DLMF:1.16.E50 (Q658)

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DLMF:1.16.E50
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    Statements

    Defining formula
    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),}}
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    DLMF id
    DLMF:1.16.E50
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    Symbols used
    Heaviside function
    Defining formula
    H ⁡ ( x ) Heaviside-H 𝑥 {\displaystyle{\displaystyle H\left(\NVar{x}\right)}}
    xml-id
    C1.S16.E13.m2aidec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2akdec
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    Q10908
    Defining formula
    δ x Dirac-delta-distribution 𝑥 {\displaystyle{\displaystyle\delta_{x}}}
    xml-id
    C1.S16.SS3.m1aldec
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    Fourier transform of a tempered distribution
    Defining formula
    ℱ ⁡ ( u ) distributional-Fourier-transform 𝑢 {\displaystyle{\displaystyle\mathscr{F}\left(\NVar{u}\right)}}
    xml-id
    C1.S16.E35.m2akdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2akdec
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    Q10897
    Defining formula
    sign ⁡ x sign 𝑥 {\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}
    xml-id
    introduction.Sx4.p2.t1.r18.m6afdec
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