There are the different definition of Fourier transform (i.e scaling constant or sign on the kernel). How can I show if the following is a correct Fourier transform pair:
$\hat{f}(\omega)=\int_{-\infty }^{\infty }f(t)e^{i\omega t}\frac{dt}{\sqrt{2\pi}}$
$f(t)=\int_{-\infty }^{\infty }\hat{f}(\omega)e^{-i\omega t}\frac{dw}{\sqrt{2\pi}}$
How I can show that this this is/or not a valid Fourier transform pair?
Thanks in advance!!
To get the constants exactly right in a particular choice of Fourier transform and inverse transform, I think there's no truly elementary heuristic... nor truly elementary proof... apart from remembering one correct pair, and then changing variables.
The immemorability of constants is illustrated, I think, by $\int_{\mathbb R} e^{2\pi i\xi x}\,dx=\delta(\xi)$, equivalent to many heuristics for Fourier inversion. Why the $2\pi$? I do not have any good rationalization for it.