DLMF:10.22.E44 (Q3418): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: Bessel function of the second kind / qualifier
 
Defining formula:

Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselY{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the second kind / qualifier
 
xml-id: C10.S2.E3.m2aedec

Revision as of 13:20, 2 January 2020

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DLMF:10.22.E44
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    Statements

    Defining formula
    ∫ 0 ∞ t μ ⁢ Y ν ⁡ ( t ) ⁢ d t = 2 μ π ⁢ Γ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ) ⁢ sin ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν ) ⁢ π , superscript subscript 0 superscript 𝑡 𝜇 Bessel-Y-Weber 𝜈 𝑡 𝑡 superscript 2 𝜇 𝜋 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 1 2 𝜇 1 2 𝜈 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu}Y_{\nu}\left(t\right)% \mathrm{d}t=\frac{2^{\mu}}{\pi}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+% \tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}% \right)\sin\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\pi,}}
    0 references
    DLMF id
    DLMF:10.22.E44
    0 references
    constraint
    ℜ ⁡ ( μ ± ν ) > - 1 plus-or-minus 𝜇 𝜈 1 {\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}}
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    ℜ ⁡ μ < 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu<\tfrac{1}{2}}}
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    Symbols used
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2aedec
    0 references
     
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