DLMF:10.22.E24 (Q3398)

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DLMF:10.22.E24
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    Statements

    Defining formula
    ∫ 0 1 2 ⁢ π J μ ⁡ ( z ⁢ sin 2 ⁡ θ ) ⁢ J ν ⁡ ( z ⁢ cos 2 ⁡ θ ) ⁢ cot ⁡ θ ⁢ d θ = 1 2 ⁢ μ - 1 ⁢ J μ + ν ⁡ ( z ) , superscript subscript 0 1 2 𝜋 Bessel-J 𝜇 𝑧 2 𝜃 Bessel-J 𝜈 𝑧 2 𝜃 𝜃 𝜃 1 2 superscript 𝜇 1 Bessel-J 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}% \theta\right)J_{\nu}\left(z{\cos^{2}}\theta\right)\cot\theta\mathrm{d}\theta=% \tfrac{1}{2}\mu^{-1}J_{\mu+\nu}\left(z\right),}}
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    DLMF id
    DLMF:10.22.E24
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    constraint
    ℜ ⁡ μ > 0 , ℜ ⁡ ν > - 1 formulae-sequence 𝜇 0 𝜈 1 {\displaystyle{\displaystyle\Re\mu>0,\Re\nu>-1}}
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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.m2apdec
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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.m2aldec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2ajdec
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    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2aadec
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