Digamma Function of One

Theorem

$\map \psi 1 = -\gamma$

where:

$\psi$ denotes the digamma function

$\gamma$ denotes the Euler-Mascheroni constant.

Proof

\(\ds \map \psi z\)

\(=\)

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

Definition of Digamma Function

\(\ds \)

\(=\)

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

Reciprocal times Derivative of Gamma Function

\(\ds \leadsto \ \ \)

\(\ds \map \psi 1\)

\(=\)

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

$z \gets 1$

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds -\gamma\)

noting that the summation is an empty sum

$\blacksquare$