Digamma Function of Three Fourths

Theorem

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

where:

$\psi$ denotes the digamma function

$\gamma$ denotes the Euler-Mascheroni constant.

Proof

\(\ds \map \psi {\frac 1 4} - \map \psi {\frac 3 4}\)

\(=\)

\(\ds -\pi \map \cot {\frac \pi 4}\)

Digamma Reflection Formula

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 3 4}\)

\(=\)

\(\ds \pi \map \cot {\frac \pi 4} + \map \psi {\frac 1 4}\)

rearranging

\(\ds \)

\(=\)

\(\ds \pi \times 1 + \paren {-\gamma - 3 \ln 2 - \dfrac \pi 2}\)

Cotangent of $45 \degrees$ and Digamma Function of One Fourth

\(\ds \)

\(=\)

\(\ds -\gamma - 3 \ln 2 + \dfrac \pi 2\)

rearranging

$\blacksquare$