Difference between revisions of "Value of polygamma at 1"
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| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Integral representation of polygamma for Re(z) greater than 0|next=Value of polygamma at positive integer}}: 6.4.2 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Integral representation of polygamma for Re(z) greater than 0|next=Value of polygamma at positive integer}}: $6.4.2$ |
Latest revision as of 22:45, 17 March 2017
Theorem
The following formula holds for $m=1,2,3,\ldots$: $$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ where $\psi^{(m)}$ denotes the polygamma, $m!$ denotes the factorial, and $\zeta$ denotes the Riemann zeta function.
Proof
Reference
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.2$
