DLMF:10.2.E2 (Q3006)

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DLMF:10.2.E2
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    Statements

    Defining formula
    J ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ ( 1 4 ⁢ z 2 ) k k ! ⁢ Γ ⁡ ( ν + k + 1 ) . Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript subscript 𝑘 0 superscript 1 𝑘 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 Euler-Gamma 𝜈 𝑘 1 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{% k=0}^{\infty}(-1)^{k}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1% \right)}.}}
    0 references
    DLMF id
    DLMF:10.2.E2
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1dec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1adec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1adec
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