DLMF:25.11.E19 (Q7693): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1agdec

Revision as of 13:39, 2 January 2020

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DLMF:25.11.E19
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    Statements

    Defining formula
    ζ ′ ⁡ ( s , a ) = - ln ⁡ a a s ⁢ ( 1 2 + a s - 1 ) - a 1 - s ( s - 1 ) 2 + s ⁢ ( s + 1 ) 2 ⁢ ∫ 0 ∞ ( B ~ 2 ⁡ ( x ) - B 2 ) ⁢ ln ⁡ ( x + a ) ( x + a ) s + 2 ⁢ d x - ( 2 ⁢ s + 1 ) 2 ⁢ ∫ 0 ∞ B ~ 2 ⁡ ( x ) - B 2 ( x + a ) s + 2 ⁢ d x , diffop Hurwitz-zeta 1 𝑠 𝑎 𝑎 superscript 𝑎 𝑠 1 2 𝑎 𝑠 1 superscript 𝑎 1 𝑠 superscript 𝑠 1 2 𝑠 𝑠 1 2 superscript subscript 0 periodic-Bernoulli-polynomial-B 2 𝑥 Bernoulli-number-B 2 𝑥 𝑎 superscript 𝑥 𝑎 𝑠 2 𝑥 2 𝑠 1 2 superscript subscript 0 periodic-Bernoulli-polynomial-B 2 𝑥 Bernoulli-number-B 2 superscript 𝑥 𝑎 𝑠 2 𝑥 {\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,}}
    0 references
    DLMF id
    DLMF:25.11.E19
    0 references
    constraint
    ℜ ⁡ s > - 1 𝑠 1 {\displaystyle{\displaystyle\Re s>-1}}
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    s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}}
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    a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}}
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    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1abdec
    0 references
    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2atdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q11110
    Defining formula
    B ~ n ⁡ ( x ) periodic-Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS3.m1abdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1agdec
    0 references
     
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