DLMF:1.14.E42 (Q534): Difference between revisions

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DLMF:1.14.E42
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    Statements

    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,}}
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    DLMF id
    DLMF:1.14.E42
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