Editing Mellin transform (section)

===Cahen–Mellin integral===
{{anchor|Cahen–Mellin}}<!-- This Anchor tag serves to provide a permanent target for incoming section links. Please do not remove it, nor modify it, except to add another appropriate anchor.  If you modify the section title, please anchor the old title. It is always best to anchor an old section header that has been changed so that links to it will not be broken. See [[Template:Anchor]] for details. This template is {{subst:Anchor comment}} -->The Mellin transform of the function <math> f(x) = e^{-x} </math> is
<math display="block">\Gamma(s) = \int_0^\infty x^{s-1}e^{-x} dx </math>
where <math>\Gamma(s)</math> is the [[gamma function]].  <math>\Gamma(s)</math> is a [[meromorphic function]] with simple [[zeros and poles|poles]] at <math>z = 0, -1, -2, \dots</math>.<ref>{{cite book |first1=E.T. |last1=Whittaker |author-link1=E. T. Whittaker|first2=G.N. |last2=Watson|author-link2=G. N. Watson |title=[[A Course of Modern Analysis]] |year=1996 |publisher=Cambridge University Press}}</ref>  Therefore, <math>\Gamma(s)</math> is analytic for <math>\Re(s)>0</math>.  Thus, letting <math>c>0</math> and <math>z^{-s}</math> on the [[principal branch]], the inverse transform gives
<math display="block"> e^{-z}= \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \Gamma(s) z^{-s} \; ds .</math>

This integral is known as the Cahen–Mellin integral.<ref>{{cite journal |first1=G. H. |last1=Hardy|author-link1=G. H. Hardy |first2=J. E. |last2=Littlewood|author-link2=J. E. Littlewood |title=Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes |journal=[[Acta Mathematica]] |volume=41 |issue=1 |year=1916 |pages=119–196 |doi=10.1007/BF02422942 |url=https://zenodo.org/record/2294397 |doi-access=free }} ''(See notes therein for further references to Cahen's and Mellin's work, including Cahen's thesis.)''</ref>