# Formula:DLMF:25.5:E19

$\displaystyle {\displaystyle \RiemannZeta@{m+s} = \opminus^{m-1} \frac{\EulerGamma@{s} \sin@{\cpi s}}{\cpi \EulerGamma@{m+s}} \* \int_0^\infty \digamma^{(m)}@{1+x} x^{-s} \diff{x} }$

## Constraint(s)

$\displaystyle {\displaystyle m = 1,2,3,\dots}$ &
$\displaystyle {\displaystyle 0 < \realpart{s} < 1}$

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

## Symbols List

& : logical and
$\displaystyle {\displaystyle \zeta}$  : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
$\displaystyle {\displaystyle (-1)}$  : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
$\displaystyle {\displaystyle \Gamma}$  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
$\displaystyle {\displaystyle \mathrm{sin}}$  : sine function : http://dlmf.nist.gov/4.14#E1
$\displaystyle {\displaystyle \pi}$  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
$\displaystyle {\displaystyle \int}$  : integral : http://dlmf.nist.gov/1.4#iv
$\displaystyle {\displaystyle \psi}$  : psi (or digamma) function : http://dlmf.nist.gov/5.2#E2
$\displaystyle {\displaystyle \mathrm{d}^nx}$  : differential : http://dlmf.nist.gov/1.4#iv
$\displaystyle {\displaystyle \Re {z}}$  : real part : http://dlmf.nist.gov/1.9#E2