DLMF:1.16.E38 (Q646)

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DLMF:1.16.E38
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    Statements

    test
    ⟨ 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
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    Q10908
    test
    δ x Dirac-delta-distribution 𝑥 {\displaystyle{\displaystyle\delta_{x}}}
    xml-id
    C1.S16.SS3.m1afdec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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