TITLE: $f:\mathbb{C}\rightarrow\mathbb{C}$ entire function and $f(z)=u(x)+iv(y)$ then $f$ is a polynomial
QUESTION [6 upvotes]: I was going through my introduction to complex analysis homework, when I came across this exercise:

If $f:\mathbb{C} \rightarrow \mathbb{C}$ is an entire function of the form $f(z)=u(x)+iv(y)$, prove that $f$ is a polynomial.

I've got completely stuck on this one. I think it might be something to do with $f$ being analytical, but I'm not so sure, because I was sick and couldn't watch the class. ):
Any hints are appreciated, thanks.

REPLY [10 votes]: Hint : Write down the Cauchy-Riemann equations and see what pops out.