Digamma Function of One Sixth
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Theorem
- $\map \psi {\dfrac 1 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2$
where:
- $\psi$ denotes the digamma function
- $\gamma$ denotes the Euler-Mascheroni constant.
Proof 1
| \(\ds \map \psi {\frac 1 6}\) | \(=\) | \(\ds -\gamma - \ln 12 - \frac \pi 2 \map \cot {\frac 1 6 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {6 / 2} - 1} \map \cos {\frac {2 \pi n} 6} \map \ln {\map \sin {\frac {\pi n} 6} }\) | Gauss's Digamma Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\gamma - 2 \ln 2 - \ln 3 - \frac \pi 2 \times \sqrt 3 + 2 \paren {\frac 1 2 \times \map \ln {\frac 1 2} + \paren {-\frac 1 2} \times \map \ln {\frac {\sqrt 3} 2} }\) | Cotangent of $30 \degrees$, Cosine of $60 \degrees$, Cosine of $120 \degrees$, Sine of $30 \degrees$, Sine of $60 \degrees$ and Sum of Logarithms | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\gamma - 2 \ln 2 - \ln 3 - \dfrac {\pi \sqrt 3} 2 + \paren {\map \ln {\frac 1 2} - \map \ln {\frac {\sqrt 3} 2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\gamma - 2 \ln 2 - \ln 3 - \dfrac {\pi \sqrt 3} 2 + \paren {\paren {\ln 1 - \ln 2} - \paren {\frac 1 2 \ln 3 - \ln 2} }\) | Difference of Logarithms and Logarithm of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2\) | Logarithm of 1 is 0 |
$\blacksquare$
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 | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^{6 - 1} \map \psi {\frac k 6}\) | \(=\) | \(\ds -\paren {6 - 1} \gamma - 6 \ln 6\) | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \map \psi {\frac 1 6} + \map \psi {\frac 2 6} + \map \psi {\frac 3 6} + \map \psi {\frac 4 6} + \map \psi {\frac 5 6}\) | \(=\) | \(\ds -5 \gamma - 6 \ln 6\) | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds \map \psi {\frac 1 6} - \map \psi {\frac 5 6}\) | \(=\) | \(\ds -\pi \map \cot {\frac \pi 6}\) | Digamma Reflection Formula | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 \map \psi {\frac 1 6} + \map \psi {\frac 2 6} + \map \psi {\frac 3 6} + \map \psi {\frac 4 6}\) | \(=\) | \(\ds -5 \gamma - 6 \ln 6 - \pi \map \cot {\frac \pi 6}\) | adding lines $1$ and $2$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 \map \psi {\frac 1 6}\) | \(=\) | \(\ds -5 \gamma - 6 \ln 6 - \pi \map \cot {\frac \pi 6} - \map \psi {\frac 1 3} - \map \psi {\frac 1 2} - \map \psi {\frac 2 3}\) | rearranging | ||||||||||
| \(\ds \) | \(=\) | \(\ds -5 \gamma - 6 \ln 2 - 6 \ln 3 - \pi \times \sqrt 3\) | Sum of Logarithms, Cotangent of $30 \degrees$, | |||||||||||
| \(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {-\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3} }\) | Digamma Function of One Third, | ||||||||||
| \(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {-\gamma - 2 \ln 2}\) | Digamma Function of One Half, | ||||||||||
| \(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {-\gamma - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3} }\) | and Digamma Function of Two Thirds | ||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \gamma - 4 \ln 2 - 3 \ln 3 - \pi \sqrt 3\) | simplifying | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \psi {\frac 1 6}\) | \(=\) | \(\ds -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2\) | dividing by $2$ |
$\blacksquare$