Landau distribution

Definition

The probability density function, as written originally by Landau, is defined by the complex integral:

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

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 ${\displaystyle \log }$ refers to the natural logarithm. In other words it is the Laplace transform of the function ${\displaystyle s^s}$.

The following real integral is equivalent to the above:

${\displaystyle p(x)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}{e}^{-t\log (t)-xt}\sin (\pi t)dt.}$

The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters ${\displaystyle \alpha =1}$ and ${\displaystyle \beta =1}$, with characteristic function:

${\displaystyle \phi (t;\mu ,c)=\exp \left(it\mu -\frac{2ict}{\pi }\log |t|-c|t|\right)}$

where ${\displaystyle c\in (0,\mathrm{\infty })}$ and ${\displaystyle \mu \in (-\mathrm{\infty },\mathrm{\infty })}$, which yields a density function:

${\displaystyle p(x;\mu ,c)=\frac{1}{\pi c}{\int }_0^{\mathrm{\infty }}{e}^{-t}\cos \left(t\left(\frac{x-\mu }{c}\right)+\frac{2t}{\pi }\log \left(\frac{t}{c}\right)\right)dt,}$

Taking ${\displaystyle \mu =0}$ and ${\displaystyle c=\frac{\pi }{2}}$ we get the original form of ${\displaystyle p(x)}$ above.