Digamma Function of One Fourth/Proof 2

Theorem

$\map \psi {\dfrac 1 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2$

Proof

\(\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}\)

\(=\)

\(\ds -\paren {n - 1} \gamma - n \ln n\)

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 $2$

$\blacksquare$