Context: probability theory, statistics
In probability theory, the characteristic function (φ(s)) of a random variable X is defined as the expected value of eisX, where e is the base of the natural logarithm, and i is the imaginary number.
If X comes from an absolutely continuous distribution, i.e. one that can be completely specified by a probability density function, then the characteristic function of X is the same as the Fourier transform of the density of X.
Unlike the moment generating function, which is not defined for subexponential distributions, the characteristic function exists for all distributions, and shares many of the same properties of moment generating functions.