DLMF:10.9.E2 (Q3067)

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DLMF:10.9.E2
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    Statements

    Defining formula
    J n ⁡ ( z ) = 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ sin ⁡ θ - n ⁢ θ ) ⁢ d θ = i - n π ⁢ ∫ 0 π e i ⁢ z ⁢ cos ⁡ θ ⁢ cos ⁡ ( n ⁢ θ ) ⁢ d θ , Bessel-J 𝑛 𝑧 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝑛 𝜃 𝜃 superscript 𝑖 𝑛 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑖 𝑧 𝜃 𝑛 𝜃 𝜃 {\displaystyle{\displaystyle J_{n}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \cos\left(z\sin\theta-n\theta\right)\mathrm{d}\theta=\frac{i^{-n}}{\pi}\int_{0% }^{\pi}e^{iz\cos\theta}\cos\left(n\theta\right)\mathrm{d}\theta,}}
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    DLMF id
    DLMF:10.9.E2
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    constraint
    n ∈ ℤ 𝑛 {\displaystyle{\displaystyle n\in\mathbb{Z}}}
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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.m2aadec
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