# Entire function invariant by translation is constant [duplicate]

I need to apply Liouville theorem ("entire bounded complex functions are constant") to prove that an entire function satisfying:

$$f(z)=f(z+1)=f(z+i)$$

for all complex numbers $$z$$ is constant. I'm really not sure on how to proceed, I've tried expanding in Taylor series but I've got confused with calculations and I can't proceed any further.

## marked as duplicate by Martin R, user10354138, Paul Frost, Community♦Dec 9 '18 at 18:06

By induction, $$f(z)=f(z+m+in)$$ for all integers $$m,n$$. This means that for each $$z\in\mathbb{C}$$ there is $$w\in[0,1]\times[0,1]$$ such that $$f(z)=f(w)$$. Continuous functions are bounded on compact sets. It follows that $$f$$ is bounded on $$\mathbb{C}$$. Now apply Liouville.