The CharacteristicFunction function computes the Characteristic function of the specified random variable at the specified point.

The first parameter can be a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

By default, all computations involving random variables are performed symbolically (see option numeric below).

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

The options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

numeric=truefalse -- By default, the Characteristic function is computed using exact arithmetic. To compute the Characteristic function numerically, specify the numeric or numeric = true option.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$\mathrm{CharacteristicFunction}\left(\'\mathrm{\Β}\'\left(p\,q\right)\,t\right)$

$\mathrm{hypergeom}\left(\left[p\right]\,\left[p\+q\right]\,\mathrm{I}t\right)$

$T≔\mathrm{Distribution}\left(\mathrm{PDF}\&equals;\left(t\&rarr;\mathrm{piecewise}\left(t<0\&comma;0\&comma;t<1\&comma;6t\left(1-t\right)\&comma;0\right)\right)\right)\&colon;$

$X≔\mathrm{RandomVariable}\left(T\right)\&colon;$

$\mathrm{CDF}\left(X\&comma;t\right)$

$\&lcub;\begin{array}{cc}0& t\le 0\\ -2t^3\&plus;3t^2& t\le 1\\ 1& 1<t\end{array}$

$\mathrm{CharacteristicFunction}\left(X\&comma;t\right)$

$\frac{6\left(-2\mathrm{I}{\&ExponentialE;}^{\mathrm{I}t}-{\&ExponentialE;}^{\mathrm{I}t}t\&plus;2\mathrm{I}-t\right)}{t^3}$

$U≔\mathrm{Distribution}\left(\mathrm{CDF}\&equals;\left(t\&rarr;F\left(t\right)\right)\&comma;\mathrm{PDF}\&equals;\left(t\&rarr;f\left(t\right)\right)\right)\&colon;$

$Y≔\mathrm{RandomVariable}\left(U\right)\&colon;$

$\mathrm{CDF}\left(Y\&comma;t\right)$

$F\left(t\right)$

$\mathrm{CharacteristicFunction}\left(Y\&comma;t\right)$

$2\mathrm{\&pi;}\left(\genfrac{}{}{0ex}{}{\mathrm{invfourier}\left(f\left(u\right)\&comma;u\&comma;v\right)}{\phantom{v\&equals;t}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{invfourier}\left(f\left(u\right)\&comma;u\&comma;v\right)}}{v\&equals;t}\right)$

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998.  Vol. 1: Distribution Theory.