DLMF:36.13.E8 (Q10709)

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DLMF:36.13.E8
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    Statements

    DLMF 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
    constraint
    ρ → ∞ → 𝜌 {\displaystyle{\displaystyle\rho\to\infty}}
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    Symbols used
    Q10893
    DLMF defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1adec
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    order not exceeding
    DLMF 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
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    Q10751
    DLMF defining formula
    ( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6aadec
    0 references
    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    real parameter
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C36.S1.XMD11.m1adec
    0 references
    function
    DLMF defining formula
    u ⁢ ( ϕ ) 𝑢 italic-ϕ {\displaystyle{\displaystyle u(\phi)}}
    xml-id
    C36.S13.XMD4.m1adec
    0 references
    function
    DLMF defining formula
    v ⁢ ( ϕ ) 𝑣 italic-ϕ {\displaystyle{\displaystyle v(\phi)}}
    xml-id
    C36.S13.XMD5.m1adec
    0 references

    Identifiers

     
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