DLMF:10.27.E9 (Q3499)

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DLMF:10.27.E9
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    Statements

    Defining formula
    π ⁢ i ⁢ J ν ⁡ ( z ) = e - ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( z ⁢ e - π ⁢ i / 2 ) - e ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( z ⁢ e π ⁢ i / 2 ) , 𝜋 𝑖 Bessel-J 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle\pi iJ_{\nu}\left(z\right)=e^{-\nu\pi i/2}K_{\nu}% \left(ze^{-\pi i/2}\right)-e^{\nu\pi i/2}K_{\nu}\left(ze^{\pi i/2}\right),}}
    0 references
    DLMF id
    DLMF:10.27.E9
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    constraint
    | ph ⁡ z | ≤ 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{2}\pi}}
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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.m2aadec
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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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    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2aedec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1hdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1gdec
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