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