DLMF:10.22.E63 (Q3437)

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DLMF:10.22.E63
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    Statements

    Defining formula
    ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J μ - 1 ⁡ ( b ⁢ t ) ⁢ d t = { b μ - 1 ⁢ a - μ , 0 < b < a , ( 2 ⁢ b ) - 1 , b = a ( > 0 ) , 0 , 0 < a < b . superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜇 1 𝑏 𝑡 𝑡 cases superscript 𝑏 𝜇 1 superscript 𝑎 𝜇 0 𝑏 𝑎 superscript 2 𝑏 1 𝑏 annotated 𝑎 absent 0 0 0 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\mu-1}% \left(bt\right)\mathrm{d}t=\begin{cases}b^{\mu-1}a^{-\mu},&0<b<a,\\ (2b)^{-1},&b=a(>0),\\ 0,&0<a<b.\end{cases}}}
    0 references
    DLMF id
    DLMF:10.22.E63
    0 references
    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.m2aavdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abjdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abjdec
    0 references
     
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