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

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Property / Symbols used
 
Property / Symbols used: Q10908 / rank
 
Normal rank
Property / Symbols used: Q10908 / qualifier
 
test:

δ x Dirac-delta-distribution 𝑥 {\displaystyle{\displaystyle\delta_{x}}}

\deltaDistribution[x]
Property / Symbols used: Q10908 / qualifier
 
xml-id: C1.S16.SS3.m1ahdec
Property / Symbols used
 
Property / Symbols used: Q10905 / rank
 
Normal rank
Property / Symbols used: Q10905 / qualifier
 
test:

⟨ Λ , ϕ ⟩ distribution Λ italic-ϕ {\displaystyle{\displaystyle\left\langle\NVar{\Lambda},\NVar{\phi}\right% \rangle}}

\intinnerprod{\NVar{\Lambda}}{\NVar{\phi}}
Property / Symbols used: Q10905 / qualifier
 
xml-id: C1.S16.SS1.p5.m5aqdec
Property / Symbols used
 
Property / Symbols used: Fourier transform of a tempered distribution / rank
 
Normal rank
Property / Symbols used: Fourier transform of a tempered distribution / qualifier
 
test:

ℱ ⁡ ( u ) distributional-Fourier-transform 𝑢 {\displaystyle{\displaystyle\mathscr{F}\left(\NVar{u}\right)}}

\Fouriertransdist@{\NVar{u}}
Property / Symbols used: Fourier transform of a tempered distribution / qualifier
 
xml-id: C1.S16.E35.m2aedec
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:

ℱ ⁢ ( ϕ ) ℱ 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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Language Label Description Also known as
English
DLMF:1.16.E41
No description defined

    Statements

    test
    ⟨ ℱ ⁡ ( 1 ) , ϕ ⟩ = 2 ⁢ π ⁢ ⟨ ℱ ⁡ ( ℱ ⁡ ( δ ) ) , ϕ ⟩ = 2 ⁢ π ⁢ ⟨ ℱ ⁡ ( δ ) , ℱ ⁢ ( ϕ ) ⟩ = 2 ⁢ π ⁢ ⟨ δ , ℱ ⁢ ( ℱ ⁢ ( 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 ℱ italic-ϕ 2 distribution delta-distribution ℱ ℱ 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),}}
    0 references
    DLMF id
    DLMF:1.16.E41
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    Q10908
    test
    δ x Dirac-delta-distribution 𝑥 {\displaystyle{\displaystyle\delta_{x}}}
    xml-id
    C1.S16.SS3.m1ahdec
    0 references
    Q10905
    test
    ⟨ Λ , ϕ ⟩ distribution Λ italic-ϕ {\displaystyle{\displaystyle\left\langle\NVar{\Lambda},\NVar{\phi}\right% \rangle}}
    xml-id
    C1.S16.SS1.p5.m5aqdec
    0 references
    Fourier transform of a tempered distribution
    test
    ℱ ⁡ ( u ) distributional-Fourier-transform 𝑢 {\displaystyle{\displaystyle\mathscr{F}\left(\NVar{u}\right)}}
    xml-id
    C1.S16.E35.m2aedec
    0 references
    Fourier transform of a test function (locally)
    test
    ℱ ⁢ ( ϕ ) ℱ italic-ϕ {\displaystyle{\displaystyle\mathscr{F}(\phi)}}
    xml-id
    C1.S16.XMD15.m1edec
    0 references
    Q10910
    test
    ϕ ⁢ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}
    xml-id
    C1.S16.XMD7.m1cdec
    0 references
     
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