DLMF:25.11.E19 (Q7693)

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Language	Label	Description	Also known as
English	DLMF:25.11.E19	No description defined

Statements

Defining formula

{\displaystyle{\displaystyle\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(% \frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}% \int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a% \right)}{(x+a)^{s+2}}\mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{% \widetilde{B}_{2}\left(x\right)-B_{2}}{(x+a)^{s+2}}\mathrm{d}x,}}

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DLMF:25.11.E19

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{\displaystyle{\displaystyle\Re s>-1}}

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{\displaystyle{\displaystyle s\neq 1}}

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{\displaystyle{\displaystyle a>0}}

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Defining formula

{\displaystyle{\displaystyle B_{\NVar{n}}}}

C24.S2.SS1.m1abdec

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Hurwitz zeta function

Defining formula

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

C25.S11.E1.m2atdec

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Defining formula

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

C1.S4.SS4.m1acdec

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Defining formula

{\displaystyle{\displaystyle\int}}

C1.S4.SS4.m3acdec

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principal branch of logarithm function

Defining formula

{\displaystyle{\displaystyle\ln\NVar{z}}}

C4.S2.E2.m2aadec

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Defining formula

{\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}

C24.S2.SS3.m1abdec

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Defining formula

{\displaystyle{\displaystyle\Re}}

C1.S9.E2.m1agdec

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Sitelinks

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