DLMF:13.23.E5 (Q4639): 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:

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

\WhittakerconfhyperW{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E3.m2aadec

Revision as of 01:02, 2 January 2020

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DLMF:13.23.E5
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    Statements

    Defining formula
    ∫ 0 ∞ e 1 2 ⁢ t ⁢ t ν - 1 ⁢ W κ , μ ⁡ ( t ) ⁢ d t = Γ ⁡ ( 1 2 + μ + ν ) ⁢ Γ ⁡ ( 1 2 - μ + ν ) ⁢ Γ ⁡ ( - κ - ν ) Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) , superscript subscript 0 superscript 𝑒 1 2 𝑡 superscript 𝑡 𝜈 1 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑡 𝑡 Euler-Gamma 1 2 𝜇 𝜈 Euler-Gamma 1 2 𝜇 𝜈 Euler-Gamma 𝜅 𝜈 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\nu-1}W_{% \kappa,\mu}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\frac{1}{2}+\mu+\nu% \right)\Gamma\left(\frac{1}{2}-\mu+\nu\right)\Gamma\left(-\kappa-\nu\right)}{% \Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa% \right)},}}
    0 references
    DLMF id
    DLMF:13.23.E5
    0 references
    constraint
    | ℜ ⁡ μ | - 1 2 < ℜ ⁡ ν < - ℜ ⁡ κ 𝜇 1 2 𝜈 𝜅 {\displaystyle{\displaystyle|\Re\mu|-\tfrac{1}{2}<\Re\nu<-\Re\kappa}}
    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.m2addec
    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.m2aadec
    0 references
     
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