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.
Approximationsedit
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.
Referencesedit
^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. ISBN 978-0-387-00178-4.
^Zolotarev, V.M. (1986). One-dimensional stable distributions. Providence, R.I.: American Mathematical Society. ISBN 0-8218-4519-5.
^"LandauDistribution—Wolfram Language Documentation".
^Behrens, S. E.; Melissinos, A.C. Univ. of Rochester Preprint UR-776 (1981).