DLMF:10.9.E1 (Q3066)

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DLMF:10.9.E1
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    Statements

    Defining formula
    J 0 ⁡ ( z ) = 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ sin ⁡ θ ) ⁢ d θ = 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ cos ⁡ θ ) ⁢ d θ , Bessel-J 0 𝑧 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜃 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜃 {\displaystyle{\displaystyle J_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \cos\left(z\sin\theta\right)\mathrm{d}\theta=\frac{1}{\pi}\int_{0}^{\pi}\cos% \left(z\cos\theta\right)\mathrm{d}\theta,}}
    0 references
    DLMF id
    DLMF:10.9.E1
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    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.m2adec
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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.m2adec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1dec
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