Digamma Function of One Third
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Theorem
- $\map \psi {\dfrac 1 3} = -\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$
where:
- $\psi$ denotes the digamma function
- $\gamma$ denotes the Euler-Mascheroni constant.
Proof 1
\(\ds \map \psi {\frac 1 3}\) | \(=\) | \(\ds -\gamma - \ln 6 - \frac \pi 2 \map \cot {\frac 1 3 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {3 / 2} - 1} \map \cos {\frac {2 \pi n} 3} \map \ln {\map \sin {\frac {\pi n} 3} }\) | Gauss's Digamma Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - \ln 2 - \ln 3 - \frac \pi 2 \paren {\frac 1 {\sqrt 3} } + 2 \paren {-\frac 1 2} \map \ln {\frac {\sqrt 3} 2 }\) | Cotangent of $60 \degrees$, Cosine of $120 \degrees$, Sine of $60 \degrees$ and Sum of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - \ln 2 - \ln 3 - \frac \pi 2 \paren {\frac 1 {\sqrt 3} } - \paren {\frac 1 2 \ln 3 - \ln 2}\) | Difference of Logarithms and Logarithm of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - \dfrac 3 2 \ln 3 - \dfrac {\pi} {2 \sqrt 3}\) |
$\blacksquare$
Proof 2
\(\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}\) | \(=\) | \(\ds -\paren {n - 1} \gamma - n \ln n\) | Digamma Additive Formula: Corollary | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^{3 - 1} \map \psi {\frac k 3}\) | \(=\) | \(\ds -\paren {3 - 1} \gamma - 3 \ln 3\) | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \map \psi {\frac 1 3} + \map \psi {\frac 2 3}\) | \(=\) | \(\ds -2 \gamma - 3 \ln 3\) | ||||||||||
\(\text {(2)}: \quad\) | \(\ds \map \psi {\frac 1 3} - \map \psi {\frac 2 3}\) | \(=\) | \(\ds -\pi \map \cot {\frac \pi 3}\) | Digamma Reflection Formula | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \map \psi {\frac 1 3}\) | \(=\) | \(\ds -2 \gamma - 3 \ln 3 - \pi \map \cot {\frac \pi 3}\) | adding lines $1$ and $2$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \psi {\frac 1 3}\) | \(=\) | \(\ds -\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}\) | dividing by $2$ and Cotangent of $60 \degrees$ |
$\blacksquare$