The CauchyDistribution method returns a Distribution object spanned by the Cauchy vector fields of input Distribution dist.

A vector field C is a Cauchy vector field of distribution dist if C within dist, then [C,X] likewise lies within dist.  That is, C lies in dist and is a symmetry of dist. Such vector fields span a distribution, which is always in involution.

This method is of little interest if the input Distribution dist is involutive, since in that case CauchyDistribution(dist) will simply return dist itself.

This use of the term 'Cauchy distribution' is unrelated to its use in statistics (see Statistics[Distributions][Cauchy]).

This method is associated with the Distribution object. For more detail see Overview of the Distribution object.

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$\mathrm{V1}≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right]\,\mathrm{space}=\left[x\,y\,z\,w\right]\right)$

$\mathrm{V1}≔\frac{\ⅆ}{\ⅆx}$

$\mathrm{V2}≔\mathrm{VectorField}\left(\mathrm{D}\left[z\right]\,\mathrm{space}=\left[x\,y\,z\,w\right]\right)$

$\mathrm{V2}≔\frac{\ⅆ}{\ⅆz}$

$\mathrm{V3}≔\mathrm{VectorField}\left(\mathrm{D}\left[y\right]+z\mathrm{D}\left[w\right]\,\mathrm{space}=\left[x\,y\,z\,w\right]\right)$

$\mathrm{V3}≔\frac{\ⅆ}{\ⅆy}+z\left(\frac{\ⅆ}{\ⅆw}\right)$

$\mathrm{\Sigma }≔\mathrm{Distribution}\left(\mathrm{V1}\,\mathrm{V2}\,\mathrm{V3}\right)$

$\mathrm{\Sigma }≔\left\{\frac{\ⅆ}{\ⅆx}\,\frac{\ⅆ}{\ⅆz}\,\frac{\frac{\ⅆ}{\ⅆy}}{z}+\frac{\ⅆ}{\ⅆw}\right\}$

$\mathrm{CauchyDistribution}\left(\mathrm{\Sigma }\right)$

$\left\{\frac{\ⅆ}{\ⅆx}\right\}$

$\mathrm{IsInvolutive}\left(\mathrm{\Sigma }\right)$

$\mathrm{false}$

The CauchyDistribution command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.