Digamma Function of Five Sixths
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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 |
$\blacksquare$