Find all entire functions with $f(z) = f(\frac{1}{z})$ Find all entire functions with $f(z) = f(\frac{1}{z})$, for all $z\ne 0$.
I tried to use the power series of $f$ but this did not help. I also tried to use Liouville's Theorem on $\frac{f(z)}{f(\frac{1}{z})}$ but then I have to deal with possible singularities.
Can someone help me with this problem?

I presume your $f$ is defined on the whole of $\mathbb C$, but the equality $f(z) = f(1/z)$ is only meant to hold for $ z \in \mathbb C \backslash \{ 0 \}$?
Well, $f$ is holomorphic, so it is bounded on $\{ z \in \mathbb C : |z | \leq 1 \}$. But since $f(z) = f(1/z)$, this means that $f$ is also bounded on $\{ | z | \geq 1 \}$. Do you see where this is going?
