DLMF:13.23.E7 (Q4641): Difference between revisions

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

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2afdec

Revision as of 01:02, 2 January 2020

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DLMF:13.23.E7
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    Statements

    Defining formula
    1 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e z ⁢ t + 1 2 ⁢ t - 1 ⁢ t κ ⁢ W κ , μ ⁡ ( t - 1 ) ⁢ d t = 2 ⁢ z - κ - 1 2 Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ K 2 ⁢ μ ⁡ ( 2 ⁢ z ) , 1 2 𝜋 imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑡 1 2 superscript 𝑡 1 superscript 𝑡 𝜅 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 superscript 𝑡 1 𝑡 2 superscript 𝑧 𝜅 1 2 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 modified-Bessel-second-kind 2 𝜇 2 𝑧 {\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt% +\frac{1}{2}t^{-1}}t^{\kappa}W_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=% \frac{2z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)% \Gamma\left(\frac{1}{2}-\mu-\kappa\right)}K_{2\mu}\left(2\sqrt{z}\right),}}
    0 references
    DLMF id
    DLMF:13.23.E7
    0 references
    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
    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.m2afdec
    0 references
     
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