Digamma Function of One Half/Proof 2

Theorem

$\map \psi {\dfrac 1 2} = -\gamma - 2 \ln 2$

Proof

\(\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}\)

\(=\)

\(\ds -\paren {n - 1} \gamma - n \ln n\)

Digamma Additive Formula: Corollary

\(\ds \leadsto \ \ \)

\(\ds \sum_{k \mathop = 1}^{2 - 1} \map \psi {\frac k 2}\)

\(=\)

\(\ds -\paren {2 - 1} \gamma - 2 \ln 2\)

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 1 2}\)

\(=\)

\(\ds -\gamma - 2 \ln 2\)

$\blacksquare$