DLMF:10.9.E4 (Q3069)

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DLMF:10.9.E4
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    Statements

    Defining formula
    J ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 π cos ⁡ ( z ⁢ cos ⁡ θ ) ⁢ ( sin ⁡ θ ) 2 ⁢ ν ⁢ d θ = 2 ⁢ ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 1 ( 1 - t 2 ) ν - 1 2 ⁢ cos ⁡ ( z ⁢ t ) ⁢ d t , Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 𝜋 𝑧 𝜃 superscript 𝜃 2 𝜈 𝜃 2 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 1 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% }{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left% (z\cos\theta\right)(\sin\theta)^{2\nu}\mathrm{d}\theta=\frac{2(\tfrac{1}{2}z)^% {\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{% 2})^{\nu-\frac{1}{2}}\cos\left(zt\right)\mathrm{d}t,}}
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    DLMF id
    DLMF:10.9.E4
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    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
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