DLMF:10.9.E29 (Q3099)

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DLMF:10.9.E29
No description defined

    Statements

    Defining formula
    J μ ⁡ ( x ) ⁢ J ν ⁡ ( x ) = 1 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( - t ) ⁢ Γ ⁡ ( 2 ⁢ t + μ + ν + 1 ) ⁢ ( 1 2 ⁢ x ) μ + ν + 2 ⁢ t Γ ⁡ ( t + μ + 1 ) ⁢ Γ ⁡ ( t + ν + 1 ) ⁢ Γ ⁡ ( t + μ + ν + 1 ) ⁢ d t , Bessel-J 𝜇 𝑥 Bessel-J 𝜈 𝑥 1 2 𝜋 𝑖 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 2 𝑡 𝜇 𝜈 1 superscript 1 2 𝑥 𝜇 𝜈 2 𝑡 Euler-Gamma 𝑡 𝜇 1 Euler-Gamma 𝑡 𝜈 1 Euler-Gamma 𝑡 𝜇 𝜈 1 𝑡 {\displaystyle{\displaystyle J_{\mu}\left(x\right)J_{\nu}\left(x\right)=\frac{% 1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(-t\right)\Gamma\left(2t+% \mu+\nu+1\right)(\tfrac{1}{2}x)^{\mu+\nu+2t}}{\Gamma\left(t+\mu+1\right)\Gamma% \left(t+\nu+1\right)\Gamma\left(t+\mu+\nu+1\right)}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.9.E29
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    constraint
    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.m2apdec
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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.m2aidec
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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.m2aabdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aabdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aodec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aabdec
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    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1ddec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1xdec
    0 references
     
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