# Prove that an analytic function with a constant imaginary part is constant itself

I want to prove that if $f$ is an analytic function for which $\Im f$ is constant, this implies that $f$ itself is constant.

So to start off, it's not given that the function is entire or anything, ruling out Liouville. Rather, I'm guessing the maximum modulus theorem will prove useful here (?).

If I were to show using that the real part of $f$ was constant, proving the statement would be easy (just take the absolute value and investigate $e^f$), but in this case I'm none the wiser from that. Should I use the same parametrization $e^f$ and apply the Cauchy-Riemann equations somewhere? Any suggestions welcome.

• If you want to pursue your own idea, look at $e^{if}$.
– mrf
Commented Dec 25, 2015 at 20:56

Hint: use the Cauchy-Riemann equations, $f(x,y)=g(x,y)+ih(x,y)$ if $h(x,y)$ is constant, $\partial_xg=\partial_yh=0$ and $\partial_yg=-\partial_xh=0$ thus $g$ is also constant. done.
The image of $f$ will be a horizontal line in $\mathbb{C}$. Now, consider the open mapping theorem.
• (Given that $f$ is defined on a "nice" domain, i.e. a simply connected region). Commented Dec 25, 2015 at 17:50