In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Mellin transform is an
integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
that may be regarded as the
multiplicative version of the
two-sided Laplace transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin t ...
. This integral transform is closely connected to the theory of
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analy ...
, and is
often used in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
,
mathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...
, and the theory of
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
s; it is closely related to the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
and the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, and the theory of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
and allied
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined b ...
s.
The Mellin transform of a function is
:
The inverse transform is
:
The notation implies this is a
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, al ...
taken over a vertical line in the complex plane, whose real part ''c'' need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the
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.
Met ...
.
The transform is named after the
Finnish
Finnish may refer to:
* Something or someone from, or related to Finland
* Culture of Finland
* Finnish people or Finns, the primary ethnic group in Finland
* Finnish language, the national language of the Finnish people
* Finnish cuisine
See also ...
mathematician
Hjalmar Mellin
Robert Hjalmar Mellin (19 June 1854 – 5 April 1933) was a Finnish mathematician and function theorist.
Biography
Mellin studied at the University of Helsinki and later in Berlin under Karl Weierstrass. He is chiefly remembered as the develope ...
, who introduced it in a paper published 1897 in ''Acta Societatis Scientiarum Fennicæ.''
Relationship to other transforms
The
two-sided Laplace transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin t ...
may be defined in terms of the Mellin transform by
:
and conversely we can get the Mellin transform from the two-sided Laplace transform by
:
The Mellin transform may be thought of as integrating using a kernel ''x''
''s'' with respect to the multiplicative
Haar measure,
, which is invariant
under dilation
, so that
the two-sided Laplace transform integrates with respect to the additive Haar measure
, which is translation invariant, so that
.
We also may define the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
in terms of the Mellin transform and vice versa; in terms of the Mellin transform and of the two-sided Laplace transform defined above
:
We may also reverse the process and obtain
:
The Mellin transform also connects the
Newton series
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
or
binomial transform In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to t ...
together with the
Poisson generating function, by means of the
Poisson–Mellin–Newton cycle.
The Mellin transform may also be viewed as the
Gelfand transform In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things:
* a way of representing commutative Banach algebras as algebras of continuous functions;
* the fact that for commutative C*-al ...
for the
convolution algebra
In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are ...
of the
locally compact abelian group
In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the ...
of positive real numbers with multiplication.
Examples
Cahen–Mellin integral
The Mellin transform of the function
is
:
where
is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.
is a
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
with simple
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
at
. Therefore,
is analytic for
. Thus, letting
and
on the
principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane.
Examples
Trigonometric inverses
Principal branches are use ...
, the inverse transform gives
:
.
This integral is known as the Cahen–Mellin integral.
Polynomial functions
Since
is not convergent for any value of
, the Mellin transform is not defined for polynomial functions defined on the whole positive real axis. However, by defining it to be zero on different sections of the real axis, it is possible to take the Mellin transform. For example, if
:
then
:
Thus
has a simple pole at
and is thus defined for
. Similarly, if
:
then
:
Thus
has a simple pole at
and is thus defined for
.
Exponential functions
For
, let
. Then
:
Zeta function
It is possible to use the Mellin transform to produce one of the fundamental formulas for the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
,
. Let
. Then
:
Thus,
:
Generalized Gaussian
For
, let
(i.e.
is a
generalized Gaussian distribution
The generalized normal distribution or generalized Gaussian distribution (GGD) is either of two families of parametric statistics, parametric continuous probability distributions on the real number, real line. Both families add a shape parameter ...
without the scaling factor.) Then
:
In particular, setting
recovers the following form of the gamma function
:
Fundamental strip
For
, let the open strip
be defined to be all
such that
with
The fundamental strip of
is defined to be the largest open strip on which it is defined. For example, for
the fundamental strip of
:
is
As seen by this example, the asymptotics of the function as
define the left endpoint of its fundamental strip, and the asymptotics of the function as
define its right endpoint. To summarize using
Big O notation, if
is
as
and
as
then
is defined in the strip
An application of this can be seen in the gamma function,
Since
is
as
and
for all
then
should be defined in the strip
which confirms that
is analytic for
Properties
The properties in this table may be found in and .
Parseval's theorem and Plancherel's theorem
Let
and
be functions with well-defined
Mellin transforms
in the fundamental strips
.
Let
with