DLMF:25.5.E12 (Q7625): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

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

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2agdec

Revision as of 12:27, 2 January 2020

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DLMF:25.5.E12
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    Statements

    test
    ζ ⁡ ( s ) = 2 s - 1 s - 1 - 2 s ⁢ ∫ 0 ∞ sin ⁡ ( s ⁢ arctan ⁡ x ) ( 1 + x 2 ) s / 2 ⁢ ( e π ⁢ x + 1 ) ⁢ d x . Riemann-zeta 𝑠 superscript 2 𝑠 1 𝑠 1 superscript 2 𝑠 superscript subscript 0 𝑠 𝑥 superscript 1 superscript 𝑥 2 𝑠 2 superscript 𝑒 𝜋 𝑥 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{s-1}-2^{s}\int_% {0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^% {\pi x}+1)}\mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:25.5.E12
    0 references
    Symbols used
    Riemann zeta function
    test
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2akdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
     
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