Editing Mellin inversion theorem

In [[mathematics]], the '''Mellin inversion formula''' (named after [[Hjalmar Mellin]]) tells us conditions under
which the inverse [[Mellin transform]], or equivalently the inverse [[two-sided Laplace transform]], are defined and recover the transformed function.

== Method ==

If <math>\varphi(s)</math> is analytic in the strip <math>a < \Re(s) < b</math>,
and if it tends to zero uniformly as  <math>  \Im(s) \to \pm \infty  </math> for any real value ''c'' between ''a'' and ''b'', with its integral along such a line converging absolutely, then if

:<math>f(x)= \{ \mathcal{M}^{-1} \varphi \} = \frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \varphi(s)\, ds</math>

we have that

:<math>\varphi(s)= \{ \mathcal{M} f \} = \int_0^{\infty} x^{s-1} f(x)\,dx.</math>

Conversely, suppose <math>f(x)</math> is piecewise continuous on the [[positive real numbers]], taking a value halfway between the limit values at any jump discontinuities, and suppose the integral

:<math>\varphi(s)=\int_0^{\infty} x^{s-1} f(x)\,dx</math>

is absolutely convergent when <math>a < \Re(s) < b</math>. Then <math>f</math> is recoverable via the inverse Mellin transform from its Mellin transform <math>\varphi</math>. These results can be obtained by relating the Mellin transform to the [[Fourier transform]] by a change of variables and then applying an appropriate version of the [[Fourier inversion theorem]].<ref>{{Cite book |first=Lokenath |last=Debnath |url=http://worldcat.org/oclc/919711727 |title=Integral transforms and their applications |date=2015 |publisher=CRC Press |isbn=978-1-4822-2357-6 |oclc=919711727}}</ref>

== Boundedness condition ==

The boundedness condition on <math>\varphi(s)</math> can be strengthened if 
<math>f(x)</math> is continuous. If <math>\varphi(s)</math> is analytic in the strip <math>a < \Re(s) < b</math>, and if <math>|\varphi(s)| < K |s|^{-2}</math>, where ''K'' is a positive constant, then <math>f(x)</math> as defined by the inversion integral exists and is continuous; moreover the Mellin transform of <math>f</math> is <math>\varphi</math> for at least <math>a < \Re(s) < b</math>.

On the other hand, if we are willing to accept an original <math>f</math> which is a 
[[generalized function]], we may relax the boundedness condition on 
<math>\varphi</math> to
simply make it of polynomial growth in any closed strip contained in the open strip <math>a < \Re(s) < b</math>.

We may also define a [[Banach space]] version of this theorem. If we call by
<math>L_{\nu, p}(R^{+})</math> the weighted [[Lp space|L<sup>p</sup> space]] of complex valued functions <math>f</math> on the positive reals such that

:<math>\|f\| = \left(\int_0^\infty |x^\nu f(x)|^p\, \frac{dx}{x}\right)^{1/p} < \infty</math>

where ν and ''p'' are fixed real numbers with <math>p>1</math>, then if <math>f(x)</math>
is in <math>L_{\nu, p}(R^{+})</math> with <math>1 < p \le 2</math>, then
<math>\varphi(s)</math> belongs to <math>L_{\nu, q}(R^{+})</math> with <math>q = p/(p-1)</math> and

:<math>f(x)=\frac{1}{2 \pi i} \int_{\nu-i \infty}^{\nu+i \infty} x^{-s} \varphi(s)\,ds.</math>

Here functions, identical everywhere except on a set of measure zero, are identified.

Since the two-sided Laplace transform can be defined as

:<math> \left\{\mathcal{B} f\right\}(s) = \left\{\mathcal{M} f(- \ln x) \right\}(s)</math>

these theorems can be immediately applied to it also.

==See also==
*[[Mellin transform]]
*[[Nachbin's theorem]]

==References==
{{Reflist}}
*{{cite journal |first1=P. |last1=Flajolet |authorlink=Philippe Flajolet |first2=X. |last2=Gourdon |first3=P. |last3=Dumas |title=Mellin transforms and asymptotics: Harmonic sums |journal=[[Theoretical Computer Science (journal)|Theoretical Computer Science]] |volume=144 |issue=1–2 |pages=3–58 |year=1995 |doi=10.1016/0304-3975(95)00002-E |url=https://hal.inria.fr/inria-00074307/file/RR-2369.pdf }}
*{{cite book |last=McLachlan |first=N. W. |title=Complex Variable Theory and Transform Calculus |publisher=Cambridge University Press |year=1953 }}
*{{cite book |last1=Polyanin |first1=A. D. |last2=Manzhirov |first2=A. V. |title=Handbook of Integral Equations |publisher=CRC Press |location=Boca Raton |year=1998 |isbn=0-8493-2876-4 }}
*{{cite book |last=Titchmarsh |first=E. C. |authorlink=Edward Charles Titchmarsh |title=Introduction to the Theory of Fourier Integrals |publisher=Oxford University Press |edition=Second |year=1948 }}
*{{cite book |last=Yakubovich |first=S. B. |title=Index Transforms |publisher=World Scientific |year=1996 |isbn=981-02-2216-5 }}
*{{cite book |last=Zemanian |first=A. H. |title=Generalized Integral Transforms |publisher=John Wiley & Sons |year=1968 }}

== External links==
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.

[[Category:Integral transforms]]
[[Category:Theorems in complex analysis]]
[[Category:Laplace transforms]]