Definition of the Fourier Transform

Question

I've seen a few different definitions of the transform going from $f(t)$ to $F(\omega)$ with a bunch of $2\pi$s or $\sqrt{2\pi}$s thrown around, each giving different results. Same thing happens with the $\delta$ function, with WolframAlpha saying it is the Fourier Transform of $\frac{1}{\sqrt{2\pi}}$, which means $\delta(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-it\omega}dt$, but as stated on some other sources, the Dirac Delta is defined as $\delta(\omega)=\int_{-\infty}^{\infty}e^{+ikx}dx$. Also according to WolframAlpha, the forward and inverse transforms of $1$ are equal to the same thing, which would imply that $e^{-i\omega t}=e^{+i\omega t}$. These results have me very confused and it would be ideal if someone can give me a general form for the transform and the Dirac Delta, and it'd be even better if someone could explain to me why Wolfram gives the above results.

Answer 1

The factors $2\pi$ (and possible roots of them) are just a convention and sadly enough there a lot of different conventions. As for the choice of a minus sign in the exponential function, this is again a convention.