DLMF:10.19#Ex20 (Q3238): Difference between revisions

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Property / constraint
 

| ph ν | 1 2 π - δ phase 𝜈 1 2 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta}}

|\phase@@{\nu}|\leq\tfrac{1}{2}\pi-\delta
Property / constraint: | ph ν | 1 2 π - δ phase 𝜈 1 2 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta}} / rank
 
Normal rank

Latest revision as of 16:18, 30 December 2019

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DLMF:10.19#Ex20
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    Statements

    J ν ( ν + a ν 1 3 ) - 2 2 3 ν 2 3 Ai ( - 2 1 3 a ) k = 0 R k ( a ) ν 2 k / 3 + 2 1 3 ν 4 3 Ai ( - 2 1 3 a ) k = 0 S k ( a ) ν 2 k / 3 , asymptotic-expansion diffop Bessel-J 𝜈 1 𝜈 𝑎 superscript 𝜈 1 3 superscript 2 2 3 superscript 𝜈 2 3 diffop Airy-Ai 1 superscript 2 1 3 𝑎 superscript subscript 𝑘 0 subscript 𝑅 𝑘 𝑎 superscript 𝜈 2 𝑘 3 superscript 2 1 3 superscript 𝜈 4 3 Airy-Ai superscript 2 1 3 𝑎 superscript subscript 𝑘 0 subscript 𝑆 𝑘 𝑎 superscript 𝜈 2 𝑘 3 {\displaystyle{\displaystyle J_{\nu}'\left(\nu+a\nu^{\frac{1}{3}}\right)\sim-% \frac{2^{\frac{2}{3}}}{\nu^{\frac{2}{3}}}\mathrm{Ai}'\left(-2^{\frac{1}{3}}a% \right)\sum_{k=0}^{\infty}\frac{R_{k}(a)}{\nu^{2k/3}}+\frac{2^{\frac{1}{3}}}{% \nu^{\frac{4}{3}}}\mathrm{Ai}\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}% \frac{S_{k}(a)}{\nu^{2k/3}},}}
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    | ph ν | 1 2 π - δ phase 𝜈 1 2 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta}}
    0 references