DLMF:1.14.E44 (Q536)

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DLMF:1.14.E44
No description defined

    Statements

    Defining formula
    f ⁢ ( x ) = - 1 π ⁢ ⨍ - ∞ ∞ ℋ ⁡ f ⁡ ( u ) u - x ⁢ d u . 𝑓 𝑥 1 𝜋 subscript superscript principal-value-integral Hilbert-transform 𝑓 𝑢 𝑢 𝑥 𝑢 {\displaystyle{\displaystyle f(x)=-\frac{1}{\pi}\pvint^{\infty}_{-\infty}\frac% {\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(u\right)}{u-x}\mathrm{d}u.}}
    0 references
    DLMF id
    DLMF:1.14.E44
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    Symbols used
    Q10895
    Defining formula
    ℋ ⁡ ( f ) ⁡ ( x ) Hilbert-transform 𝑓 𝑥 {\displaystyle{\displaystyle\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}% \right)}}
    xml-id
    C1.S14.SS5.p1.m2acdec
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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.m2asdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aagdec
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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.m2abdec
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