DLMF:9.12.E29 (Q2957)

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DLMF:9.12.E29
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    Statements

    test
    Hi ⁡ ( z ) ∼ - 1 π ⁢ z ⁢ ∑ k = 0 ∞ ( 3 ⁢ k ) ! k ! ⁢ ( 3 ⁢ z 3 ) k + e ζ π ⁢ z 1 / 4 ⁢ ∑ k = 0 ∞ u k ζ k , asymptotic-expansion Scorer-Hi 𝑧 1 𝜋 𝑧 superscript subscript 𝑘 0 3 𝑘 𝑘 superscript 3 superscript 𝑧 3 𝑘 superscript 𝑒 𝜁 𝜋 superscript 𝑧 1 4 superscript subscript 𝑘 0 subscript 𝑢 𝑘 superscript 𝜁 𝑘 {\displaystyle{\displaystyle\mathrm{Hi}\left(z\right)\sim-\frac{1}{\pi z}\sum_% {k=0}^{\infty}\frac{(3k)!}{k!(3z^{3})^{k}}+\frac{e^{\zeta}}{\sqrt{\pi}z^{1/4}}% \sum_{k=0}^{\infty}\frac{u_{k}}{\zeta^{k}},}}
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    DLMF id
    DLMF:9.12.E29
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    constraint
    | ph ⁡ z | ≤ π - δ phase 𝑧 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\pi-\delta}}
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    Symbols used
    Scorer function (inhomogeneous Airy function)
    test
    Hi ⁡ ( z ) Scorer-Hi 𝑧 {\displaystyle{\displaystyle\mathrm{Hi}\left(\NVar{z}\right)}}
    xml-id
    C9.S12.E5.m2ardec
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    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11addec
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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.m2avdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
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    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5ahdec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ahdec
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    Q11385
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C9.S1.XMD1.m1hdec
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    Q11381
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1xdec
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    Q11386
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C9.S1.XMD4.m1ddec
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    change of variable (locally)
    test
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S7.XMD1.m1dec
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    Q11387
    test
    u s subscript 𝑢 𝑠 {\displaystyle{\displaystyle u_{s}}}
    xml-id
    C9.S7.XMD2.m1dec
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