DLMF:13.14.E29 (Q4521)

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DLMF:13.14.E29
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    Statements

    Defining formula
    đ’Č ⁥ { M Îș , - ÎŒ ⁥ ( z ) , W - Îș , ÎŒ ⁥ ( e ± π ⁹ i ⁹ z ) } = Γ ⁥ ( 1 - 2 ⁹ ÎŒ ) Γ ⁥ ( 1 2 - ÎŒ + Îș ) ⁹ e ∓ ( 1 2 - ÎŒ ) ⁹ π ⁹ i , Wronskian Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 superscript 𝑒 plus-or-minus 𝜋 imaginary-unit 𝑧 Euler-Gamma 1 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 superscript 𝑒 minus-or-plus 1 2 𝜇 𝜋 imaginary-unit {\displaystyle{\displaystyle\mathscr{W}\left\{M_{\kappa,-\mu}\left(z\right),W_% {-\kappa,\mu}\left(e^{\pm\pi\mathrm{i}}z\right)\right\}=\frac{\Gamma\left(1-2% \mu\right)}{\Gamma\left(\frac{1}{2}-\mu+\kappa\right)}e^{\mp(\frac{1}{2}-\mu)% \pi\mathrm{i}},}}
    0 references
    DLMF id
    DLMF:13.14.E29
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aldec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    M Îș , ÎŒ ⁥ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2andec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W Îș , ÎŒ ⁥ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2aqdec
    0 references
    Q10865
    Defining formula
    đ’Č Wronskian {\displaystyle{\displaystyle\mathscr{W}}}
    xml-id
    C1.S13.Px2.p1.m3addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aqdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2agdec
    0 references
     
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