General Harmonic Numbers/Examples/Order 1/Minus One Half
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Example of General Harmonic Number
- $\harm 1 {-\dfrac 1 2} = -2 \ln 2$
where $\harm 1 {-\dfrac 1 2}$ denotes the general harmonic number of order $1$ evaluated at $-\dfrac 1 2$.
Proof
| \(\ds \harm 1 z\) | \(=\) | \(\ds \map \psi {z + 1} + \gamma\) | Reciprocal times Derivative of Gamma Function: Corollary $3$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \harm 1 {-\dfrac 1 2}\) | \(=\) | \(\ds \map \psi {-\dfrac 1 2 + 1} + \gamma\) | setting $z := -\dfrac 1 2$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds \map \psi {\dfrac 1 2} + \gamma\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-\gamma - 2 \ln 2} + \gamma\) | Digamma Function of One Half | |||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \ln 2\) |
$\blacksquare$