DLMF:1.16.E49 (Q657): Difference between revisions

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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.m2ajdec

Revision as of 15:41, 1 January 2020

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DLMF:1.16.E49
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    Statements

    test
    ⟨ ℱ ⁡ ( sign ) , ϕ ⟩ = i ⁢ 2 π ⁢ ⨍ - ∞ ∞ ϕ ⁢ ( x ) x ⁢ d x . distribution distributional-Fourier-transform sign italic-ϕ imaginary-unit 2 subscript superscript principal-value-integral italic-ϕ 𝑥 𝑥 𝑥 {\displaystyle{\displaystyle\left\langle\mathscr{F}\left(\operatorname{sign}% \right),\phi\right\rangle=\mathrm{i}\sqrt{\frac{2}{\pi}}\pvint^{\infty}_{-% \infty}\frac{\phi(x)}{x}\mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:1.16.E49
    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.m2ajdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
    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.m5ardec
    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.m2ajdec
    0 references
     
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