DLMF:13.14.E32 (Q4524)

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DLMF:13.14.E32
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    Statements

    Defining formula
    1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( z ) = e ± ( κ - μ - 1 2 ) ⁢ π ⁢ i Γ ⁡ ( 1 2 + μ + κ ) ⁢ W κ , μ ⁡ ( z ) + e ± κ ⁢ π ⁢ i Γ ⁡ ( 1 2 + μ - κ ) ⁢ W - κ , μ ⁡ ( e ± π ⁢ i ⁢ z ) . 1 Euler-Gamma 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 superscript 𝑒 plus-or-minus 𝜅 𝜇 1 2 𝜋 imaginary-unit Euler-Gamma 1 2 𝜇 𝜅 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 plus-or-minus 𝜅 𝜋 imaginary-unit Euler-Gamma 1 2 𝜇 𝜅 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 superscript 𝑒 plus-or-minus 𝜋 imaginary-unit 𝑧 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}% \left(z\right)=\frac{e^{\pm(\kappa-\mu-\frac{1}{2})\pi\mathrm{i}}}{\Gamma\left% (\frac{1}{2}+\mu+\kappa\right)}W_{\kappa,\mu}\left(z\right)+\frac{e^{\pm\kappa% \pi\mathrm{i}}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}W_{-\kappa,\mu}\left% (e^{\pm\pi\mathrm{i}}z\right).}}
    0 references
    DLMF id
    DLMF:13.14.E32
    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.m2amdec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2aodec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2akdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2asdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aidec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1aedec
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    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.m2atdec
    0 references
     
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