Integral representation of polygamma for Re(z) greater than 0

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.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.1$