DLMF:36.2.E11 (Q9847)

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DLMF:36.2.E11
No description defined

    Statements

    Defining formula
    Ψ ( U ) ⁡ ( 𝐱 ; k ) = k ⁢ ∫ - ∞ ∞ ∫ - ∞ ∞ exp ⁡ ( i ⁢ k ⁢ Φ ( U ) ⁡ ( s , t ; 𝐱 ) ) ⁢ d s ⁢ d t , diffraction-umbilic-catastrophe 𝐱 𝑘 𝑘 superscript subscript superscript subscript 𝑖 𝑘 umbilic-catastrophe 𝑠 𝑡 𝐱 𝑠 𝑡 {\displaystyle{\displaystyle\Psi^{(\mathrm{U})}(\mathbf{x};k)=k\int_{-\infty}^% {\infty}\int_{-\infty}^{\infty}\exp\left(ik\Phi^{(\mathrm{U})}\left(s,t;% \mathbf{x}\right)\right)\mathrm{d}s\mathrm{d}t,}}
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    DLMF id
    DLMF:36.2.E11
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    constraint
    k > 0 𝑘 0 {\displaystyle{\displaystyle k>0}}
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    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2agdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2agdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
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    Q12749
    Defining formula
    Φ ( U ) ⁡ ( s , t ; 𝐱 ) umbilic-catastrophe 𝑠 𝑡 𝐱 {\displaystyle{\displaystyle\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{% \mathbf{x}}\right)}}
    xml-id
    C36.S2.Px2.m2aadec
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    Q12750
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C36.S1.XMD4.m1adec
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    Q12742
    Defining formula
    t 𝑡 {\displaystyle{\displaystyle t}}
    xml-id
    C36.S1.XMD5.m1fdec
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    Q12747
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C36.S1.XMD6.m1ddec
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