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DLMF:1.14.E42 - MaRDI portal

DLMF:1.14.E42 (Q503)

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DLMF:1.14.E42
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    Statements

    DLMF defining formula
    ℋ ⁡ f ⁡ ( x ) = lim y → 0 + ⁡ 1 π ⁢ ∫ - ∞ ∞ t - x ( t - x ) 2 + y 2 ⁢ f ⁢ ( t ) ⁢ d t , Hilbert-transform 𝑓 𝑥 subscript → 𝑦 limit-from 0 1 𝜋 subscript superscript 𝑡 𝑥 superscript 𝑡 𝑥 2 superscript 𝑦 2 𝑓 𝑡 𝑡 {\displaystyle{\displaystyle\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x% \right)=\lim_{y\to 0+}\frac{1}{\pi}\int^{\infty}_{-\infty}\frac{t-x}{(t-x)^{2}% +y^{2}}f(t)\mathrm{d}t,}}
    0 references
    Symbols used
    Q10869
    DLMF 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.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aqdec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaedec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aacdec
    0 references

    Identifiers

     
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