Difference between revisions of "Value of polygamma at 1"
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==Theorem== | ==Theorem== | ||
| − | The following formula holds: | + | The following formula holds for $m=1,2,3,\ldots$: |
$$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ | $$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ | ||
| − | where $\psi^{(m)}$ denotes the [[polygamma]], $m!$ denotes the [[factorial]] | + | where $\psi^{(m)}$ denotes the [[polygamma]], $m!$ denotes the [[factorial]], and $\zeta$ denotes the [[Riemann zeta]] function. |
==Proof== | ==Proof== | ||
==Reference== | ==Reference== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=}}: 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): $6.4.2$
