Sum of complex digamma functions

Question

It seems that the sum of the digamma function of $z$ and the digamma function of its conjugate $z^*$ is always real-valued. $$\psi(z)+\psi(z^*)=\frac{\Gamma'(z)}{\Gamma(z)}+\frac{\Gamma'(z^*)}{\Gamma(z^*)}\in\mathbb{R}$$ Why is this so? Is there a simpler identity for the above?

Answer 1

$\psi(z) - \overline{\psi(\overline{z})}$ is analytic (except perhaps at the singularities of $\psi$) and is $0$ on the positive real axis (because $\psi$ is real there), so by analytic continuation it must be $0$ everywhere. Thus $\psi(z) + \psi(\overline{z}) = \psi(z) + \overline{\psi(z)}$ is always real.