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.}}
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    DLMF id
    DLMF:1.16.E51
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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.m2ajdec
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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.m2aldec
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