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DLMF:10.22.E30 - MaRDI portal

DLMF:10.22.E30 (Q4101)

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DLMF:10.22.E30
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    Statements

    DLMF defining formula
    ∫ 0 x J n ⁡ ( t ) ⁢ J n + 1 ⁡ ( t ) ⁢ d t = 1 2 ⁢ ( 1 - J 0 2 ⁡ ( x ) ) - ∑ k = 1 n J k 2 ⁡ ( x ) = ∑ k = n + 1 ∞ J k 2 ⁡ ( x ) , superscript subscript 0 𝑥 Bessel-J 𝑛 𝑡 Bessel-J 𝑛 1 𝑡 𝑡 1 2 1 Bessel-J 0 2 𝑥 superscript subscript 𝑘 1 𝑛 Bessel-J 𝑘 2 𝑥 superscript subscript 𝑘 𝑛 1 Bessel-J 𝑘 2 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}J_{n}\left(t\right)J_{n+1}\left(t% \right)\mathrm{d}t=\tfrac{1}{2}\left(1-{J_{0}^{2}}\left(x\right)\right)-\sum_{% k=1}^{n}{J_{k}^{2}}\left(x\right)=\sum_{k=n+1}^{\infty}{J_{k}^{2}}\left(x% \right),}}
    0 references
    constraint
    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\ldots}}
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    Symbols used
    Bessel function of the first kind
    DLMF defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2avdec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aacdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aacdec
    0 references
    integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1edec
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    nonnegative integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1fdec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1hdec
    0 references
     
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