Formula:DLMF:25.11:E35

<< Formula:DLMF:25.11:E34

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E36 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}{\mathrm{e}^{-ax}}}{1+{\mathrm{e}^{-x}}}\mathrm{d}x=2^{-s}\left(\HurwitzZeta@{s}{\tfrac{1}{2}a}-\HurwitzZeta@{s}{\tfrac{1}{2}(1+a)}\right)}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}$, ${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ &
or ${\displaystyle{\displaystyle{\displaystyle\Re{a}=0}}}$, ${\displaystyle{\displaystyle{\displaystyle\Im{a}\neq 0}}}$, ${\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<1}}}$

Proof

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

Use

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}{\mathrm{e}^{-ax}}}{1-{\mathrm{e}^{-x}}}\mathrm{d}x}}}$
and
${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{\tfrac{1}{2}a}=\HurwitzZeta@{s}{\tfrac{1}{2}a+\tfrac{1}{2}}+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}}}}$.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2
${\displaystyle{\displaystyle{\displaystyle\Im{z}}}}$ : imaginary part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (35), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E34

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E36 >>