DLMF:9.7.E10 (Q2817): Difference between revisions

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Property / constraint
 

| ph z | 2 3 π - δ phase 𝑧 2 3 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta}}

|\phase@@{z}|\leq\tfrac{2}{3}\pi-\delta
Property / constraint: | ph z | 2 3 π - δ phase 𝑧 2 3 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta}} / rank
 
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Revision as of 16:13, 30 December 2019

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DLMF:9.7.E10
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    Statements

    Ai ( - z ) z 1 / 4 π ( sin ( ζ - 1 4 π ) k = 0 ( - 1 ) k v 2 k ζ 2 k - cos ( ζ - 1 4 π ) k = 0 ( - 1 ) k v 2 k + 1 ζ 2 k + 1 ) , asymptotic-expansion diffop Airy-Ai 1 𝑧 superscript 𝑧 1 4 𝜋 𝜁 1 4 𝜋 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑣 2 𝑘 superscript 𝜁 2 𝑘 𝜁 1 4 𝜋 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑣 2 𝑘 1 superscript 𝜁 2 𝑘 1 {\displaystyle{\displaystyle\mathrm{Ai}'\left(-z\right)\sim\frac{z^{1/4}}{% \sqrt{\pi}}\left(\sin\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)% ^{k}\frac{v_{2k}}{\zeta^{2k}}-\cos\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}% ^{\infty}(-1)^{k}\frac{v_{2k+1}}{\zeta^{2k+1}}\right),}}
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    | ph z | 2 3 π - δ phase 𝑧 2 3 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta}}
    0 references