Digamma Function/Examples/Digamma Function of Four Thirds

Example of Use of Recurrence Relation for Digamma Function

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

Proof

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

\(=\)

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

Recurrence Relation for Digamma Function

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 1 3 + 1}\)

\(=\)

\(\ds \map \psi {\frac 1 3} + 3\)

$z := \dfrac 1 3$

\(\ds \leadsto \ \ \)

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

\(=\)

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

Digamma Function of One Third

$\blacksquare$