## Characteristic Function

The characteristic function is defined as the Fourier Transform of the Probability Density Function,

 (1) (2) (3) (4)

where (sometimes also denoted ) is the th Moment about 0 and . The characteristic function can therefore be used to generate Moments about 0,
 (5)

or the Cumulants ,
 (6)

A Distribution is not uniquely specified by its Moments, but is uniquely specified by its characteristic function.

Kenney, J. F. and Keeping, E. S. Moment-Generating and Characteristic Functions,'' Some Examples of Moment-Generating Functions,'' and Uniqueness Theorem for Characteristic Functions.'' §4.6-4.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 72-77, 1951.