Digamma Function in terms of General Harmonic Number

Corollary to Reciprocal times Derivative of Gamma Function

Let $z \in \C \setminus \Z_{\le 0}$.

Then:

$\ds \map \psi {z + 1} = -\gamma + \harm 1 z$

where:

$\psi$ is the digamma function

$\gamma$ denotes the Euler-Mascheroni constant

$\harm 1 z$ denotes the general harmonic number of order $1$ evaluated at $z$.

Proof

\(\ds \dfrac {\map {\Gamma'} {z + 1} } {\map \Gamma {z + 1} }\)

\(=\)

\(\ds -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {\paren {z + 1} + n - 1} }\)

Reciprocal times Derivative of Gamma Function

\(\ds \leadsto \ \ \)

\(\ds \map \psi {z + 1}\)

\(=\)

\(\ds -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n} }\)

Definition of Digamma Function

\(\ds \)

\(=\)

\(\ds -\gamma + \harm 1 z\)

Definition of General Harmonic Numbers

$\blacksquare$