Digamma Function of One Third/Proof 2

Theorem

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

$\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}$

$=$

$\ds -\paren {n - 1} \gamma - n \ln n$

$\ds \leadsto \ \ $

$\ds \sum_{k \mathop = 1}^{3 - 1} \map \psi {\frac k 3}$

$=$

$\ds -\paren {3 - 1} \gamma - 3 \ln 3$

$\text {(1)}: \quad$

$\ds \leadsto \ \ $

$\ds \map \psi {\frac 1 3} + \map \psi {\frac 2 3}$

$=$

$\ds -2 \gamma - 3 \ln 3$

$\text {(2)}: \quad$

$\ds \map \psi {\frac 1 3} - \map \psi {\frac 2 3}$

$=$

$\ds -\pi \map \cot {\frac \pi 3}$

$\ds \leadsto \ \ $

$\ds 2 \map \psi {\frac 1 3}$

$=$

$\ds -2 \gamma - 3 \ln 3 - \pi \map \cot {\frac \pi 3}$

adding lines $1$ and $2$

$\ds \leadsto \ \ $

$\ds \map \psi {\frac 1 3}$

$=$

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

$\blacksquare$