[KvanteKat]

You can think of it like this:

- The characteristic function of a random variable X is defined as the function that maps t --> ExpectedValue[ exp( i * t * X ) ]

- Computing this expected value is the same as regarding t as a constant and integrating the function x --> exp( i * t * x) with respect to the distribution of X, i.e. if X has the density f, we compute the integral of f(x) * exp( i * t * x) with respect to x over the domain of f.

- on the other hand: computing the Fourier transform of f (here representing the density of X) and evaluating it at point t (i.e. computing (F(f))(t) if F represents the Fourier transform) is the same as fixing t and computing the integral of f(x) * exp( -i * t * x) with respect to x.

- Rearranging the integrand in the previous expression to f(x) * exp( i * -t * x), we see that it is the same as the integrand used in the characteristic function, only with a -t instead of a t.

Hope that helps :)