DLMF:10.9.E22 (Q3092)

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DLMF:10.9.E22
No description defined

    Statements

    Defining formula
    J ν ⁡ ( x ) = 1 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( - t ) ⁢ ( 1 2 ⁢ x ) ν + 2 ⁢ t Γ ⁡ ( ν + t + 1 ) ⁢ d t , Bessel-J 𝜈 𝑥 1 2 𝜋 𝑖 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 superscript 1 2 𝑥 𝜈 2 𝑡 Euler-Gamma 𝜈 𝑡 1 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{1}{2\pi i}\int_{-i% \infty}^{i\infty}\frac{\Gamma\left(-t\right)(\tfrac{1}{2}x)^{\nu+2t}}{\Gamma% \left(\nu+t+1\right)}\mathrm{d}t,}}
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    DLMF id
    DLMF:10.9.E22
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    constraint
    ℜ ⁡ ν > 0 𝜈 0 {\displaystyle{\displaystyle\Re\nu>0}}
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    ℜ ⁡ ν > 0 𝜈 0 {\displaystyle{\displaystyle\Re\nu>0}}
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    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2akdec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2audec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1audec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ajdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3audec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1cdec
    0 references
     
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