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
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2atdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aaidec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2aldec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2akdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aafdec
Property / Symbols used
 
Property / Symbols used: Q10897 / rank
 
Normal rank
Property / Symbols used: Q10897 / qualifier
 
test:

sign ⁡ x sign 𝑥 {\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}

\sign@@{\NVar{x}}
Property / Symbols used: Q10897 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r18.m6adec

Latest 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
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2atdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaidec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aldec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2akdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aafdec
    0 references
    Q10897
    test
    sign ⁡ x sign 𝑥 {\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}
    xml-id
    introduction.Sx4.p2.t1.r18.m6adec
    0 references
     
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