TITLE: What happens if I remove the sign from the exponential of Fourier Transform?
QUESTION [8 upvotes]: Forward Fourier Transform $\hat f(x)$ is defined such as
$$ \hat f(x) = \int_{-\infty}^{\infty}{f(t) e^{-2\pi t x i} dt} $$
but I am wondering what happens if I define it 
$$ \hat f(x) = \int_{-\infty}^{\infty}{f(t) e^{2\pi t x i} dt} $$
Note the missing minus sign from the exponential.
Obviously, if I do this I will have to define Inverse Fourier Transform with a minus sign in order to get the original signal back.
My guess is that this will provide negative frequencies but nothing else would change; am I right?
And then, why does this happen? I would expect that $e^{ai}$ to create a positive frequency while $e^{-ai}$ create a negative but the Fourier Transform seems to use them the other way around.

REPLY [6 votes]: This is just a matter of choice. The theory and applications of the Fourier transform can be carried out with any of the signs. Something similar occurs with the $\pi$ in the exponent.