Digamma Function of One Fourth/Proof 1

Theorem

$\map \psi {\dfrac 1 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2$

Proof

\(\ds \map \psi {\frac 1 4}\)

\(=\)

\(\ds -\gamma - \ln 8 - \frac \pi 2 \map \cot {\frac 1 4 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {4 / 2} - 1} \map \cos {\frac {2 \pi n} 4} \map \ln {\map \sin {\frac {\pi n} 4} }\)

Gauss's Digamma Theorem

\(\ds \)

\(=\)

\(\ds -\gamma - 3 \ln 2 - \frac \pi 2 \times 1 + 2 \times 0 \times \map \ln {\frac {\sqrt 2} 2}\)

Cotangent of $45 \degrees$, Cosine of $90 \degrees$, Sine of $45 \degrees$ and Sum of Logarithms

\(\ds \)

\(=\)

\(\ds -\gamma - 3 \ln 2 - \dfrac \pi 2\)

$\blacksquare$