Digamma Function of One Half
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Theorem
- $\map \psi {\dfrac 1 2} = -\gamma - 2 \ln 2$
where:
- $\psi$ denotes the digamma function
- $\gamma$ denotes the Euler-Mascheroni constant.
Proof 1
\(\ds \map \psi {\frac 1 2}\) | \(=\) | \(\ds -\gamma - \ln 4 - \frac \pi 2 \map \cot {\frac 1 2 \pi} + 2 \sum_{n \mathop = 1}^0 \map \cos {\frac {2 \pi n} 2} \map \ln {\map \sin {\frac {\pi n} 2} }\) | Gauss's Digamma Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - \ln 4\) | Cotangent of Right Angle, noting also that the summation is an empty sum | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - 2 \ln 2\) | Logarithm of Power |
$\blacksquare$
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 | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^{2 - 1} \map \psi {\frac k 2}\) | \(=\) | \(\ds -\paren {2 - 1} \gamma - 2 \ln 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \psi {\frac 1 2}\) | \(=\) | \(\ds -\gamma - 2 \ln 2\) |
$\blacksquare$