DLMF:10.22.E45 (Q3419): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2ardec

Revision as of 12:21, 2 January 2020

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DLMF:10.22.E45
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    Statements

    test
    ∫ 0 ∞ 1 - J 0 ⁡ ( t ) t μ ⁢ d t = - π ⁢ sec ⁡ ( 1 2 ⁢ μ ⁢ π ) 2 μ ⁢ Γ 2 ⁡ ( 1 2 ⁢ μ + 1 2 ) , superscript subscript 0 1 Bessel-J 0 𝑡 superscript 𝑡 𝜇 𝑡 𝜋 1 2 𝜇 𝜋 superscript 2 𝜇 Euler-Gamma 2 1 2 𝜇 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{1-J_{0}\left(t\right)}{t^{% \mu}}\mathrm{d}t=-\frac{\pi\sec\left(\frac{1}{2}\mu\pi\right)}{2^{\mu}{\Gamma^% {2}}\left(\frac{1}{2}\mu+\frac{1}{2}\right)},}}
    0 references
    DLMF id
    DLMF:10.22.E45
    0 references
    constraint
    1 < ℜ ⁡ μ < 3 1 𝜇 3 {\displaystyle{\displaystyle 1<\Re\mu<3}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aahdec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.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.m2ardec
    0 references
     
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