— scale parameter In probability theory, the Landau distribution[1] is a probability distribution named after Lev Landau.

Because of the distribution's "fat" tail, the moments of the distribution, such as mean or variance, are undefined.

The probability density function, as written originally by Landau, is defined by the complex integral: where a is an arbitrary positive real number, meaning that the integration path can be any parallel to the imaginary axis, intersecting the real positive semi-axis, and

refers to the natural logarithm.

In other words it is the Laplace transform of the function

The following real integral is equivalent to the above: The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters

, which yields a density function: Taking

we get the original form of

These properties can all be derived from the characteristic function.

Together they imply that the Landau distributions are closed under affine transformations.

, the pdf can be approximated[4] using Lindhard theory which says: where

A similar approximation [5] of