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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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10818
    Defining formula
    ℤ {\displaystyle{\displaystyle\mathbb{Z}}}
    xml-id
    introduction.Sx4.p2.t1.r20.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
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