DLMF:1.14.E46 (Q538)

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DLMF:1.14.E46
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    1 2 ⁢ π ⁢ ∫ - ∞ ∞ ℋ ⁡ f ⁡ ( u ) ⁢ e i ⁢ u ⁢ x ⁢ d u = - i ⁢ ( sign ⁡ x ) ⁢ ℱ ⁡ f ⁡ ( x ) , 1 2 𝜋 superscript subscript Hilbert-transform 𝑓 𝑢 superscript 𝑒 𝑖 𝑢 𝑥 𝑢 imaginary-unit sign 𝑥 Fourier-transform 𝑓 𝑥 {\displaystyle{\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}% \mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(u\right)e^{iux}\mathrm{d}u=-% \mathrm{i}(\operatorname{sign}x)\mathscr{F}\mskip-3.0mu f\mskip 3.0mu \left(x% \right),}}
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    DLMF id
    DLMF:1.14.E46
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