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The polygamma function may be represented as $$\begin{align} \psi^{(m)}(z)&= (-1)^{m+1}\int_0^\infty\frac{t^m e^{-zt}} {1-e^{-t}}\ dt\\ &=-\int_0^1\frac{t^{z-1}}{1-t}\ln^mt\ dt \end{align}$$ which holds for $Re z >0$ and $m > 0$. For $m = 0$ see the digamma function definition.

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