DLMF:10.16.E5 (Q3164)

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DLMF:10.16.E5
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    Statements

    test
    J ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν ⁢ e ∓ i ⁢ z Γ ⁡ ( ν + 1 ) ⁢ M ⁡ ( ν + 1 2 , 2 ⁢ ν + 1 , ± 2 ⁢ i ⁢ z ) , Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝑒 minus-or-plus 𝑖 𝑧 Euler-Gamma 𝜈 1 Kummer-confluent-hypergeometric-M 𝜈 1 2 2 𝜈 1 plus-or-minus 2 𝑖 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% e^{\mp iz}}{\Gamma\left(\nu+1\right)}M\left(\nu+\tfrac{1}{2},2\nu+1,\pm 2iz% \right),}}
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    DLMF id
    DLMF:10.16.E5
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    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2acdec
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    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    test
    M ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E2.m2adec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1ddec
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