DLMF:13.14.E27 (Q4519)

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DLMF:13.14.E27
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    Statements

    test
    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}},}}
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    DLMF id
    DLMF:13.14.E27
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ajdec
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    Whittaker confluent hypergeometric function
    test
    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.m2aldec
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    Whittaker confluent hypergeometric function
    test
    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.m2aodec
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    Q10865
    test
    đ’Č Wronskian {\displaystyle{\displaystyle\mathscr{W}}}
    xml-id
    C1.S13.Px2.p1.m3abdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2apdec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
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    Q11557
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1zdec
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    Q11557
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1zdec
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