DLMF:10.22.E72 (Q3446)

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DLMF:10.22.E72
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ J ν ⁡ ( c ⁢ t ) ⁢ t 1 - μ ⁢ d t = ( b ⁢ c ) μ - 1 ⁢ sin ⁡ ( ( μ - ν ) ⁢ π ) ⁢ ( sinh ⁡ χ ) μ - 1 2 ( 1 2 ⁢ π 3 ) 1 2 ⁢ a μ ⁢ e ( μ - 1 2 ) ⁢ i ⁢ π ⁢ Q ν - 1 2 1 2 - μ ⁡ ( cosh ⁡ χ ) , superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 Bessel-J 𝜈 𝑐 𝑡 superscript 𝑡 1 𝜇 𝑡 superscript 𝑏 𝑐 𝜇 1 𝜇 𝜈 superscript 𝜒 𝜇 1 2 superscript 1 2 superscript 𝜋 3 1 2 superscript 𝑎 𝜇 𝜇 1 2 imaginary-unit Legendre-Q-second-kind 1 2 𝜇 𝜈 1 2 𝜒 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}% \left(bt\right)J_{\nu}\left(ct\right)t^{1-\mu}\mathrm{d}t=\frac{(bc)^{\mu-1}% \sin\left((\mu-\nu)\pi\right)(\sinh\chi)^{\mu-\frac{1}{2}}}{(\frac{1}{2}\pi^{3% })^{\frac{1}{2}}a^{\mu}}{\mathrm{e}^{(\mu-\frac{1}{2})\mathrm{i}\pi}}Q^{\frac{% 1}{2}-\mu}_{\nu-\frac{1}{2}}\left(\cosh\chi\right),}}
    0 references
    DLMF id
    DLMF:10.22.E72
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    constraint
    ℜ ⁡ μ > - 1 2 , ℜ ⁡ ν > - 1 , a > b + c , cosh ⁡ χ = ( a 2 - b 2 - c 2 ) / ( 2 ⁢ b ⁢ c ) formulae-sequence 𝜇 1 2 formulae-sequence 𝜈 1 formulae-sequence 𝑎 𝑏 𝑐 𝜒 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 2 𝑏 𝑐 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-1,a>b+c,\cosh\chi=(a^% {2}-b^{2}-c^{2})/(2bc)}}
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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.m2abedec
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    Q11470
    Defining formula
    Q ν μ ⁡ ( z ) Legendre-Q-second-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m2adec
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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.m2aaddec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aodec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1absdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aadec
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    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2agdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3absdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aasdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aodec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1axdec
    0 references
     
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