DLMF:13.21.E4 (Q4612)

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DLMF:13.21.E4
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    Statements

    Defining formula
    W - κ , μ ⁡ ( x ⁢ e π ⁢ i ) = π ⁢ x Γ ⁡ ( κ + 1 2 ) ⁢ e - μ ⁢ π ⁢ i ⁢ ( H 2 ⁢ μ ( 2 ) ⁡ ( 2 ⁢ x ⁢ κ ) + env ⁢ Y 2 ⁢ μ ⁡ ( 2 ⁢ x ⁢ κ ) ⁢ O ⁡ ( κ - 1 2 ) ) , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑥 superscript 𝑒 𝜋 imaginary-unit 𝜋 𝑥 Euler-Gamma 𝜅 1 2 superscript 𝑒 𝜇 𝜋 imaginary-unit Hankel-H-2-Bessel-third-kind 2 𝜇 2 𝑥 𝜅 envelope-Bessel-Y 2 𝜇 2 𝑥 𝜅 Big-O superscript 𝜅 1 2 {\displaystyle{\displaystyle W_{-\kappa,\mu}\left(xe^{\pi\mathrm{i}}\right)=% \frac{\pi\sqrt{x}}{\Gamma\left(\kappa+\tfrac{1}{2}\right)}e^{-\mu\pi\mathrm{i}% }\*\left({H^{(2)}_{2\mu}}\left(2\sqrt{x\kappa}\right)+\mathrm{env}\mskip-2.0mu% Y_{2\mu}\left(2\sqrt{x\kappa}\right)O\left(\kappa^{-\frac{1}{2}}\right)\right% ),}}
    0 references
    DLMF id
    DLMF:13.21.E4
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2acdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    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
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q11091
    Defining formula
    env ⁢ Y ν ⁡ ( x ) envelope-Bessel-Y 𝜈 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu Y_{\NVar{\nu}}\left(\NVar% {x}\right)}}
    xml-id
    C2.S8.SS4.p5.m4abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q11566
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1cdec
    0 references
     
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