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, like mean or variance, are undefined. The distribution is a particular case of stable distribution.
Definition
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
and
,[2] with characteristic function:[3]

where
and
, which yields a density function:

Taking
and
we get the original form of
above.
Properties
The approximation function for
- Translation: If
then
.
- Scaling: If
then
.
- Sum: If
and
then
.
These properties can all be derived from the characteristic function.
Together they imply that the Landau distribution is closed under affine transformations.
Approximations
In the "standard" case
and
, the pdf can be approximated[4] using Lindhard theory which says:

where
is Euler's constant.
A similar approximation [5] of
for
and
is:
