Digamma Function/Examples/Digamma Function of Three Halves

Example of Use of Recurrence Relation for Digamma Function

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

Proof

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

\(=\)

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

Recurrence Relation for Digamma Function

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 1 2 + 1}\)

\(=\)

\(\ds \map \psi {\frac 1 2} + 2\)

$z := \dfrac 1 2$

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 3 2}\)

\(=\)

\(\ds \paren {-\gamma - 2 \ln 2} + 2\)

Digamma Function of One Half

\(\ds \)

\(=\)

\(\ds -\gamma - 2 \ln 2 + 2\)

$\blacksquare$