DLMF:1.17.E14 (Q673)

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DLMF:1.17.E14
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    Statements

    Defining formula
    δ ⁡ ( x - a ) = 2 ⁢ x ⁢ a π ⁢ ∫ 0 ∞ t 2 ⁢ 𝗃 ℓ ⁡ ( x ⁢ t ) ⁢ 𝗃 ℓ ⁡ ( a ⁢ t ) ⁢ d t , Dirac-delta 𝑥 𝑎 2 𝑥 𝑎 𝜋 superscript subscript 0 superscript 𝑡 2 spherical-Bessel-J ℓ 𝑥 𝑡 spherical-Bessel-J ℓ 𝑎 𝑡 𝑡 {\displaystyle{\displaystyle\delta\left(x-a\right)=\frac{2xa}{\pi}\int_{0}^{% \infty}t^{2}\mathsf{j}_{\ell}\left(xt\right)\mathsf{j}_{\ell}\left(at\right)% \mathrm{d}t,}}
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    DLMF id
    DLMF:1.17.E14
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    constraint
    a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}}
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    Symbols used
    Q10915
    Defining formula
    δ ⁡ ( x - a ) Dirac-delta 𝑥 𝑎 {\displaystyle{\displaystyle\delta\left(\NVar{x-a}\right)}}
    xml-id
    C1.S17.SS1.p1.m2aedec
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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.m2ahdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
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    spherical Bessel function of the first kind
    Defining formula
    𝗃 n ⁡ ( z ) spherical-Bessel-J 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C10.S47.E3.m2adec
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