Digamma Function in terms of General Harmonic Number
(Redirected from Reciprocal times Derivative of Gamma Function/Corollary 3)
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Corollary to Reciprocal times Derivative of Gamma Function
Let $z \in \C \setminus \Z_{\le 0}$.
Then:
- $\ds \map \psi {z + 1} = -\gamma + \harm 1 z$
where:
- $\psi$ is the digamma function
- $\gamma$ denotes the Euler-Mascheroni constant
- $\harm 1 z$ denotes the general harmonic number of order $1$ evaluated at $z$.
Proof
\(\ds \dfrac {\map {\Gamma'} {z + 1} } {\map \Gamma {z + 1} }\) | \(=\) | \(\ds -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {\paren {z + 1} + n - 1} }\) | Reciprocal times Derivative of Gamma Function | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \psi {z + 1}\) | \(=\) | \(\ds -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n} }\) | Definition of Digamma Function | ||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma + \harm 1 z\) | Definition of General Harmonic Numbers |
$\blacksquare$