Mellin inversion theorem

If $\backslash varphi(s)$ is analytic in the strip ,

and if it tends to zero uniformly as $\backslash Im(s)\; \backslash to\; \backslash pm\; \backslash infty$ for any real value c between a and b, with its integral along such a line converging absolutely, then if

:$f(x)=\; \backslash \{\; \backslash mathcal\{M\}^\{-1\}\; \backslash varphi\; \backslash \}\; =\; \backslash frac\{1\}\{2\; \backslash pi\; i\}\; \backslash int\_\{c-i\; \backslash infty\}^\{c+i\; \backslash infty\}\; x^\{-s\}\; \backslash varphi(s)\backslash ,\; ds$

we have that

:$\backslash varphi(s)=\; \backslash \{\; \backslash mathcal\{M\}\; f\; \backslash \}\; =\; \backslash int\_0^\{\backslash infty\}\; x^\{s-1\}\; f(x)\backslash ,dx.$

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

:$\backslash varphi(s)=\backslash int\_0^\{\backslash infty\}\; x^\{s-1\}\; f(x)\backslash ,dx$

is absolutely convergent when . Then $f$ is recoverable via the inverse Mellin transform from its Mellin transform $\backslash varphi$. 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.{{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}}

The boundedness condition on $\backslash varphi(s)$ can be strengthened if

$f(x)$ is continuous. If $\backslash varphi(s)$ is analytic in the strip , and if , where K is a positive constant, then $f(x)$ as defined by the inversion integral exists and is continuous; moreover the Mellin transform of $f$ is $\backslash varphi$ for at least .

On the other hand, if we are willing to accept an original $f$ which is a

generalized function, we may relax the boundedness condition on

$\backslash varphi$ to

simply make it of polynomial growth in any closed strip contained in the open strip .

We may also define a Banach space version of this theorem. If we call by

$L\_\{\backslash nu,\; p\}(R^\{+\})$ the weighted L^p space of complex valued functions $f$ on the positive reals such that

:

where ν and p are fixed real numbers with $p>1$, then if $f(x)$

is in $L\_\{\backslash nu,\; p\}(R^\{+\})$ with , then

$\backslash varphi(s)$ belongs to $L\_\{\backslash nu,\; q\}(R^\{+\})$ with $q\; =\; p/(p-1)$ and

:$f(x)=\backslash frac\{1\}\{2\; \backslash pi\; i\}\; \backslash int\_\{\backslash nu-i\; \backslash infty\}^\{\backslash nu+i\; \backslash infty\}\; x^\{-s\}\; \backslash varphi(s)\backslash ,ds.$

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

Since the two-sided Laplace transform can be defined as

:$\backslash left\backslash \{\backslash mathcal\{B\}\; f\backslash right\backslash \}(s)\; =\; \backslash left\backslash \{\backslash mathcal\{M\}\; f(-\; \backslash ln\; x)\; \backslash right\backslash \}(s)$

these theorems can be immediately applied to it also.

• Mellin transform
• Nachbin's theorem

{{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 |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 }}

• [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