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

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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.m2ajdec
Property / Symbols used
 
Property / Symbols used: Cauchy principal value / rank
 
Normal rank
Property / Symbols used: Cauchy principal value / qualifier
 
test:

⨍ a b superscript subscript principal-value-integral 𝑎 𝑏 {\displaystyle{\displaystyle\pvint_{\NVar{a}}^{\NVar{b}}}}

\pvint_{\NVar{a}}^{\NVar{b}}
Property / Symbols used: Cauchy principal value / qualifier
 
xml-id: C1.S4.E24.m2adec
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.m6aedec
Property / Symbols used
 
Property / Symbols used: Q10910 / rank
 
Normal rank
Property / Symbols used: Q10910 / qualifier
 
test:

ϕ ⁢ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}

\phi(x)
Property / Symbols used: Q10910 / qualifier
 
xml-id: C1.S16.XMD7.m1ddec

Latest revision as of 15:41, 1 January 2020

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DLMF:1.16.E49
No description defined

    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
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ajdec
    0 references
    Cauchy principal value
    test
    ⨍ a b superscript subscript principal-value-integral 𝑎 𝑏 {\displaystyle{\displaystyle\pvint_{\NVar{a}}^{\NVar{b}}}}
    xml-id
    C1.S4.E24.m2adec
    0 references
    Q10897
    test
    sign ⁡ x sign 𝑥 {\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}
    xml-id
    introduction.Sx4.p2.t1.r18.m6aedec
    0 references
    Q10910
    test
    ϕ ⁢ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}
    xml-id
    C1.S16.XMD7.m1ddec
    0 references
     
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