Difference between revisions of "Value of derivative of trigamma at positive integer plus 1/2"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\psi^{(m)} \left( n + \dfrac{1}{2} \right)=\dfrac{\pi^2}{2} - 4 \displaystyle\sum_{k=1}^n \dfrac{1}{(2k-1)^2},$$ | + | $$\psi^{(m)}' \left( n + \dfrac{1}{2} \right)=\dfrac{\pi^2}{2} - 4 \displaystyle\sum_{k=1}^n \dfrac{1}{(2k-1)^2},$$ |
where $\psi^{(m)}$ denotes the [[polygamma]] and $\pi$ denotes [[pi]]. | where $\psi^{(m)}$ denotes the [[polygamma]] and $\pi$ denotes [[pi]]. | ||
Revision as of 19:46, 11 June 2016
Theorem
The following formula holds: $$\psi^{(m)}' \left( n + \dfrac{1}{2} \right)=\dfrac{\pi^2}{2} - 4 \displaystyle\sum_{k=1}^n \dfrac{1}{(2k-1)^2},$$ 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
