Formula:628

Digital Library of Mathematical Functions ID 1.16.E50

${\displaystyle{\displaystyle\mathscr{F}\left(H\right)=\frac{1}{2}\mathscr{F}\left(1+\operatorname{sign}\right)=\frac{1}{2}\left[\mathscr{F}\left(1\right)+\mathscr{F}\left(\operatorname{sign}\right)\right]=\sqrt{\frac{\pi}{2}}\left(\delta+\frac{\mathrm{i}}{\pi x}\right),}}$

Symbols List

• ${\displaystyle{\displaystyle H\left(\NVar{x}\right)}}$: Heaviside function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\delta_{x}}}$:
• ${\displaystyle{\displaystyle\mathscr{F}\left(\NVar{u}\right)}}$: Fourier transform of a tempered distribution
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}$:
 

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