Digamma Function/Examples/Digamma Function of Five Fourths
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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 | ||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - 3 \ln 2 - \dfrac \pi 2 + 4\) |
$\blacksquare$