Digamma Function of Five Sixths

Theorem

$\map \psi {\dfrac 5 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 + \dfrac {\pi \sqrt 3} 2$

where:

$\psi$ denotes the digamma function

$\gamma$ denotes the Euler-Mascheroni constant.

Proof

\(\ds \map \psi {\frac 1 6} - \map \psi {\frac 5 6}\)

\(=\)

\(\ds -\pi \map \cot {\frac \pi 6}\)

Digamma Reflection Formula

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 5 6}\)

\(=\)

\(\ds \pi \map \cot {\frac \pi 6} + \map \psi {\frac 1 6}\)

rearranging

\(\ds \)

\(=\)

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

Cotangent of $30 \degrees$ and Digamma Function of One Sixth

\(\ds \)

\(=\)

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

rearranging

$\blacksquare$