Editing Mellin transform (section)

==In probability theory==
In probability theory, the Mellin transform is an essential tool in studying the distributions of products of random variables.<ref>{{harvtxt|Galambos|Simonelli|2004|p=15}}</ref> If ''X'' is a random variable, and {{nowrap|1=''X''<sup>+</sup> = max{''X'',0}}} denotes its positive part, while {{nowrap|1=''X''<sup>&thinsp;−</sup> = max{−''X'',0}}} is its negative part, then the ''Mellin transform'' of ''X'' is defined as<ref name="GalSim16">{{harvtxt|Galambos|Simonelli|2004|p=16}}</ref>
<math display="block"> \mathcal{M}_X(s) = \int_0^\infty  x^s dF_{X^+}(x) + \gamma\int_0^\infty x^s dF_{X^-}(x), </math>
where ''γ'' is a formal indeterminate with {{nowrap|''γ''<sup>2</sup> {{=}} 1}}. This transform exists for all ''s'' in some complex strip {{nowrap|''D'' {{=}} {''s''&nbsp;: ''a'' ≤ Re(''s'') ≤ ''b''} }}, where {{nowrap|''a'' ≤ 0 ≤ ''b''}}.<ref name="GalSim16" />

The Mellin transform <math>\mathcal{M}_X(it)</math> of a random variable ''X'' uniquely determines its distribution function ''F<sub>X</sub>''.<ref name="GalSim16" /> The importance of the Mellin transform in probability theory lies in the fact that if ''X'' and ''Y'' are two independent random variables, then the Mellin transform of their product is equal to the product of the Mellin transforms of ''X'' and ''Y'':<ref>{{harvtxt|Galambos|Simonelli|2004|p=23}}</ref>
<math display="block">\mathcal{M}_{XY}(s) = \mathcal{M}_X(s)\mathcal{M}_Y(s) </math>