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: | ||
| − | $$\psi^{ | + | $$\psi^{(1)} \left( n + \dfrac{1}{2} \right)=\dfrac{\pi^2}{2} - 4 \displaystyle\sum_{k=1}^n \dfrac{1}{(2k-1)^2},$$ |
| − | where $\psi^{( | + | where $\psi^{(1)}$ denotes the [[trigamma]] and $\pi$ denotes [[pi]]. |
==Proof== | ==Proof== | ||
==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Value of polygamma at 1/2|next=Polygamma recurrence relation}}: 6.4.5 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Value of polygamma at 1/2|next=Polygamma recurrence relation}}: $6.4.5$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 22:45, 17 March 2017
Theorem
The following formula holds: $$\psi^{(1)} \left( n + \dfrac{1}{2} \right)=\dfrac{\pi^2}{2} - 4 \displaystyle\sum_{k=1}^n \dfrac{1}{(2k-1)^2},$$ where $\psi^{(1)}$ denotes the trigamma and $\pi$ denotes pi.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.5$
