Digamma Function of One Sixth/Proof 1
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Theorem
- $\map \psi {\dfrac 1 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2$
Proof
| \(\ds \map \psi {\frac 1 6}\) | \(=\) | \(\ds -\gamma - \ln 12 - \frac \pi 2 \map \cot {\frac 1 6 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {6 / 2} - 1} \map \cos {\frac {2 \pi n} 6} \map \ln {\map \sin {\frac {\pi n} 6} }\) | 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 |
$\blacksquare$