DLMF:10.22.E47 (Q3421)

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DLMF:10.22.E47
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ t ν ⁢ Y ν ⁡ ( a ⁢ t ) t 2 + b 2 ⁢ d t = - b ν - 1 ⁢ K ν ⁡ ( a ⁢ b ) , superscript subscript 0 superscript 𝑡 𝜈 Bessel-Y-Weber 𝜈 𝑎 𝑡 superscript 𝑡 2 superscript 𝑏 2 𝑡 superscript 𝑏 𝜈 1 modified-Bessel-second-kind 𝜈 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{\nu}Y_{\nu}\left(at% \right)}{t^{2}+b^{2}}\mathrm{d}t=-b^{\nu-1}K_{\nu}\left(ab\right),}}
    0 references
    DLMF id
    DLMF:10.22.E47
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    constraint
    a > 0 , ℜ ⁡ b > 0 , - 1 2 < ℜ ⁡ ν < 5 2 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 1 2 𝜈 5 2 {\displaystyle{\displaystyle a>0,\Re b>0,-\tfrac{1}{2}<\Re\nu<\tfrac{5}{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.m2afdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aatdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aatdec
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    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2aadec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aabdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1afdec
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