Formula:4308

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Digital Library of Mathematical Functions ID 10.41.E14

J ν ( ν z ) = ( 4 ζ 1 - z 2 ) 1 4 ( Ai ( ν 2 3 ζ ) ν 1 3 ( k = 0 A k ( ζ ) ν 2 k + O ( 1 ζ 3 + 3 ) ) + Ai ( ν 2 3 ζ ) ν 5 3 ( k = 0 - 1 B k ( ζ ) ν 2 k + O ( 1 ζ 3 + 1 ) ) ) , Bessel-J 𝜈 𝜈 𝑧 superscript 4 𝜁 1 superscript 𝑧 2 1 4 Airy-Ai superscript 𝜈 2 3 𝜁 superscript 𝜈 1 3 superscript subscript 𝑘 0 subscript 𝐴 𝑘 𝜁 superscript 𝜈 2 𝑘 Big-O 1 superscript 𝜁 3 3 diffop Airy-Ai 1 superscript 𝜈 2 3 𝜁 superscript 𝜈 5 3 superscript subscript 𝑘 0 1 subscript 𝐵 𝑘 𝜁 superscript 𝜈 2 𝑘 Big-O 1 superscript 𝜁 3 1 {\displaystyle{\displaystyle J_{\nu}\left(\nu z\right)=\left(\frac{4\zeta}{1-z% ^{2}}\right)^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left(\nu^{\frac{2}{3}}\zeta% \right)}{\nu^{\frac{1}{3}}}\left(\sum_{k=0}^{\ell}\frac{A_{k}(\zeta)}{\nu^{2k}% }+O\left(\frac{1}{\zeta^{3\ell+3}}\right)\right)+\frac{\mathrm{Ai}'\left(\nu^{% \frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\left(\sum_{k=0}^{\ell-1}\frac{B_{% k}(\zeta)}{\nu^{2k}}+O\left(\frac{1}{\zeta^{3\ell+1}}\right)\right)\right),}}


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