Digamma Function of One Third/Proof 1

Theorem

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

Proof

\(\ds \map \psi {\frac 1 3}\)

\(=\)

\(\ds -\gamma - \ln 6 - \frac \pi 2 \map \cot {\frac 1 3 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {3 / 2} - 1} \map \cos {\frac {2 \pi n} 3} \map \ln {\map \sin {\frac {\pi n} 3} }\)

Gauss's Digamma Theorem

\(\ds \)

\(=\)

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

Cotangent of $60 \degrees$, Cosine of $120 \degrees$, Sine of $60 \degrees$ and Sum of Logarithms

\(\ds \)

\(=\)

\(\ds -\gamma - \ln 2 - \ln 3 - \frac \pi 2 \paren {\frac 1 {\sqrt 3} } - \paren {\frac 1 2 \ln 3 - \ln 2}\)

Difference of Logarithms and Logarithm of Power

\(\ds \)

\(=\)

\(\ds -\gamma - \dfrac 3 2 \ln 3 - \dfrac {\pi} {2 \sqrt 3}\)

$\blacksquare$