The basic mathematical inconvenience is that (for functions where the Fourier and Inverse Fourier Transforms exist)
\begin{equation}
\int_{-\infty}^\infty d\omega e^{-i \omega T} \int_{-\infty}^\infty dt e^{i \omega t} f(t) = 2\pi f(T)
\end{equation}
So you need to put that $2\pi$ somewhere in your definition of a Fourier transform, if you want the Inverse Fourier Transform of the Fourier Transform to give you back the original function.
There are two common conventions in physics for dealing with this annoying factor of $2\pi$.
(1) Define the FT and IFT "symmetrically"
\begin{eqnarray}
\tilde{F}(\omega) &=& \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty d t e^{i \omega t}f(t) \\
f(t) &=& \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty d \omega e^{-i\omega t} \tilde{F}(\omega)
\end{eqnarray}
(2) Always choose to associate factors of $2\pi$ with the "momentum space" measure, $d\omega$ or $dk$
\begin{eqnarray}
\tilde{F}(\omega) &=& \int_{-\infty}^\infty d t e^{i \omega t}f(t) \\
f(t) &=& \int_{-\infty}^\infty \frac{d \omega}{2\pi} e^{-i\omega t} \tilde{F}(\omega)
\end{eqnarray}
In my experience in quantum field theory and cosmology (others may differ), option (1) is common in teaching quantum mechanics, but at a research level option (2) is almost always used.
