Digamma Function/Examples/Digamma Function of Seven Sixths
Jump to navigation
Jump to search
Example of Use of Recurrence Relation for Digamma Function
- $\map \psi {\dfrac 7 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2 + 6$
where $\psi$ denotes the digamma function.
Proof
| \(\ds \map \psi {z + 1}\) | \(=\) | \(\ds \map \psi z + \frac 1 z\) | Recurrence Relation for Digamma Function | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \psi {\frac 1 6 + 1}\) | \(=\) | \(\ds \map \psi {\frac 1 6} + 6\) | $z := \dfrac 1 6$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \psi {\frac 7 6}\) | \(=\) | \(\ds \paren {-\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2} + 6\) | Digamma Function of One Sixth | ||||||||||
| \(\ds \) | \(=\) | \(\ds -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2 + 6\) |
$\blacksquare$