DLMF:1.16.E50 (Q658): Difference between revisions

From testwiki
Jump to navigation Jump to search
imported>Admin
 
imported>Admin
 
(5 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: Heaviside function / rank
 
Normal rank
Property / Symbols used: Heaviside function / qualifier
 
test:

H ⁡ ( x ) Heaviside-H 𝑥 {\displaystyle{\displaystyle H\left(\NVar{x}\right)}}

\HeavisideH@{\NVar{x}}
Property / Symbols used: Heaviside function / qualifier
 
xml-id: C1.S16.E13.m2aidec
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.m2akdec
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.m1aldec
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.m2akdec
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: 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.m6afdec

Latest revision as of 15:41, 1 January 2020

No description defined
Language Label Description Also known as
English
DLMF:1.16.E50
No description defined

    Statements

    test
    ℱ ⁡ ( H ) = 1 2 ⁢ ℱ ⁡ ( 1 + sign ) = 1 2 ⁢ [ ℱ ⁡ ( 1 ) + ℱ ⁡ ( sign ) ] = π 2 ⁢ ( δ + i π ⁢ x ) , distributional-Fourier-transform Heaviside-H 1 2 distributional-Fourier-transform 1 sign 1 2 delimited-[] distributional-Fourier-transform 1 distributional-Fourier-transform sign 2 delta-distribution imaginary-unit 𝑥 {\displaystyle{\displaystyle\mathscr{F}\left(H\right)=\frac{1}{2}\mathscr{F}% \left(1+\operatorname{sign}\right)=\frac{1}{2}\left[\mathscr{F}\left(1\right)+% \mathscr{F}\left(\operatorname{sign}\right)\right]=\sqrt{\frac{\pi}{2}}\left(% \delta+\frac{\mathrm{i}}{\pi x}\right),}}
    0 references
    DLMF id
    DLMF:1.16.E50
    0 references
    Symbols used
    Heaviside function
    test
    H ⁡ ( x ) Heaviside-H 𝑥 {\displaystyle{\displaystyle H\left(\NVar{x}\right)}}
    xml-id
    C1.S16.E13.m2aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2akdec
    0 references
    Q10908
    test
    δ x Dirac-delta-distribution 𝑥 {\displaystyle{\displaystyle\delta_{x}}}
    xml-id
    C1.S16.SS3.m1aldec
    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.m2akdec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2akdec
    0 references
    Q10897
    test
    sign ⁡ x sign 𝑥 {\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}
    xml-id
    introduction.Sx4.p2.t1.r18.m6afdec
    0 references
     
      Retrieved from "https://lct.wmflabs.org/w/index.php?title=Item:Q658&oldid=3511"