If $Re(f)$ is a polynomial then $f$ is a polynomial, where$f$ is entire

Question

If $f$ is entire, and $Re(f)$ is a polynomial in x,y I am trying to show that f is a polynomial in z.

We can write $f(z) = u(z) + iv(z)$. where $Re(f) = u(z)$

I have solved problems similar to this one before, except usually the real part is bounded which allows me to use Liouville's theorem, here since there is no such condition I am unsure of how to proceed.

I also know that since v is the harmonic conjugate of u, it is uniquely determined up to addition of a constant, I believe this fact is helpful but not sure how to begin with this.

Answer 1

$f$ can be written as power series $f(z) = \sum_{n=0}^\infty a_n z^n$ with radius of convergence = $\infty$. If $f$ is not a polynomial, then neccessarily infinitely many $a_n \ne 0$. In that case $Re(f)$ is not a polynomial.