How to know that $\text{Im}(A\cosh(z)+\frac\pi z)$ is harmonic on domain $\{z|0\lt\text{Im }z\lt \pi\}$ where $A\in\Bbb R$?
I am not sure how I would verify Laplace's equation here(which I imagine is how I am meant to solve this).
I tried reducing it to it's imaginary components, but it seemed to be the wrong method.
For completeness, here's an answer (since I could not find it written anywhere yet).
The real and imaginary parts of a holomorphic function are harmonic.
Proof: Suppose $f = u+iv$ is holomorphic ($u,v$ real). The Cauchy-Riemann equations say $u_x=v_y$ and $u_y=-v_x$. Therefore, $$u_{xx}+u_{yy}=v_{yx}-v_{xy}=0$$ and $$v_{xx}+v_{yy}=-u_{yx}+u_{xy}=0$$ as claimed.