DLMF:10.22.E64 (Q3438)

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DLMF:10.22.E64
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ J μ + 2 ⁢ n + 1 ⁡ ( a ⁢ t ) ⁢ J μ ⁡ ( b ⁢ t ) ⁢ d t = { b μ ⁢ Γ ⁡ ( μ + n + 1 ) a μ + 1 ⁢ n ! ⁢ 𝐅 ⁡ ( - n , μ + n + 1 ; μ + 1 ; b 2 a 2 ) , 0 < b < a , ( - 1 ) n / ( 2 ⁢ a ) , b = a ( > 0 ) , 0 , 0 < a < b . superscript subscript 0 Bessel-J 𝜇 2 𝑛 1 𝑎 𝑡 Bessel-J 𝜇 𝑏 𝑡 𝑡 cases superscript 𝑏 𝜇 Euler-Gamma 𝜇 𝑛 1 superscript 𝑎 𝜇 1 𝑛 scaled-hypergeometric-bold-F 𝑛 𝜇 𝑛 1 𝜇 1 superscript 𝑏 2 superscript 𝑎 2 0 𝑏 𝑎 superscript 1 𝑛 2 𝑎 𝑏 annotated 𝑎 absent 0 0 0 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu+2n+1}\left(at\right)J_{\mu% }\left(bt\right)\mathrm{d}t=\begin{cases}\dfrac{b^{\mu}\Gamma\left(\mu+n+1% \right)}{a^{\mu+1}n!}\mathbf{F}\left(-n,\mu+n+1;\mu+1;\dfrac{b^{2}}{a^{2}}% \right),&0<b<a,\\ (-1)^{n}/(2a),&b=a(>0),\\ 0,&0<a<b.\end{cases}}}
    0 references
    DLMF id
    DLMF:10.22.E64
    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.m2aawdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2audec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abkdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abkdec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2addec
    0 references
    Q11429
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1fdec
    0 references
     
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