Digamma Function of Two Thirds

Theorem

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

where:

$\psi$ denotes the digamma function

$\gamma$ denotes the Euler-Mascheroni constant.

Proof

\(\ds \map \psi {\frac 1 3} - \map \psi {\frac 2 3}\)

\(=\)

\(\ds -\pi \map \cot {\frac \pi 3}\)

Digamma Reflection Formula

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 2 3}\)

\(=\)

\(\ds \pi \map \cot {\frac \pi 3} + \map \psi {\frac 1 3}\)

rearranging

\(\ds \)

\(=\)

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

Cotangent of $60 \degrees$ and Digamma Function of One Third

\(\ds \)

\(=\)

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

rearranging

$\blacksquare$