DLMF:9.13.E10 (Q2972)

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DLMF:9.13.E10
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    Statements

    Defining formula
    A n ⁡ ( - z ) = { 2 ⁢ p / π ⁢ cos ⁡ ( 1 2 ⁢ p ⁢ π ) ⁢ z - n / 4 ⁢ ( cos ⁡ ( ζ - 1 4 ⁢ π ) + e | ℑ ⁡ ζ | ⁢ O ⁡ ( ζ - 1 ) ) , | ph ⁡ z | ≤ 2 ⁢ p ⁢ π - δ ,  n  odd , p / π ⁢ z - n / 4 ⁢ e ζ ⁢ ( 1 + O ⁡ ( ζ - 1 ) ) , | ph ⁡ z | ≤ p ⁢ π - δ ,  n  even , ODE-generalized-Airy-A 𝑛 𝑧 cases 2 𝑝 𝜋 1 2 𝑝 𝜋 superscript 𝑧 𝑛 4 𝜁 1 4 𝜋 superscript 𝑒 𝜁 Big-O superscript 𝜁 1 | ph ⁡ z | ≤ 2 ⁢ p ⁢ π - δ ,  n  odd 𝑝 𝜋 superscript 𝑧 𝑛 4 superscript 𝑒 𝜁 1 Big-O superscript 𝜁 1 | ph ⁡ z | ≤ p ⁢ π - δ ,  n  even {\displaystyle{\displaystyle A_{n}\left(-z\right)=\begin{cases}2\sqrt{p/\pi}% \cos\left(\tfrac{1}{2}p\pi\right)z^{-n/4}\left(\cos\left(\zeta-\tfrac{1}{4}\pi% \right)+e^{|\Im\zeta|}O\left(\zeta^{-1}\right)\right),&\text{$|\operatorname{% ph}z|\leq 2p\pi-\delta$, $n$ odd},\\ \sqrt{p/\pi}z^{-n/4}e^{\zeta}\left(1+O\left(\zeta^{-1}\right)\right),&\text{$|% \operatorname{ph}z|\leq p\pi-\delta$, $n$ even},\end{cases}}}
    0 references
    DLMF id
    DLMF:9.13.E10
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    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aadec
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    Q11416
    Defining formula
    A n ⁡ ( z ) ODE-generalized-Airy-A 𝑛 𝑧 {\displaystyle{\displaystyle A_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C9.S13.SS1.p2.m3afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
     
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