# Landau distribution

File:Landau distribution.png
Landau distribution for a most-probable-value of 2 and sigma of 1

In probability theory, the Landau distribution is a probability distribution named after Lev Landau. It is defined by the complex integral

${\displaystyle p(x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }\!e^{s\log s+xs}\,ds,}$

where ${\displaystyle c}$ is any positive real number, and log refers to the logarithm base e, the natural logarithm.

For numerical purposes it is more convenient to use the following equivalent form of the integral,

${\displaystyle p(x)={\frac {1}{\pi }}\int _{0}^{\infty }\!e^{-t\log t-xt}\sin(\pi t)\,dt.}$

where log refers to the logarithm base e, the natural logarithm.

The Landau distribution is used in physics to describe the fluctuations in the energy loss of a charged particle passing through a thin layer of matter.

From GSL manual, used under GFDL.