DLMF:10.16.E7 (Q3166)

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DLMF:10.16.E7
No description defined

    Statements

    Defining formula
    J ν ⁡ ( z ) = e ∓ ( 2 ⁢ ν + 1 ) ⁢ π ⁢ i / 4 2 2 ⁢ ν ⁢ Γ ⁡ ( ν + 1 ) ⁢ ( 2 ⁢ z ) - 1 2 ⁢ M 0 , ν ⁡ ( ± 2 ⁢ i ⁢ z ) , Bessel-J 𝜈 𝑧 superscript 𝑒 minus-or-plus 2 𝜈 1 𝜋 𝑖 4 superscript 2 2 𝜈 Euler-Gamma 𝜈 1 superscript 2 𝑧 1 2 Whittaker-confluent-hypergeometric-M 0 𝜈 plus-or-minus 2 𝑖 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{e^{\mp(2\nu+1)\pi i/4% }}{2^{2\nu}\Gamma\left(\nu+1\right)}(2z)^{-\frac{1}{2}}M_{0,\nu}\left(\pm 2iz% \right),}}
    0 references
    DLMF id
    DLMF:10.16.E7
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    constraint
    2 ⁢ ν ≠ - 1 , - 2 , - 3 , … 2 𝜈 1 2 3 … {\displaystyle{\displaystyle 2\nu\neq-1,-2,-3,\ldots}}
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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.m2addec
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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.m2aadec
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    Whittaker confluent hypergeometric function
    Defining formula
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2adec
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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.m2aedec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1fdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1bdec
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