Editing Polygamma function (section)

==Integral representation==
{{see also|Digamma function#Integral representations}}
When {{math|''m'' > 0}} and {{math|Re ''z'' > 0}}, the polygamma function equals

:<math>\begin{align}
\psi^{(m)}(z)
&= (-1)^{m+1}\int_0^\infty \frac{t^m e^{-zt}}{1-e^{-t}}\,\mathrm{d}t \\
&= -\int_0^1 \frac{t^{z-1}}{1-t}(\ln t)^m\,\mathrm{d}t\\
&= (-1)^{m+1}m!\zeta(m+1,z)
\end{align}</math>

where <math>\zeta(s,q)</math> is the [[Hurwitz zeta function]].

This expresses the polygamma function as the [[Laplace transform]] of {{math|{{sfrac|(−1)<sup>''m''+1</sup> ''t<sup>m</sup>''|1 − ''e''<sup>−''t''</sup>}}}}.  It follows from [[Bernstein's theorem on monotone functions]] that, for {{math|''m'' > 0}} and {{math|''x''}} real and non-negative, {{math|(−1)<sup>''m''+1</sup> ''ψ''<sup>(''m'')</sup>(''x'')}} is a completely monotone function.

Setting {{math|''m'' {{=}} 0}} in the above formula does not give an integral representation of the digamma function.  The digamma function has an integral representation, due to Gauss, which is similar to the {{math|''m'' {{=}} 0}} case above but which has an extra term {{math|{{sfrac|''e''<sup>−''t''</sup>|''t''}}}}.