DLMF:10.17.E5 (Q3174)

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DLMF:10.17.E5
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    Statements

    Defining formula
    H ν ( 1 ) ⁡ ( z ) ∼ ( 2 π ⁢ z ) 1 2 ⁢ e i ⁢ ω ⁢ ∑ k = 0 ∞ i k ⁢ a k ⁢ ( ν ) z k , asymptotic-expansion Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 2 𝜋 𝑧 1 2 superscript 𝑒 𝑖 𝜔 superscript subscript 𝑘 0 superscript 𝑖 𝑘 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)\sim\left(\frac{2}{% \pi z}\right)^{\frac{1}{2}}e^{i\omega}\sum_{k=0}^{\infty}i^{k}\frac{a_{k}(\nu)% }{z^{k}},}}
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    DLMF id
    DLMF:10.17.E5
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    constraint
    - π + δ ≤ ph ⁡ z ≤ 2 ⁢ π - δ 𝜋 𝛿 phase 𝑧 2 𝜋 𝛿 {\displaystyle{\displaystyle-\pi+\delta\leq\operatorname{ph}z\leq 2\pi-\delta}}
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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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    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
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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.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
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    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1cdec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1cdec
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