DLMF:13.16.E10 (Q4561): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
Defining formula:

M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}

\WhittakerconfhyperM{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E2.m2aedec

Revision as of 16:11, 2 January 2020

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DLMF:13.16.E10
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    Statements

    Defining formula
    1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( e ± π ⁢ i ⁢ z ) = e 1 2 ⁢ z ± ( 1 2 + μ ) ⁢ π ⁢ i 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 2 + μ - κ ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t - κ ) ⁢ Γ ⁡ ( 1 2 + μ - t ) Γ ⁡ ( 1 2 + μ + t ) ⁢ z t ⁢ d t , 1 Euler-Gamma 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 superscript 𝑒 plus-or-minus 𝜋 imaginary-unit 𝑧 superscript 𝑒 plus-or-minus 1 2 𝑧 1 2 𝜇 𝜋 imaginary-unit 2 𝜋 imaginary-unit Euler-Gamma 1 2 𝜇 𝜅 superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑡 𝜅 Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 1 2 𝜇 𝑡 superscript 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}% \left(e^{\pm\pi\mathrm{i}}z\right)=\frac{e^{\frac{1}{2}z\pm(\frac{1}{2}+\mu)% \pi\mathrm{i}}}{2\pi\mathrm{i}\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*\int% _{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(t-\kappa\right)\Gamma% \left(\frac{1}{2}+\mu-t\right)}{\Gamma\left(\frac{1}{2}+\mu+t\right)}z^{t}% \mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.16.E10
    0 references
    constraint
    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\tfrac{1}{2}\pi}}
    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.m2ahdec
    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.m2aedec
    0 references
     
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