DLMF:25.5.E11 (Q7624): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3ajdec

Revision as of 13:27, 2 January 2020

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DLMF:25.5.E11
No description defined

    Statements

    Defining formula
    ζ ⁡ ( s ) = 1 2 + 1 s - 1 + 2 ⁢ ∫ 0 ∞ sin ⁡ ( s ⁢ arctan ⁡ x ) ( 1 + x 2 ) s / 2 ⁢ ( e 2 ⁢ π ⁢ x - 1 ) ⁢ d x . Riemann-zeta 𝑠 1 2 1 𝑠 1 2 superscript subscript 0 𝑠 𝑥 superscript 1 superscript 𝑥 2 𝑠 2 superscript 𝑒 2 𝜋 𝑥 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2% \int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/% 2}(e^{2\pi x}-1)}\mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:25.5.E11
    0 references
    Symbols used
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2ajdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
    0 references
     
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