DLMF:10.22.E16 (Q3390)

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DLMF:10.22.E16
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    Statements

    Defining formula
    ∫ 0 1 2 ⁢ π J 0 ⁡ ( 2 ⁢ z ⁢ sin ⁡ θ ) ⁢ cos ⁡ ( 2 ⁢ n ⁢ θ ) ⁢ d θ = 1 2 ⁢ π ⁢ J n 2 ⁡ ( z ) , superscript subscript 0 1 2 𝜋 Bessel-J 0 2 𝑧 𝜃 2 𝑛 𝜃 𝜃 1 2 𝜋 Bessel-J 𝑛 2 𝑧 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{0}\left(2z\sin\theta% \right)\cos\left(2n\theta\right)\mathrm{d}\theta=\tfrac{1}{2}\pi{J_{n}^{2}}% \left(z\right),}}
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    DLMF id
    DLMF:10.22.E16
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    constraint
    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\ldots}}
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    Symbols used
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1jdec
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    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.m2ahdec
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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.m2addec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aodec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
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    Q11429
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1cdec
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