DLMF:1.14.E46 (Q538): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10895 / rank
 
Normal rank
Property / Symbols used: Q10895 / qualifier
 
test:

ℋ ⁡ ( f ) ⁡ ( x ) Hilbert-transform 𝑓 𝑥 {\displaystyle{\displaystyle\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}% \right)}}

\Hilberttrans@{\NVar{f}}@{\NVar{x}}
Property / Symbols used: Q10895 / qualifier
 
xml-id: C1.S14.SS5.p1.m2aedec

Revision as of 15:27, 1 January 2020

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DLMF:1.14.E46
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    Statements

    test
    1 2 ⁢ π ⁢ ∫ - ∞ ∞ ℋ ⁡ f ⁡ ( u ) ⁢ e i ⁢ u ⁢ x ⁢ d u = - i ⁢ ( sign ⁡ x ) ⁢ ℱ ⁡ f ⁡ ( x ) , 1 2 𝜋 superscript subscript Hilbert-transform 𝑓 𝑢 superscript 𝑒 𝑖 𝑢 𝑥 𝑢 imaginary-unit sign 𝑥 Fourier-transform 𝑓 𝑥 {\displaystyle{\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}% \mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(u\right)e^{iux}\mathrm{d}u=-% \mathrm{i}(\operatorname{sign}x)\mathscr{F}\mskip-3.0mu f\mskip 3.0mu \left(x% \right),}}
    0 references
    DLMF id
    DLMF:1.14.E46
    0 references
    Symbols used
    Fourier transform
    test
    ℱ ⁡ ( f ) ⁡ ( s ) Fourier-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathscr{F}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E1.m2aadec
    0 references
    Q10895
    test
    ℋ ⁡ ( f ) ⁡ ( x ) Hilbert-transform 𝑓 𝑥 {\displaystyle{\displaystyle\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}% \right)}}
    xml-id
    C1.S14.SS5.p1.m2aedec
    0 references
     
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