DLMF:36.13.E8 (Q10017): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2abdec

Revision as of 15:21, 2 January 2020

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DLMF:36.13.E8
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    Statements

    Defining formula
    z ⁢ ( ρ , ϕ ) = 2 ⁢ π ⁢ ( ρ - 1 / 3 ⁢ u ⁢ ( ϕ ) ⁢ cos ⁡ ( ρ ⁢ f ~ ⁢ ( ϕ ) ) ⁢ Ai ⁡ ( - ρ 2 / 3 ⁢ Δ ⁢ ( ϕ ) ) ⁢ ( 1 + O ⁡ ( 1 / ρ ) ) + ρ - 2 / 3 ⁢ v ⁢ ( ϕ ) ⁢ sin ⁡ ( ρ ⁢ f ~ ⁢ ( ϕ ) ) ⁢ Ai ′ ⁡ ( - ρ 2 / 3 ⁢ Δ ⁢ ( ϕ ) ) ⁢ ( 1 + O ⁡ ( 1 / ρ ) ) ) , 𝑧 𝜌 italic-ϕ 2 𝜋 superscript 𝜌 1 3 𝑢 italic-ϕ 𝜌 ~ 𝑓 italic-ϕ Airy-Ai superscript 𝜌 2 3 Δ italic-ϕ 1 Big-O 1 𝜌 superscript 𝜌 2 3 𝑣 italic-ϕ 𝜌 ~ 𝑓 italic-ϕ diffop Airy-Ai 1 superscript 𝜌 2 3 Δ italic-ϕ 1 Big-O 1 𝜌 {\displaystyle{\displaystyle z(\rho,\phi)=2\pi\left(\rho^{-1/3}u(\phi)\cos% \left(\rho\widetilde{f}(\phi)\right)\mathrm{Ai}\left(-\rho^{2/3}\Delta(\phi)% \right)\*(1+O\left(1/\rho\right))+\rho^{-2/3}v(\phi)\sin\left(\rho\widetilde{f% }(\phi)\right)\mathrm{Ai}'\left(-\rho^{2/3}\Delta(\phi)\right)\*(1+O\left(1/% \rho\right))\right),}}
    0 references
    DLMF id
    DLMF:36.13.E8
    0 references
    constraint
    ρ → ∞ → 𝜌 {\displaystyle{\displaystyle\rho\to\infty}}
    0 references
    Symbols used
    Q10919
    Defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1adec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
     
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