DLMF:25.11.E35 (Q7709)

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Language	Label	Description	Also known as
English	DLMF:25.11.E35	No description defined

Statements

Defining formula

{\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}e^{-ax}}{1+e^{-x}}% \mathrm{d}x=2^{-s}\left(\zeta\left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{% 1}{2}(1+a)\right)\right),}}

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DLMF:25.11.E35

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{\displaystyle{\displaystyle\Re a>0}}

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{\displaystyle{\displaystyle\Re s>0}}

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{\displaystyle{\displaystyle\Re a=0}}

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{\displaystyle{\displaystyle\Im a\neq 0}}

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{\displaystyle{\displaystyle 0<\Re s<1}}

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Defining formula

{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

C5.S2.E1.m2aidec

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Hurwitz zeta function

Defining formula

{\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}

C25.S11.E1.m2aaidec

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Defining formula

{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

C1.S4.SS4.m1andec

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base of natural logarithm

Defining formula

{\displaystyle{\displaystyle\mathrm{e}}}

C4.S2.E11.m2afdec

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Defining formula

{\displaystyle{\displaystyle\Im}}

C1.S9.E2.m2adec

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Defining formula

{\displaystyle{\displaystyle\int}}

C1.S4.SS4.m3andec

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Defining formula

{\displaystyle{\displaystyle\Re}}

C1.S9.E2.m1ardec

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Defining formula

{\displaystyle{\displaystyle n}}

C25.S1.XMD3.m1rdec

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Defining formula

{\displaystyle{\displaystyle x}}

C25.S1.XMD5.m1odec

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Defining formula

{\displaystyle{\displaystyle a}}

C25.S1.XMD6.m1acdec

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Defining formula

{\displaystyle{\displaystyle s}}

C25.S1.XMD7.m1xdec

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Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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