DLMF:1.16.E51 (Q659)

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DLMF:1.16.E51
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    Statements

    Defining formula
    ⟨ ℱ ⁡ ( H ) , ϕ ⟩ = π 2 ⁢ ϕ ⁢ ( 0 ) + i 2 ⁢ π ⁢ ⨍ - ∞ ∞ ϕ ⁢ ( x ) x ⁢ d x . distribution distributional-Fourier-transform Heaviside-H italic-ϕ 2 italic-ϕ 0 imaginary-unit 2 subscript superscript principal-value-integral italic-ϕ 𝑥 𝑥 𝑥 {\displaystyle{\displaystyle\left\langle\mathscr{F}\left(H\right),\phi\right% \rangle=\sqrt{\frac{\pi}{2}}\phi(0)+\frac{\mathrm{i}}{\sqrt{2\pi}}\pvint^{% \infty}_{-\infty}\frac{\phi(x)}{x}\mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:1.16.E51
    0 references
    Symbols used
    Heaviside function
    Defining formula
    H ⁡ ( x ) Heaviside-H 𝑥 {\displaystyle{\displaystyle H\left(\NVar{x}\right)}}
    xml-id
    C1.S16.E13.m2ajdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aldec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
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    Q10905
    Defining formula
    ⟨ Λ , ϕ ⟩ distribution Λ italic-ϕ {\displaystyle{\displaystyle\left\langle\NVar{\Lambda},\NVar{\phi}\right% \rangle}}
    xml-id
    C1.S16.SS1.p5.m5asdec
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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.m2aldec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aldec
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    Cauchy principal value
    Defining formula
    ⨍ a b superscript subscript principal-value-integral 𝑎 𝑏 {\displaystyle{\displaystyle\pvint_{\NVar{a}}^{\NVar{b}}}}
    xml-id
    C1.S4.E24.m2aadec
    0 references
    Q10910
    Defining formula
    ϕ ⁢ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}
    xml-id
    C1.S16.XMD7.m1edec
    0 references
     
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