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| In [[mathematics]], the '''Mellin transform''' is an [[integral transform]] that may be regarded as the [[multiplicative group|multiplicative]] version of the [[two-sided Laplace transform]]. This integral transform is closely connected to the theory of [[Dirichlet series]], and is
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| often used in [[number theory]] and the theory of [[asymptotic expansion]]s; it is closely related to the [[Laplace transform]] and the [[Fourier transform]], and the theory of the [[gamma function]] and allied [[special function]]s.
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| The Mellin transform of a function ''f'' is
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| :<math>\left\{\mathcal{M}f\right\}(s) = \varphi(s)=\int_0^{\infty} x^{s-1} f(x)dx.</math>
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| The inverse transform is
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| :<math>\left\{\mathcal{M}^{-1}\varphi\right\}(x) = f(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \varphi(s)\, ds.</math>
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| The notation implies this is a [[line integral]] taken over a vertical line in the complex plane. Conditions under which this inversion is valid are given in the [[Mellin inversion theorem]].
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| The transform is named after the [[Finland|Finnish]] mathematician [[Hjalmar Mellin]].
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| ==Relationship to other transforms==
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| The [[two-sided Laplace transform]] may be defined in terms of the Mellin
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| transform by
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| :<math> \left\{\mathcal{B} f\right\}(s) = \left\{\mathcal{M} f(-\ln x) \right\}(s)</math>
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| and conversely we can get the Mellin transform from the two-sided Laplace transform by
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| :<math>\left\{\mathcal{M} f\right\}(s) = \left\{\mathcal{B} f(e^{-x})\right\}(s).</math>
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| The Mellin transform may be thought of as integrating using a kernel ''x''<sup>''s''</sup> with respect to the multiplicative [[Haar measure]],
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| <math>\frac{dx}{x}</math>, which is invariant
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| under dilation <math>x \mapsto ax</math>, so that
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| <math>\frac{d(ax)}{ax} = \frac{dx}{x}</math>; the two-sided Laplace transform integrates with respect to the additive Haar measure <math>dx</math>, which is translation invariant, so that <math>d(x+a) = dx</math>.
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| We also may define the [[Fourier transform]] in terms of the Mellin transform and vice-versa; if we define the two-sided Laplace transform as above, then
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| :<math>\left\{\mathcal{F} f\right\}(-s) = \left\{\mathcal{B} f\right\}(-is) | |
| = \left\{\mathcal{M} f(-\ln x)\right\}(-is).</math>
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| We may also reverse the process and obtain
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| :<math>\left\{\mathcal{M} f\right\}(s) = \left\{\mathcal{B}
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| f(e^{-x})\right\}(s) = \left\{\mathcal{F} f(e^{-x})\right\}(-is).</math>
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| The Mellin transform also connects the [[Newton series]] or [[binomial transform]] together with the [[Poisson generating function]], by means of the [[Poisson–Mellin–Newton cycle]].
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| ==Examples==
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| ===Cahen–Mellin integral===
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| For <math>c>0</math>, <math>\Re(y)>0</math> and <math>y^{-s}</math> on the [[principal branch]], one has
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| :<math>e^{-y}= \frac{1}{2\pi i}
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| \int_{c-i\infty}^{c+i\infty} \Gamma(s) y^{-s}\;ds</math>
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| where <math>\Gamma(s)</math> is the [[gamma function]]. This integral is known as the Cahen-Mellin integral.<ref>{{cite journal |first=G. H. |last=Hardy |first2=J. E. |last2=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 }} ''(See notes therein for further references to Cahen's and Mellin's work, including Cahen's thesis.)''</ref>
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| ===Number theory===
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| An important application in number theory includes the simple function <math>f(x)=\begin{cases} 0 & x < 1, \\ x^{a} & x > 1, \end{cases},</math>
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| for which
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| :<math>\mathcal M f (s)= - \frac 1 {s+a}, </math>
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| assuming
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| <math>\Re (s+a)<0.</math>
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| == As a unitary operator on ''L''<sup>2</sup> ==
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| In the study of [[Hilbert space]]s, the Mellin transform is often posed in a slightly different way. For functions in <math>L^2(0,\infty)</math> (see [[Lp space]]) the fundamental strip always includes <math>\tfrac{1}{2}+i\mathbb{R}</math>, so we may define a [[linear operator]] <math>\tilde{\mathcal{M}}</math> as
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| :<math>\tilde{\mathcal{M}}\colon L^2(0,\infty)\to L^2(-\infty,\infty), \{\tilde{\mathcal{M}}f\}(s) := \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2}+is} f(x)\,dx. </math>
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| In other words we have set
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| :<math>\{\tilde{\mathcal{M}}f\}(s):=\tfrac{1}{\sqrt{2\pi}}\{\mathcal{M}f\}(\tfrac{1}{2}+is).</math>
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| This operator is usually denoted by just plain <math>\mathcal{M}</math> and called the "Mellin transform", but <math>\tilde{\mathcal{M}}</math> is used here to distinguish from the definition used elsewhere in this article. The [[Mellin inversion theorem]] then shows that <math>\tilde{\mathcal{M}}</math> is invertible with inverse
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| :<math>\tilde{\mathcal{M}}^{-1}\colon L^2(-\infty,\infty) \to L^2(0,\infty), \{\tilde{\mathcal{M}}^{-1}\varphi\}(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} x^{-\frac{1}{2}-is} \varphi(s)\,ds. </math>
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| Furthermore this operator is an [[isometry]], that is to say <math>\|\tilde{\mathcal{M}} f\|_{L^2(-\infty,\infty)}=\|f\|_{L^2(0,\infty)}</math> for all <math>f\in L^2(0,\infty)</math> (this explains why the factor of <math>1/\sqrt{2\pi}</math> was used). Thus <math>\tilde{\mathcal{M}}</math> is a [[unitary operator]].
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| ==In probability theory==
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| 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|''X''<sup>+</sup> {{=}} max{''X'',0}}} denotes its positive part, while {{nowrap|''X''<sup> −</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>
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| : <math>
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| \mathcal{M}_X(s) = \int_0^\infty x^s dF_{X^+}(x) + \gamma\int_0^\infty x^s dF_{X^-}(x),
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| </math>
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| 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'': ''a'' ≤ Re(''s'') ≤ ''b''}}}, where {{nowrap|''a'' ≤ 0 ≤ ''b''}}.<ref name="GalSim16"/>
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| The Mellin transform <math>\scriptstyle\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 products is equal to the product of the Mellin transforms of ''X'' and ''Y'':<ref>{{harvtxt|Galambos|Simonelli|2004|p=23}}</ref>
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| : <math>
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| \mathcal{M}_{XY}(s) = \mathcal{M}_X(s)\mathcal{M}_Y(s)
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| </math>
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| ==Applications==
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| The Mellin Transform is widely used in computer science for the analysis of algorithms because of its scale invariance property. The magnitude of the Mellin Transform of a scaled function is identical to the magnitude of the original function. This scale invariance property is analogous to the Fourier Transform's shift invariance property. The magnitude of a Fourier transform of a time-shifted function is identical to the original function.
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| This property is useful in image recognition. An image of an object is easily scaled when the object is moved towards or away from the camera.
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| ==Examples==
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| * [[Perron's formula]] describes the inverse Mellin transform applied to a [[Dirichlet series]].
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| * The Mellin transform is used in analysis of the [[prime-counting function]] and occurs in discussions of the [[Riemann zeta function]].
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| * Inverse Mellin transforms commonly occur in [[Riesz mean]]s.
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| * The Mellin transform can be used in [[Audio timescale-pitch modification]] (needs substantive reference).
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| ==See also==
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| *[[Mellin inversion theorem]]
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| *[[Perron's formula]]
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| ==Notes==
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| <references />
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| ==References==
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| {{refbegin}}
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| *{{cite book
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| | last1 = Galambos | first1 = Janos
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| | last2 = Simonelli | first2 = Italo
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| | year = 2004
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| | title = Products of random variables: applications to problems of physics and to arithmetical functions
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| | publisher = Marcel Dekker, Inc.
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| | isbn = 0-8247-5402-6
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| | ref = harv }}
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| *{{cite book |last=Paris |first=R. B. |last2=Kaminski |first2=D. |title=Asymptotics and Mellin-Barnes Integrals |location= |publisher=Cambridge University Press |year=2001 |isbn= }}
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| *{{cite book |first=A. D. |last=Polyanin |first2=A. V. |last2=Manzhirov |title=Handbook of Integral Equations |publisher=CRC Press |location=Boca Raton |year=1998 |isbn=0-8493-2876-4 }}
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| *{{cite journal |first=P. |last=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= }}
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| * [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
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| *{{springer|title=Mellin transform|id=p/m063380}}
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| *{{mathworld|urlname=MellinTransform|title=Mellin Transform}}
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| {{refend}}
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| == External links ==
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| * Philippe Flajolet, Xavier Gourdon, Philippe Dumas, ''[http://algo.inria.fr/flajolet/Publications/mellin-harm.pdf Mellin Transforms and Asymptotics: Harmonic sums.]''
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| * Antonio Gonzáles, Marko Riedel ''[http://groups.google.com/group/es.ciencia.matematicas/browse_thread/thread/cfcc11fbd5eeaa48/eab2e1423902ced1#eab2e1423902ced1 Celebrando un clásico], newsgroup es.ciencia.matematicas''
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| * Juan Sacerdoti, ''[http://www.fi.uba.ar/materias/61107/Apuntes/Eu00.pdf Funciones Eulerianas]'' (in Spanish).
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| * [http://dlmf.nist.gov/2.5 Mellin Transform Methods], [[Digital Library of Mathematical Functions]], 2011-08-29, [[National Institute of Standards and Technology]]
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| * Antonio De Sena and Davide Rocchesso, ''[http://www.di.univr.it/documenti/ArticoloConferenza/allegato/allegato082603.pdf A FAST MELLIN TRANSFORM WITH APPLICATIONS IN DAFX]''
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| {{DEFAULTSORT:Mellin Transform}}
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| [[Category:Complex analysis]]
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| [[Category:Integral transforms]]
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