Fourier Transforms: Exp(-iwt) or Exp(iwt)?

[zheng89120]

I have a simple question. Is the kernel of a Fourier transform exp(-iwt) or exp(iwt). It feels like my professor sometimes uses one, and sometimes uses the other.

 

[Mute]

It can be either. There's no strict convention. The only thing you have to make sure of is that you are consistent between the Fourier transform and the inverse Fourier transform. Whichever sign you pick for the forward transform, the inverse transform must have the opposite sign.

Note that you do not even have to use the same sign convention in multidimensional Fourier transforms. e.g., one could write

$$F(k_x,k_y,k_z,\omega) = \int_{-\infty}^\infty dx~dy~dz~dt~e^{i(k_x x + k_y y + k_z z - \omega t)} f(x,y,z,t)$$

Some places adopt this sign convention so that it follows the sign convention of the minkowski metric in special relativity.

The point is the sign is up to you to choose, but the inverse transform has to have the opposite sign.

 

[matonski]

Ditto about the $2\pi$ 's.