Difference between revisions of "Value of polygamma at 1"

From specialfunctionswiki
Jump to: navigation, search
Line 7: Line 7:
  
 
==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

Revision as of 19:34, 11 June 2016

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