Value of derivative of trigamma at positive integer plus 1/2
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Revision as of 19:48, 11 June 2016 by Tom (talk | contribs) (Tom moved page Value of derivative of polygamma at positive integer plus 1/2 to Value of derivative of digamma at positive integer plus 1/2)
Theorem
The following formula holds: $$\psi^{\prime} \left( n + \dfrac{1}{2} \right)=\dfrac{\pi^2}{2} - 4 \displaystyle\sum_{k=1}^n \dfrac{1}{(2k-1)^2},$$ where $\psi^{(m)}$ denotes the polygamma and $\pi$ denotes pi.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous): 6.4.5
