Then$$\left|e^{g(x+yi}\right|=e^{\operatorname{Re}g(x+yi)}=e^{x^3-3xy^2}.$$

Answer 2

Hint:

Observe that

$$\log|f|=\log\left(e^{x^3-3xy^2}\right)=\Re(z^3).$$

You can retrieve the imaginary part from the Cauchy-Riemann equations and you can expect some resemblance to $\Im(z^3)$.