1.17.E8 (Q637)

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1.17.E8
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    Statements

    DLMF defining formula
    1 2 ⁢ π ⁢ ∫ - ∞ ∞ e - i ⁢ a ⁢ t ⁢ ( ∫ - ∞ ∞ ϕ ⁢ ( x ) ⁢ e i ⁢ t ⁢ x ⁢ d x ) ⁢ d t = ϕ ⁢ ( a ) 1 2 𝜋 superscript subscript superscript 𝑒 𝑖 𝑎 𝑡 superscript subscript italic-ϕ 𝑥 superscript 𝑒 𝑖 𝑡 𝑥 𝑥 𝑡 italic-ϕ 𝑎 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iat}\left% (\int_{-\infty}^{\infty}\phi(x)e^{itx}\mathrm{d}x\right)\mathrm{d}t=\phi(a)}}
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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.m2acdec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
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    continuous function
    DLMF defining formula
    ϕ ⁢ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}
    xml-id
    C1.S17.XMD1.m1ddec
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    Identifiers

     
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