DLMF:36.11.E8 (Q10683)

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DLMF:36.11.E8
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    Statements

    DLMF defining formula
    Ψ ( H ) ⁡ ( 0 , 0 , z ) = 2 ⁢ π z ⁢ ( 1 - i 3 ⁢ exp ⁡ ( 1 27 ⁢ i ⁢ z 3 ) + o ⁡ ( 1 ) ) , hyperbolic-umbilic-canonical-integral 0 0 𝑧 2 𝜋 𝑧 1 𝑖 3 1 27 𝑖 superscript 𝑧 3 little-o 1 {\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{% z}\left(1-\frac{i}{\sqrt{3}}\exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right% )\right),}}
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    constraint
    z → ± ∞ → 𝑧 plus-or-minus {\displaystyle{\displaystyle z\to\pm\infty}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
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    exponential function
    DLMF defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2aedec
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    hyperbolic umbilic canonical integral function
    DLMF defining formula
    Ψ ( H ) ⁡ ( 𝐱 ) hyperbolic-umbilic-canonical-integral 𝐱 {\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)}}
    xml-id
    C36.S2.E5.m4adec
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    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
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    order less than
    DLMF defining formula
    o ⁡ ( x ) little-o 𝑥 {\displaystyle{\displaystyle o\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E2.m2afdec
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    real parameter
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C36.S1.XMD11.m1bdec
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    Identifiers

     
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