Difference between revisions of "Integral representation of polygamma for Re(z) greater than 0"
From specialfunctionswiki
(→References) |
|||
| Line 7: | Line 7: | ||
==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Polygamma|next=Value of polygamma at 1}}: 6.4.1 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Polygamma|next=Value of polygamma at 1}}: $6.4.1$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 22:45, 17 March 2017
Theorem
The following formula holds for $\mathrm{Re}(z)>0$ and $m>0$: $$\psi^{(m)}(z)=(-1)^{m+1} \displaystyle\int_0^{\infty} \dfrac{t^m e^{-zt}}{1-e^{-t}} \mathrm{d}t,$$ where $\psi^{(m)}$ denotes the polygamma and $e^{-zt}$ denotes the exponential.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.1$
