DLMF:13.16.E11 (Q4562): 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.m2aidec

Revision as of 16:11, 2 January 2020

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DLMF:13.16.E11
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    Statements

    Defining formula
    W κ , μ ⁡ ( z ) = e - 1 2 ⁢ z 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( 1 2 + μ + t ) ⁢ Γ ⁡ ( 1 2 - μ + t ) ⁢ Γ ⁡ ( - κ - t ) Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ z - t ⁢ d t , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 1 2 𝑧 2 𝜋 imaginary-unit superscript subscript imaginary-unit imaginary-unit Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 𝜅 𝑡 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 superscript 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{e^{-\frac{1}{2% }z}}{2\pi\mathrm{i}}\*\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma% \left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)\Gamma\left(% -\kappa-t\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1% }{2}-\mu-\kappa\right)}z^{-t}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.16.E11
    0 references
    constraint
    | ph ⁡ z | < 3 2 ⁢ π phase 𝑧 3 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\tfrac{3}{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.m2aidec
    0 references
     
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