Characteristic function (cf) for the normal distribution:

Characteristic function for the binomial distribution:

Characteristic function for the bivariate normal distribution:

Characteristic function for the multinomial distribution:

Characteristic function for a specific continuous distribution:

Characteristic function for a specific discrete distribution:

Characteristic function at a particular value:

Characteristic function evaluated numerically:

Obtain a result at any precision:

Compute the characteristic function for a formula distribution:

Find the characteristic function for a parameter mixture distribution:

Characteristic function for the slice distribution of a random process:

Compute the raw moments for a Poisson distribution:

First 5 raw moments using derivatives of the characteristic function at the origin:

Use Moment directly:

Compute mixed raw moments for a multivariate distribution:

Use Moment to obtain raw moments directly:

Find raw moments of a Student distribution from its characteristic function:

Compute to extract moments by taking limits from the right:

Evaluate the limits from the left:

Only the first four moments are defined, as confirmed by using Moment directly:

Use inverse Fourier transform to compute the PDF corresponding to a characteristic function:

Illustrate the central limit theorem on the example of symmetric LaplaceDistribution:

Find the characteristic function of the rescaled random variate:

Compute the large limit of the cf of the sum of such i.i.d. random variates:

Compare with the characteristic function of a standard normal variate:

Use smooth characteristic function to construct the upper bound for the distribution density of ErlangDistribution:

Plot the upper bounds and the original density:

Verify that the sum where are independent identically distributed BernoulliDistribution[1/2] variates tends in distribution to UniformDistribution[] for large :

Use a combinatorial equality for product :

Evaluate the sum:

Take the limit and compare it to the characteristic function of the UniformDistribution:

CharacteristicFunction is the Expectation of for real :

The characteristic function is related to all other generating functions when they exist:

The cf of a continuous distribution is equivalent to FourierTransform of its PDF:

The cf of a discrete distribution is equivalent to FourierSequenceTransform of its PDF:

The PDF is the inverse Fourier transform of the cf for continuous distributions:

The PDF is the inverse Fourier sequence transform of the cf for discrete distributions:

Symbolic closed forms do not exist for some distributions:

Visualize real and imaginary parts of CharacteristicFunction for random instances of BinomialDistribution: