DLMF:10.22.E43 (Q3417)

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DLMF:10.22.E43
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    Statements

    Defining formula
    ∫ 0 ∞ t μ ⁢ J ν ⁡ ( t ) ⁢ d t = 2 μ ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ) Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) , superscript subscript 0 superscript 𝑡 𝜇 Bessel-J 𝜈 𝑡 𝑡 superscript 2 𝜇 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu}J_{\nu}\left(t\right)% \mathrm{d}t=2^{\mu}\frac{\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}% {2}\right)}{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)},}}
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    DLMF id
    DLMF:10.22.E43
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    constraint
    ℜ ⁡ ( μ + ν ) > - 1 𝜇 𝜈 1 {\displaystyle{\displaystyle\Re(\mu+\nu)>-1}}
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    ℜ ⁡ μ < 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu<\tfrac{1}{2}}}
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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.m2aagdec
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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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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aapdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aapdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1axdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1acdec
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