Formula:616

Digital Library of Mathematical Functions ID 1.16.E38

${\displaystyle{\displaystyle\left\langle\mathscr{F}\left(\delta\right),\phi\right\rangle=\left\langle\delta,\mathscr{F}(\phi)\right\rangle=\left\langle\delta,\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}\phi(t){\mathrm{e}^{\mathrm{i}xt}}\mathrm{d}t\right\rangle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi(t)\mathrm{d}t=\frac{1}{\sqrt{2\pi}}\left\langle 1,\phi\right\rangle,}}$

Constraint(s)

${\displaystyle{\displaystyle\phi\in\mathcal{T}}}$

Symbols List

• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\delta_{x}}}$:
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle}}$:
• ${\displaystyle{\displaystyle\mathscr{F}\left(\NVar{u}\right)}}$: Fourier transform of a tempered distribution
• ${\displaystyle{\displaystyle\in}}$:
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\mathcal{T}}}$: space of test functions
• ${\displaystyle{\displaystyle\mathscr{F}(\phi)}}$: Fourier transform of a test function (locally)
• ${\displaystyle{\displaystyle\phi(x)}}$: test function
 

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