DLMF:9.7.E11 (Q2818)

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DLMF:9.7.E11
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    Statements

    test
    Bi ⁡ ( - z ) ∼ 1 π ⁢ z 1 / 4 ⁢ ( - sin ⁡ ( ζ - 1 4 ⁢ π ) ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ u 2 ⁢ k ζ 2 ⁢ k + cos ⁡ ( ζ - 1 4 ⁢ π ) ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ u 2 ⁢ k + 1 ζ 2 ⁢ k + 1 ) , asymptotic-expansion Airy-Bi 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{Bi}\left(-z\right)\sim\frac{1}{\sqrt{\pi}z% ^{1/4}}\left(-\sin\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k% }\frac{u_{2k}}{\zeta^{2k}}+\cos\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{% \infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}}\right),}}
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    DLMF id
    DLMF:9.7.E11
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    constraint
    | ph ⁡ z | ≤ 2 3 ⁢ π - δ phase 𝑧 2 3 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta}}
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    Symbols used
    Q11081
    test
    Bi ⁡ ( z ) Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m2abdec
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    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11afdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1afdec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
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    Q11385
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C9.S1.XMD1.m1gdec
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    Q11381
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1gdec
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    Q11386
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C9.S1.XMD4.m1fdec
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    change of variable (locally)
    test
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S7.XMD1.m1gdec
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    Q11387
    test
    u s subscript 𝑢 𝑠 {\displaystyle{\displaystyle u_{s}}}
    xml-id
    C9.S7.XMD2.m1ddec
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