Show that $f$ is a polynomial regarding a given question

Question

This is in reference to this $f:\mathbb{C}\rightarrow\mathbb{C}$ entire function and $f(z)=u(x)+iv(y)$ then $f$ is a polynomial

Since $f$ is entire it satisfies Cauchy Riemann so $u_x=v_y \& 0=u_y=v_x$

But how does that show $f$ is a polynomial?

Any help please

Answer 1

$u_x$ and $v_y$ are functions only of $x$ and of $y$ respectively, so the only way they’re equal is if they are constants.