Formula:DLMF:25.11:E30

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Failed to parse (unknown function "\HurwitzZeta"): {\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{\EulerGamma@{1-s}}{2 \cpi \iunit} \int_{-\infty}^{(0+)} \frac{\expe^{az} z^{s-1}}{1 - \expe^z} \diff{z} }}

Constraint(s)

&
Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{a} > 0}} &

the integration contour is a loop around the negative real axis; it starts at , encircles the origin once in the positive direction without enclosing any of the points

Failed to parse (unknown function "\cpi"): {\displaystyle {\displaystyle z=\pm2\cpi\iunit}} , Failed to parse (unknown function "\cpi"): {\displaystyle {\displaystyle \pm4\cpi\iunit, \ldots,}} and returns to


Proof

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Assume Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{s} > 1}} , collapse the integration path onto the

real axis, apply
Failed to parse (unknown function "\HurwitzZeta"): {\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1-\expe^{-x}} \diff{x} }}
and
Failed to parse (unknown function "\EulerGamma"): {\displaystyle {\displaystyle \EulerGamma@{z} \EulerGamma@{1-z} = \cpi / \sin@{\cpi z} }}

followed by analytic continuation.


Symbols List

& : logical and
 : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
 : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
 : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
 : integral : http://dlmf.nist.gov/1.4#iv
 : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
 : differential : http://dlmf.nist.gov/1.4#iv
 : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (30), Section 25.11 of DLMF.

URL links

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