DLMF:10.22.E65 (Q3439)

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DLMF:10.22.E65
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    Statements

    Defining formula
    ∫ 0 ∞ J 0 ⁡ ( a ⁢ t ) ⁢ ( J 0 ⁡ ( b ⁢ t ) - J 0 ⁡ ( c ⁢ t ) ) ⁢ d t t = { 0 , 0 ≤ b < a , 0 < c ≤ a , ln ⁡ ( c / a ) , 0 ≤ b < a ≤ c . superscript subscript 0 Bessel-J 0 𝑎 𝑡 Bessel-J 0 𝑏 𝑡 Bessel-J 0 𝑐 𝑡 𝑡 𝑡 cases 0 formulae-sequence 0 𝑏 𝑎 0 𝑐 𝑎 𝑐 𝑎 0 𝑏 𝑎 𝑐 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{0}\left(at\right)\left(J_{0}% \left(bt\right)-J_{0}\left(ct\right)\right)\frac{\mathrm{d}t}{t}=\begin{cases}% 0,&0\leq b<a,0<c\leq a,\\ \ln\left(c/a\right),&0\leq b<a\leq c.\end{cases}}}
    0 references
    DLMF id
    DLMF:10.22.E65
    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.m2aaxdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abldec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abldec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2acdec
    0 references
     
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