Digamma Function of One Half/Proof 1

Theorem

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

Proof

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

\(=\)

\(\ds -\gamma - \ln 4 - \frac \pi 2 \map \cot {\frac 1 2 \pi} + 2 \sum_{n \mathop = 1}^0 \map \cos {\frac {2 \pi n} 2} \map \ln {\map \sin {\frac {\pi n} 2} }\)

Gauss's Digamma Theorem

\(\ds \)

\(=\)

\(\ds -\gamma - \ln 4\)

Cotangent of Right Angle, noting also that the summation is an empty sum

\(\ds \)

\(=\)

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

Logarithm of Power

$\blacksquare$