DLMF:10.27.E7 (Q3497)

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DLMF:10.27.E7
No description defined

    Statements

    Defining formula
    I ν ⁡ ( z ) = 1 2 ⁢ e ∓ ν ⁢ π ⁢ i / 2 ⁢ ( H ν ( 1 ) ⁡ ( z ⁢ e ± π ⁢ i / 2 ) + H ν ( 2 ) ⁡ ( z ⁢ e ± π ⁢ i / 2 ) ) , modified-Bessel-first-kind 𝜈 𝑧 1 2 superscript 𝑒 minus-or-plus 𝜈 𝜋 𝑖 2 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 plus-or-minus 𝜋 𝑖 2 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 plus-or-minus 𝜋 𝑖 2 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\tfrac{1}{2}e^{\mp\nu\pi i/% 2}\left({H^{(1)}_{\nu}}\left(ze^{\pm\pi i/2}\right)+{H^{(2)}_{\nu}}\left(ze^{% \pm\pi i/2}\right)\right),}}
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    DLMF id
    DLMF:10.27.E7
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    constraint
    - π ≤ ± ph ⁡ z ≤ 1 2 ⁢ π 𝜋 plus-or-minus phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle-\pi\leq\pm\operatorname{ph}z\leq\tfrac{1}{2}\pi}}
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    Symbols used
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E5.m2adec
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    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 2 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E6.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.m2acdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2aedec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aadec
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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.m1edec
    0 references
     
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