Editing Mellin transform (section)

==Fundamental strip==

For <math>\alpha,\beta\in\mathbb{R}</math>, let the open strip <math>\langle\alpha,\beta\rangle</math> be defined to be all <math>s\in\mathbb{C}</math> such that <math>s=\sigma + it</math> with <math>\alpha < \sigma < \beta.</math>  The '''fundamental strip''' of <math>\mathcal{M} f(s)</math> is defined to be the largest open strip on which it is defined.  For example, for <math>a > b</math> the fundamental strip of
<math display="block">f(x)=\begin{cases} x^a  & x < 1, \\ x^b & x > 1, \end{cases}</math>
is <math>\langle -a,-b \rangle.</math>  As seen by this example, the asymptotics of the function as <math>x\to 0^+</math> define the left endpoint of its fundamental strip, and the asymptotics of the function as <math>x\to +\infty</math> define its right endpoint.  To summarize using [[Big O notation]], if <math>f</math> is <math>O(x^a)</math> as <math>x\to 0^+</math> and <math>O(x^b)</math> as <math>x\to +\infty,</math> then <math>\mathcal{M} f(s)</math> is defined in the strip <math>\langle -a,-b \rangle.</math>{{sfnp|Flajolet|Gourdon|Dumas|1995}}

An application of this can be seen in the gamma function, <math>\Gamma(s).</math>  Since <math>f(x)=e^{-x}</math> is <math>O(x^0)</math> as <math>x\to 0^+</math> and <math>O(x^{k})</math> for all <math>k,</math> then <math>\Gamma(s) = \mathcal{M} f(s)</math> should be defined in the strip <math>\langle 0,+\infty \rangle,</math> which confirms that <math>\Gamma(s)</math> is analytic for <math>\Re(s) > 0.</math>