Digamma Function of One Fourth
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Theorem
- $\map \psi {\dfrac 1 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2$
where:
- $\psi$ denotes the digamma function
- $\gamma$ denotes the Euler-Mascheroni constant.
Proof 1
| \(\ds \map \psi {\frac 1 4}\) | \(=\) | \(\ds -\gamma - \ln 8 - \frac \pi 2 \map \cot {\frac 1 4 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {4 / 2} - 1} \map \cos {\frac {2 \pi n} 4} \map \ln {\map \sin {\frac {\pi n} 4} }\) | Gauss's Digamma Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\gamma - 3 \ln 2 - \frac \pi 2 \times 1 + 2 \times 0 \times \map \ln {\frac {\sqrt 2} 2}\) | Cotangent of $45 \degrees$, Cosine of $90 \degrees$, Sine of $45 \degrees$ and Sum of Logarithms | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\gamma - 3 \ln 2 - \dfrac \pi 2\) |
$\blacksquare$
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 | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^{4 - 1} \map \psi {\frac k 4}\) | \(=\) | \(\ds -\paren {4 - 1} \gamma - 4 \ln 4\) | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \map \psi {\frac 1 4} + \map \psi {\frac 2 4} + \map \psi {\frac 3 4}\) | \(=\) | \(\ds -3 \gamma - 4 \ln 4\) | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds \map \psi {\frac 1 4} - \map \psi {\frac 3 4}\) | \(=\) | \(\ds -\pi \map \cot {\frac \pi 4}\) | Digamma Reflection Formula | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 \map \psi {\frac 1 4} + \map \psi {\frac 1 2}\) | \(=\) | \(\ds -3 \gamma - 4 \ln 4 - \pi \map \cot {\frac \pi 4}\) | adding lines $1$ and $2$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 \map \psi {\frac 1 4}\) | \(=\) | \(\ds -3 \gamma - 4 \ln 4 - \pi \map \cot {\frac \pi 4} - \map \psi {\frac 1 2}\) | subtracting $\map \psi {\frac 1 2}$ from both sides | ||||||||||
| \(\ds \) | \(=\) | \(\ds -3 \gamma - 8 \ln 2 - \pi \paren 1 - \paren {-\gamma - 2 \ln 2}\) | Digamma Function of One Half, Logarithm of Power and Cotangent of $45 \degrees$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \gamma - 6 \ln 2 -\pi\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \psi {\frac 1 4}\) | \(=\) | \(\ds -\gamma - 3 \ln 2 - \dfrac \pi 2\) | dividing by $2$ |
$\blacksquare$