Digamma Function/Examples/Digamma Function of Three Halves
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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 | ||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - 2 \ln 2 + 2\) |
$\blacksquare$