DLMF:1.16.E38 (Q646)

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DLMF:1.16.E38
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    Statements

    Defining formula
    ⟨ ℱ ⁡ ( δ ) , ϕ ⟩ = ⟨ δ , ℱ ⁢ ( ϕ ) ⟩ = ⟨ δ , 1 2 ⁢ π ⁢ ∫ - ∞ ∞ ϕ ⁢ ( t ) ⁢ e i ⁢ x ⁢ t ⁢ d t ⟩ = 1 2 ⁢ π ⁢ ∫ - ∞ ∞ ϕ ⁢ ( t ) ⁢ d t = 1 2 ⁢ π ⁢ ⟨ 1 , ϕ ⟩ , distribution distributional-Fourier-transform delta-distribution italic-ϕ distribution delta-distribution ℱ italic-ϕ distribution delta-distribution 1 2 subscript superscript italic-ϕ 𝑡 imaginary-unit 𝑥 𝑡 𝑡 1 2 superscript subscript italic-ϕ 𝑡 𝑡 1 2 distribution 1 italic-ϕ {\displaystyle{\displaystyle\left\langle\mathscr{F}\left(\delta\right),\phi% \right\rangle=\left\langle\delta,\mathscr{F}(\phi)\right\rangle=\left\langle% \delta,\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}\phi(t){\mathrm{e}^{\mathrm% {i}xt}}\mathrm{d}t\right\rangle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}% \phi(t)\mathrm{d}t=\frac{1}{\sqrt{2\pi}}\left\langle 1,\phi\right\rangle,}}
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    DLMF id
    DLMF:1.16.E38
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    constraint
    ϕ ∈ 𝒯 italic-ϕ 𝒯 {\displaystyle{\displaystyle\phi\in\mathcal{T}}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    Q10908
    Defining formula
    δ x Dirac-delta-distribution 𝑥 {\displaystyle{\displaystyle\delta_{x}}}
    xml-id
    C1.S16.SS3.m1afdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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    Q10905
    Defining formula
    ⟨ Λ , ϕ ⟩ distribution Λ italic-ϕ {\displaystyle{\displaystyle\left\langle\NVar{\Lambda},\NVar{\phi}\right% \rangle}}
    xml-id
    C1.S16.SS1.p5.m5apdec
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    Fourier transform of a tempered distribution
    Defining formula
    ℱ ⁡ ( u ) distributional-Fourier-transform 𝑢 {\displaystyle{\displaystyle\mathscr{F}\left(\NVar{u}\right)}}
    xml-id
    C1.S16.E35.m2acdec
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    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2ahdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
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    Q10914
    Defining formula
    𝒯 𝒯 {\displaystyle{\displaystyle\mathcal{T}}}
    xml-id
    C1.S16.XMD12.m1adec
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    Fourier transform of a test function (locally)
    Defining formula
    ℱ ⁢ ( ϕ ) ℱ italic-ϕ {\displaystyle{\displaystyle\mathscr{F}(\phi)}}
    xml-id
    C1.S16.XMD15.m1ddec
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    Q10910
    Defining formula
    ϕ ⁢ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}
    xml-id
    C1.S16.XMD7.m1bdec
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