Difference between revisions of "Integral representation of polygamma 2"

From specialfunctionswiki
Jump to: navigation, search
 
Line 1: Line 1:
<div class="toccolours mw-collapsible mw-collapsed">
+
==Theorem==
<strong>[[Integral representation of polygamma 2|Theorem]]:</strong> The following formula holds for $\mathrm{Re}(z)>0$ and $m>0$:
+
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]].
<div class="mw-collapsible-content">
+
 
<strong>Proof:</strong> █
+
==Proof==
</div>
+
 
</div>
+
==References==
 +
 
 +
[[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.

Proof

References