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Formula:3920

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Digital Library of Mathematical Functions ID 10.19#Ex5

J ν ⁡ ( ν ⁢ sec ⁡ β ) ∼ ( 2 π ⁢ ν ⁢ tan ⁡ β ) 1 2 ⁢ ( cos ⁡ ξ ⁢ ∑ k = 0 ∞ U 2 ⁢ k ⁢ ( i ⁢ cot ⁡ β ) ν 2 ⁢ k - i ⁢ sin ⁡ ξ ⁢ ∑ k = 0 ∞ U 2 ⁢ k + 1 ⁢ ( i ⁢ cot ⁡ β ) ν 2 ⁢ k + 1 ) , asymptotic-expansion Bessel-J 𝜈 𝜈 𝛽 superscript 2 𝜋 𝜈 𝛽 1 2 𝜉 superscript subscript 𝑘 0 subscript 𝑈 2 𝑘 𝑖 𝛽 superscript 𝜈 2 𝑘 𝑖 𝜉 superscript subscript 𝑘 0 subscript 𝑈 2 𝑘 1 𝑖 𝛽 superscript 𝜈 2 𝑘 1 {\displaystyle{\displaystyle J_{\nu}\left(\nu\sec\beta\right)\sim\left(\frac{2% }{\pi\nu\tan\beta}\right)^{\frac{1}{2}}\*\left(\cos\xi\sum_{k=0}^{\infty}\frac% {U_{2k}(i\cot\beta)}{\nu^{2k}}-i\sin\xi\sum_{k=0}^{\infty}\frac{U_{2k+1}(i\cot% \beta)}{\nu^{2k+1}}\right),}}


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