DLMF:10.7.E7 (Q3060)

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DLMF:10.7.E7
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    Statements

    Defining formula
    H ν ( 1 ) ⁡ ( z ) ∼ - H ν ( 2 ) ⁡ ( z ) ∼ - ( i / π ) ⁢ Γ ⁡ ( ν ) ⁢ ( 1 2 ⁢ z ) - ν , asymptotically-equals Hankel-H-1-Bessel-third-kind 𝜈 𝑧 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 asymptotically-equals 𝑖 𝜋 Euler-Gamma 𝜈 superscript 1 2 𝑧 𝜈 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)\sim-{H^{(2)}_{\nu}}% \left(z\right)\sim-(i/\pi)\Gamma\left(\nu\right)(\tfrac{1}{2}z)^{-\nu},}}
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    DLMF id
    DLMF:10.7.E7
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    constraint
    ℜ ⁡ ν > 0 𝜈 0 {\displaystyle{\displaystyle\Re\nu>0}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
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    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.m2aadec
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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.m2aadec
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    asymptotic equality
    Defining formula
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.m2aedec
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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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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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