Difference between revisions of "Integral representation of polygamma 2"
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| − | + | ==Theorem== | |
| − | + | The following formula holds for $\mathrm{Re}(z)>0$ and $m>0$: | |
$$\psi^{(m)}(z)=-\displaystyle\int_0^1 \dfrac{t^{z-1}}{1-t} \log^m(t) \mathrm{d}t,$$ | $$\psi^{(m)}(z)=-\displaystyle\int_0^1 \dfrac{t^{z-1}}{1-t} \log^m(t) \mathrm{d}t,$$ | ||
where $\psi^{(m)}$ denotes the [[polygamma]] and $\log$ denotes the [[logarithm]]. | where $\psi^{(m)}$ denotes the [[polygamma]] and $\log$ denotes the [[logarithm]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 06:32, 11 June 2016
Theorem
The following formula holds for $\mathrm{Re}(z)>0$ and $m>0$: $$\psi^{(m)}(z)=-\displaystyle\int_0^1 \dfrac{t^{z-1}}{1-t} \log^m(t) \mathrm{d}t,$$ where $\psi^{(m)}$ denotes the polygamma and $\log$ denotes the logarithm.
