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:
These properties can all be derived from the characteristic function.
Together they imply that the Landau distributions are closed under affine transformations.
Approximations
In the "standard" case and , the pdf can be approximated[4] using Lindhard theory which says:
The Landau distribution is a stable distribution with stability parameter and skewness parameter both equal to 1.
References
^Landau, L. (1944). "On the energy loss of fast particles by ionization". J. Phys. (USSR). 8: 201.
^Gentle, James E. (2003). Random Number Generation and Monte Carlo Methods. Statistics and Computing (2nd ed.). New York, NY: Springer. p. 196. doi:10.1007/b97336. ISBN978-0-387-00178-4.