Difference between revisions of "Integral representation of polygamma for Re(z) greater than 0"
From specialfunctionswiki
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| − | + | ==Theorem== | |
| − | + | The following formula holds for $\mathrm{Re}(z)>0$ and $m>0$: | |
$$\psi^{(m)}(z)=(-1)^{m+1} \displaystyle\int_0^{\infty} \dfrac{t^m e^{-zt}}{1-e^{-t}} \mathrm{d}t,$$ | $$\psi^{(m)}(z)=(-1)^{m+1} \displaystyle\int_0^{\infty} \dfrac{t^m e^{-zt}}{1-e^{-t}} \mathrm{d}t,$$ | ||
where $\psi^{(m)}$ denotes the [[polygamma]] and $e^{-zt}$ denotes the [[exponential]]. | where $\psi^{(m)}$ denotes the [[polygamma]] and $e^{-zt}$ denotes the [[exponential]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Revision as of 06:31, 11 June 2016
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.
