Polygamma Reflection Formula/Lemma

Polygamma Reflection Formula: Lemma

Let $\psi$ denote the digamma function.

The expression:

$\map \psi z - \map \psi {1 - z}$

is defined on the domain $\C \setminus \Z$.

That is, on the set of complex numbers but specifically excluding the integers.

Proof

From the definition of the digamma function:

$\map \psi z$ is defined for $z \in \C \setminus \Z_{\le 0}$

and:

$\map \psi {1 - z}$ is defined for $\paren {1 - z} \in \C \setminus \Z_{\le 0}$.

Therefore, $\map \psi z - \map \psi {1 - z}$ is defined for $z \in \C \setminus \Z$.

$\blacksquare$