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 that may be regarded as the
multiplicative version of the
two-sided Laplace transform. 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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
,
mathematical statistics, and the theory of
asymptotic expansions; it is closely related to the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
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 excep ...
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, ...
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.
The transform is named after the
Finnish 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 develo ...
, who introduced it in a paper published 1897 in ''Acta Societatis Scientiarum Fennicæ.''
Relationship to other transforms
The
two-sided Laplace transform 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 In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, thou ...
,
, 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 or
binomial transform 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 for the
convolution algebra of the
locally compact abelian group 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 excep ...
.
is a
meromorphic function 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 ...
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 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
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund L ...
, 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