DLMF:10.22.E70 (Q3444)

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DLMF:10.22.E70
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    Statements

    Defining formula
    ∫ 0 ∞ Y ν ⁡ ( a ⁢ t ) ⁢ J ν + 1 ⁡ ( b ⁢ t ) ⁢ t ⁢ d t t 2 - z 2 = 1 2 ⁢ π ⁢ J ν + 1 ⁡ ( b ⁢ z ) ⁢ H ν ( 1 ) ⁡ ( a ⁢ z ) , superscript subscript 0 Bessel-Y-Weber 𝜈 𝑎 𝑡 Bessel-J 𝜈 1 𝑏 𝑡 𝑡 𝑡 superscript 𝑡 2 superscript 𝑧 2 1 2 𝜋 Bessel-J 𝜈 1 𝑏 𝑧 Hankel-H-1-Bessel-third-kind 𝜈 𝑎 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu+1}% \left(bt\right)\frac{t\mathrm{d}t}{t^{2}-z^{2}}=\frac{1}{2}\pi J_{\nu+1}\left(% bz\right){H^{(1)}_{\nu}}\left(az\right),}}
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    DLMF id
    DLMF:10.22.E70
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    constraint
    ℜ ⁡ ν > - 3 2 , ℑ ⁡ z > 0 formulae-sequence 𝜈 3 2 𝑧 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{3}{2},\Im z>0}}
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    a ≥ b > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle a\geq b>0}}
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    ℜ ⁡ ν > - 3 2 , ℑ ⁡ z > 0 formulae-sequence 𝜈 3 2 𝑧 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{3}{2},\Im z>0}}
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    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2abcdec
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    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.m2akdec
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    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E5.m2aadec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aabdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abqdec
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    Q10901
    Defining formula
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abqdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aaqdec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1vdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1avdec
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