Digamma Function of Two Thirds
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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 |
$\blacksquare$