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DLMF:10.9#Ex5 - MaRDI portal

DLMF:10.9#Ex5 (Q3777)

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DLMF:10.9#Ex5
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    DLMF defining formula
    J ν ⁡ ( x ) = 2 ⁢ ( 1 2 ⁢ x ) - ν π 1 2 ⁢ Γ ⁡ ( 1 2 - ν ) ⁢ ∫ 1 ∞ sin ⁡ ( x ⁢ t ) ⁢ d t ( t 2 - 1 ) ν + 1 2 , Bessel-J 𝜈 𝑥 2 superscript 1 2 𝑥 𝜈 superscript 𝜋 1 2 Euler-Gamma 1 2 𝜈 superscript subscript 1 𝑥 𝑡 𝑡 superscript superscript 𝑡 2 1 𝜈 1 2 {\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{2(\tfrac{1}{2}x)^{-% \nu}}{\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}% \frac{\sin\left(xt\right)\mathrm{d}t}{(t^{2}-1)^{\nu+\frac{1}{2}}},}}
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