DLMF:10.9.E14 (Q3082): 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.m2afdec

Revision as of 15:14, 2 January 2020

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DLMF:10.9.E14
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    Statements

    Defining formula
    ( z + ζ z - ζ ) 1 2 ⁢ ν ⁢ Y ν ⁡ ( ( z 2 - ζ 2 ) 1 2 ) = 1 π ⁢ ∫ 0 π e ζ ⁢ cos ⁡ θ ⁢ sin ⁡ ( z ⁢ sin ⁡ θ - ν ⁢ θ ) ⁢ d θ - 1 π ⁢ ∫ 0 ∞ ( e ν ⁢ t + ζ ⁢ cosh ⁡ t + e - ν ⁢ t - ζ ⁢ cosh ⁡ t ⁢ cos ⁡ ( ν ⁢ π ) ) ⁢ e - z ⁢ sinh ⁡ t ⁢ d t , superscript 𝑧 𝜁 𝑧 𝜁 1 2 𝜈 Bessel-Y-Weber 𝜈 superscript superscript 𝑧 2 superscript 𝜁 2 1 2 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝜁 𝜃 𝑧 𝜃 𝜈 𝜃 𝜃 1 𝜋 superscript subscript 0 superscript 𝑒 𝜈 𝑡 𝜁 𝑡 superscript 𝑒 𝜈 𝑡 𝜁 𝑡 𝜈 𝜋 superscript 𝑒 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}Y_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^% {\pi}e^{\zeta\cos\theta}\sin\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta% -\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t+\zeta\cosh t}+e^{-\nu t-\zeta% \cosh t}\cos\left(\nu\pi\right)\right)\*e^{-z\sinh t}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.9.E14
    0 references
    constraint
    ℜ ⁡ ( z ± ζ ) > 0 plus-or-minus 𝑧 𝜁 0 {\displaystyle{\displaystyle\Re(z\pm\zeta)>0}}
    0 references
    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.m2afdec
    0 references
     
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