DLMF:9.7.E13 (Q2820)

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DLMF:9.7.E13
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    Statements

    test
    Bi ⁡ ( z ⁢ e ± π ⁢ i / 3 ) ∼ 2 π ⁢ e ± π ⁢ i / 6 z 1 / 4 ⁢ ( cos ⁡ ( ζ - 1 4 ⁢ π ∓ 1 2 ⁢ i ⁢ ln ⁡ 2 ) ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ u 2 ⁢ k ζ 2 ⁢ k + sin ⁡ ( ζ - 1 4 ⁢ π ∓ 1 2 ⁢ i ⁢ ln ⁡ 2 ) ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ u 2 ⁢ k + 1 ζ 2 ⁢ k + 1 ) , asymptotic-expansion Airy-Bi 𝑧 superscript 𝑒 plus-or-minus 𝜋 𝑖 3 2 𝜋 superscript 𝑒 plus-or-minus 𝜋 𝑖 6 superscript 𝑧 1 4 minus-or-plus 𝜁 1 4 𝜋 1 2 imaginary-unit 2 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑢 2 𝑘 superscript 𝜁 2 𝑘 minus-or-plus 𝜁 1 4 𝜋 1 2 imaginary-unit 2 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑢 2 𝑘 1 superscript 𝜁 2 𝑘 1 {\displaystyle{\displaystyle\mathrm{Bi}\left(ze^{\pm\pi i/3}\right)\mathrel{% \sim}\sqrt{\frac{2}{\pi}}\frac{e^{\pm\pi i/6}}{z^{1/4}}\*\left(\cos\left(\zeta% -\tfrac{1}{4}\pi\mp\tfrac{1}{2}\mathrm{i}\ln 2\right)\sum_{k=0}^{\infty}(-1)^{% k}\frac{u_{2k}}{\zeta^{2k}}+\sin\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}% \mathrm{i}\ln 2\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}}% \right),}}
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    DLMF id
    DLMF:9.7.E13
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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.m2addec
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    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11ahdec
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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.m2ajdec
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    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ahdec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2addec
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    Q11385
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C9.S1.XMD1.m1idec
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    Q11381
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1idec
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    Q11386
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C9.S1.XMD4.m1hdec
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    change of variable (locally)
    test
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S7.XMD1.m1idec
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    Q11387
    test
    u s subscript 𝑢 𝑠 {\displaystyle{\displaystyle u_{s}}}
    xml-id
    C9.S7.XMD2.m1edec
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