Difference between revisions of "Integral representation of polygamma for Re(z) greater than 0"
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| − | <strong>[[Integral representation of polygamma|Theorem]]:</strong> The following formula holds: | + | <strong>[[Integral representation of polygamma|Theorem]]:</strong> 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]]. | ||
Revision as of 19:23, 3 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.
Proof: █
