DLMF:10.22.E22 (Q3396)

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DLMF:10.22.E22
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    Statements

    Defining formula
    ∫ 0 1 2 ⁢ π J μ ⁡ ( z ⁢ sin 2 ⁡ θ ) ⁢ J ν ⁡ ( z ⁢ cos 2 ⁡ θ ) ⁢ ( sin ⁡ θ ) 2 ⁢ μ + 1 ⁢ ( cos ⁡ θ ) 2 ⁢ ν + 1 ⁢ d θ = Γ ⁡ ( μ + 1 2 ) ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ J μ + ν + 1 2 ⁡ ( z ) ( 8 ⁢ π ⁢ z ) 1 2 ⁢ Γ ⁡ ( μ + ν + 1 ) , superscript subscript 0 1 2 𝜋 Bessel-J 𝜇 𝑧 2 𝜃 Bessel-J 𝜈 𝑧 2 𝜃 superscript 𝜃 2 𝜇 1 superscript 𝜃 2 𝜈 1 𝜃 Euler-Gamma 𝜇 1 2 Euler-Gamma 𝜈 1 2 Bessel-J 𝜇 𝜈 1 2 𝑧 superscript 8 𝜋 𝑧 1 2 Euler-Gamma 𝜇 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}% \theta\right)J_{\nu}\left(z{\cos^{2}}\theta\right)(\sin\theta)^{2\mu+1}(\cos% \theta)^{2\nu+1}\mathrm{d}\theta=\frac{\Gamma\left(\mu+\tfrac{1}{2}\right)% \Gamma\left(\nu+\tfrac{1}{2}\right)J_{\mu+\nu+\frac{1}{2}}\left(z\right)}{(8% \pi z)^{\frac{1}{2}}\Gamma\left(\mu+\nu+1\right)},}}
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    DLMF id
    DLMF:10.22.E22
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    constraint
    ℜ ⁡ μ > - 1 2 , ℜ ⁡ ν > - 1 2 formulae-sequence 𝜇 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-\tfrac{1}{2}}}
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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.m2andec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
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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.m2ajdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2ahdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1audec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3audec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aidec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2agdec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1pdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ldec
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