DLMF:10.22.E36 (Q3410)

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DLMF:10.22.E36
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    Statements

    Defining formula
    1 Γ ⁡ ( α ) ⁢ ∫ 0 x ( x - t ) α - 1 ⁢ J ν ⁡ ( t ) ⁢ d t = 2 α ⁢ ∑ k = 0 ∞ ( α ) k k ! ⁢ J ν + α + 2 ⁢ k ⁡ ( x ) , 1 Euler-Gamma 𝛼 superscript subscript 0 𝑥 superscript 𝑥 𝑡 𝛼 1 Bessel-J 𝜈 𝑡 𝑡 superscript 2 𝛼 superscript subscript 𝑘 0 subscript 𝛼 𝑘 𝑘 Bessel-J 𝜈 𝛼 2 𝑘 𝑥 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(\alpha\right)}\int_{0}^{x}(x-% t)^{\alpha-1}J_{\nu}\left(t\right)\mathrm{d}t=2^{\alpha}\sum_{k=0}^{\infty}% \frac{(\alpha)_{k}}{k!}J_{\nu+\alpha+2k}\left(x\right),}}
    0 references
    DLMF id
    DLMF:10.22.E36
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    constraint
    ℜ ⁡ α > 0 , ℜ ⁡ ν ≥ 0 formulae-sequence 𝛼 0 𝜈 0 {\displaystyle{\displaystyle\Re\alpha>0,\Re\nu\geq 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.m2aabdec
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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.m2ahdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaidec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaidec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1audec
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    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1hdec
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    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1ndec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1xdec
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    Q11469
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C10.S22.XMD1.m1adec
    0 references
     
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