DLMF:10.9.E5 (Q3070): Difference between revisions

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Property / constraint
 

ℜ ⁡ ν > - 1 2 , | ph ⁡ z | < 1 2 ⁢ π formulae-sequence 𝜈 1 2 phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},|\operatorname{ph}z|<\tfrac{1% }{2}\pi}}

\realpart@@{\nu}>-\tfrac{1}{2},|\phase@@{z}|<\tfrac{1}{2}\pi
Property / constraint: ℜ ⁡ ν > - 1 2 , | ph ⁡ z | < 1 2 ⁢ π formulae-sequence 𝜈 1 2 phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},|\operatorname{ph}z|<\tfrac{1% }{2}\pi}} / rank
 
Normal rank

Revision as of 16:15, 30 December 2019

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DLMF:10.9.E5
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    Statements

    test
    Y ν ⁡ ( z ) = 2 ⁢ ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ( ∫ 0 1 ( 1 - t 2 ) ν - 1 2 ⁢ sin ⁡ ( z ⁢ t ) ⁢ d t - ∫ 0 ∞ e - z ⁢ t ⁢ ( 1 + t 2 ) ν - 1 2 ⁢ d t ) , Bessel-Y-Weber 𝜈 𝑧 2 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 1 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑧 𝑡 𝑡 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑡 {\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu% }}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\left(\int_{0}^{1}(1-t% ^{2})^{\nu-\frac{1}{2}}\sin\left(zt\right)\mathrm{d}t-\int_{0}^{\infty}e^{-zt}% (1+t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t\right),}}
    0 references
    DLMF id
    DLMF:10.9.E5
    0 references
    constraint
    ℜ ⁡ ν > - 1 2 , | ph ⁡ z | < 1 2 ⁢ π formulae-sequence 𝜈 1 2 phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},|\operatorname{ph}z|<\tfrac{1% }{2}\pi}}
    0 references
     
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