DLMF:1.16.E41 (Q649): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Fourier transform of a test function (locally) / rank
 
Normal rank
Property / Symbols used: Fourier transform of a test function (locally) / qualifier
 
test:

( ϕ ) italic-ϕ {\displaystyle{\displaystyle\mathscr{F}(\phi)}}

\mathscr{F}(\phi)
Property / Symbols used: Fourier transform of a test function (locally) / qualifier
 
xml-id: C1.S16.XMD15.m1edec
Property / Symbols used
 
Property / Symbols used: Q10910 / rank
 
Normal rank
Property / Symbols used: Q10910 / qualifier
 
test:

ϕ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}

\phi(x)
Property / Symbols used: Q10910 / qualifier
 
xml-id: C1.S16.XMD7.m1cdec

Latest revision as of 15:40, 1 January 2020

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DLMF:1.16.E41
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    Statements

    ( 1 ) , ϕ = 2 π ( ( δ ) ) , ϕ = 2 π ( δ ) , ( ϕ ) = 2 π δ , ( ( ϕ ) ) = 2 π δ , ϕ - = 2 π ϕ ( 0 ) , distribution distributional-Fourier-transform 1 italic-ϕ 2 distribution distributional-Fourier-transform distributional-Fourier-transform delta-distribution italic-ϕ 2 distribution distributional-Fourier-transform delta-distribution italic-ϕ 2 distribution delta-distribution italic-ϕ 2 distribution delta-distribution subscript italic-ϕ 2 italic-ϕ 0 {\displaystyle{\displaystyle\left\langle\mathscr{F}\left(1\right),\phi\right% \rangle=\sqrt{2\pi}\left\langle\mathscr{F}\left(\mathscr{F}\left(\delta\right)% \right),\phi\right\rangle=\sqrt{2\pi}\left\langle\mathscr{F}\left(\delta\right% ),\mathscr{F}(\phi)\right\rangle=\sqrt{2\pi}\left\langle\delta,\mathscr{F}(% \mathscr{F}(\phi))\right\rangle=\sqrt{2\pi}\left\langle\delta,\phi_{-}\right% \rangle=\sqrt{2\pi}\phi(0),}}
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    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2aedec
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    δ x Dirac-delta-distribution 𝑥 {\displaystyle{\displaystyle\delta_{x}}}
    C1.S16.SS3.m1ahdec
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    Λ , ϕ distribution Λ italic-ϕ {\displaystyle{\displaystyle\left\langle\NVar{\Lambda},\NVar{\phi}\right% \rangle}}
    C1.S16.SS1.p5.m5aqdec
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    ( u ) distributional-Fourier-transform 𝑢 {\displaystyle{\displaystyle\mathscr{F}\left(\NVar{u}\right)}}
    C1.S16.E35.m2aedec
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    ( ϕ ) italic-ϕ {\displaystyle{\displaystyle\mathscr{F}(\phi)}}
    C1.S16.XMD15.m1edec
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    ϕ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}
    C1.S16.XMD7.m1cdec
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