DLMF:9.7.E9 (Q2816): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11386 / rank
 
Normal rank
Property / Symbols used: Q11386 / qualifier
 
test:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q11386 / qualifier
 
xml-id: C9.S1.XMD4.m1ddec
Property / Symbols used
 
Property / Symbols used: change of variable (locally) / rank
 
Normal rank
Property / Symbols used: change of variable (locally) / qualifier
 
test:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: change of variable (locally) / qualifier
 
xml-id: C9.S7.XMD1.m1edec
Property / Symbols used
 
Property / Symbols used: Q11387 / rank
 
Normal rank
Property / Symbols used: Q11387 / qualifier
 
test:

u s subscript 𝑢 𝑠 {\displaystyle{\displaystyle u_{s}}}

u_{s}
Property / Symbols used: Q11387 / qualifier
 
xml-id: C9.S7.XMD2.m1cdec

Latest revision as of 13:33, 2 January 2020

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DLMF:9.7.E9
No description defined

    Statements

    test
    Ai ⁡ ( - z ) ∼ 1 π ⁢ z 1 / 4 ⁢ ( cos ⁡ ( ζ - 1 4 ⁢ π ) ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ u 2 ⁢ k ζ 2 ⁢ k + sin ⁡ ( ζ - 1 4 ⁢ π ) ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ u 2 ⁢ k + 1 ζ 2 ⁢ k + 1 ) , asymptotic-expansion Airy-Ai 𝑧 1 𝜋 superscript 𝑧 1 4 𝜁 1 4 𝜋 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑢 2 𝑘 superscript 𝜁 2 𝑘 𝜁 1 4 𝜋 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑢 2 𝑘 1 superscript 𝜁 2 𝑘 1 {\displaystyle{\displaystyle\mathrm{Ai}\left(-z\right)\sim\frac{1}{\sqrt{\pi}z% ^{1/4}}\left(\cos\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}% \frac{u_{2k}}{\zeta^{2k}}+\sin\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{% \infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}}\right),}}
    0 references
    DLMF id
    DLMF:9.7.E9
    0 references
    constraint
    | 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
    Symbols used
    Q10919
    test
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1abdec
    0 references
    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11addec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1addec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11385
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C9.S1.XMD1.m1edec
    0 references
    Q11381
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1edec
    0 references
    Q11386
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C9.S1.XMD4.m1ddec
    0 references
    change of variable (locally)
    test
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S7.XMD1.m1edec
    0 references
    Q11387
    test
    u s subscript 𝑢 𝑠 {\displaystyle{\displaystyle u_{s}}}
    xml-id
    C9.S7.XMD2.m1cdec
    0 references
     
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