Formula:DLMF:25.11:E35

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Failed to parse (unknown function "\hiderel"): {\displaystyle {\displaystyle \sum_{n \hiderel{=} 0}^\infty \frac{\opminus^n}{(n+a)^s} \hiderel{=} \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1+\expe^{-x}} \diff{x} \hiderel{=} 2^{-s} \left( \HurwitzZeta@{s}{\tfrac{1}{2}a} - \HurwitzZeta@{s}{\tfrac{1}{2} (1+a)} \right) }}

Constraint(s)

Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{a} > 0}} , Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{s} > 0}} &
or Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{a} = 0}} , Failed to parse (unknown function "\imagpart"): {\displaystyle {\displaystyle \imagpart{a} \neq 0}} , Failed to parse (unknown function "\realpart"): {\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.

Use

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 "\HurwitzZeta"): {\displaystyle {\displaystyle \HurwitzZeta@{s}{\tfrac{1}{2} a} = \HurwitzZeta@{s}{\tfrac{1}{2} a + \tfrac{1}{2}} + 2^s \sum_{n=0}^\infty \frac{\opminus^n}{(n+a)^s} }} .


Symbols List

& : logical and
 : sum : http://drmf.wmflabs.org/wiki/Definition:sum
 : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
 : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
 : 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
 : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
 : real part : http://dlmf.nist.gov/1.9#E2
 : imaginary part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (35), Section 25.11 of DLMF.

URL links

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