Digamma Function/Examples/Digamma Function of Five Fourths

Example of Use of Recurrence Relation for Digamma Function

$\map \psi {\dfrac 5 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2 + 4$

Proof

\(\ds \map \psi {z + 1}\)

\(=\)

\(\ds \map \psi z + \frac 1 z\)

Recurrence Relation for Digamma Function

\(\ds \leadsto \ \ \)

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

\(=\)

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

$z := \dfrac 1 4$

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 5 4}\)

\(=\)

\(\ds \paren {-\gamma - 3 \ln 2 - \dfrac \pi 2} + 4\)

Digamma Function of One Fourth

\(\ds \)

\(=\)

\(\ds -\gamma - 3 \ln 2 - \dfrac \pi 2 + 4\)

$\blacksquare$