DLMF:13.21.E2 (Q4610)

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DLMF:13.21.E2
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    Statements

    Defining formula
    W κ , μ ⁡ ( x ) = x ⁢ Γ ⁡ ( κ + 1 2 ) ⁢ ( sin ⁡ ( κ ⁢ π - μ ⁢ π ) ⁢ J 2 ⁢ μ ⁡ ( 2 ⁢ x ⁢ κ ) - cos ⁡ ( κ ⁢ π - μ ⁢ π ) ⁢ Y 2 ⁢ μ ⁡ ( 2 ⁢ x ⁢ κ ) + env ⁢ Y 2 ⁢ μ ⁡ ( 2 ⁢ x ⁢ κ ) ⁢ O ⁡ ( κ - 1 2 ) ) , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑥 𝑥 Euler-Gamma 𝜅 1 2 𝜅 𝜋 𝜇 𝜋 Bessel-J 2 𝜇 2 𝑥 𝜅 𝜅 𝜋 𝜇 𝜋 Bessel-Y-Weber 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(x\right)=\sqrt{x}\Gamma\left(% \kappa+\tfrac{1}{2}\right)\*\left(\sin\left(\kappa\pi-\mu\pi\right)J_{2\mu}% \left(2\sqrt{x\kappa}\right)-\cos\left(\kappa\pi-\mu\pi\right)Y_{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.E2
    0 references
    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
    0 references
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2adec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aadec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    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.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    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.m4adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11566
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1adec
    0 references
     
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