Formula:507

Digital Library of Mathematical Functions ID 1.14.E46

${\displaystyle{\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(u\right)e^{iux}\mathrm{d}u=-\mathrm{i}(\operatorname{sign}x)\mathscr{F}\mskip-3.0mu f\mskip 3.0mu \left(x\right),}}$

Symbols List

• ${\displaystyle{\displaystyle\mathscr{F}\left(\NVar{f}\right)\left(\NVar{s}\right)}}$: Fourier transform
• ${\displaystyle{\displaystyle\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}$:
 

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