Mellin transform

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In 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, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.

The Mellin transform of a function f is

The inverse transform is

The notation implies this is a line integral 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, who introduced it in a paper published 1897 in Acta Societatis Scientiarum Fennicæ.[1]

Relationship to other transforms[edit]

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 xs 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 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.


Cahen–Mellin integral[edit]

The Mellin transform of the function is

where is the gamma function. is a meromorphic function with simple poles at .[2] Therefore, is analytic for . Thus, letting and on the principal branch, the inverse transform gives


This integral is known as the Cahen–Mellin integral.[3]

Polynomial functions[edit]

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


Thus has a simple pole at and is thus defined for . Similarly, if


Thus has a simple pole at and is thus defined for .

Exponential functions[edit]

For , let . Then

Zeta function[edit]

It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function, . Let . Then


Generalized Gaussian[edit]

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

Power series and Dirichlet series[edit]

Generally, assuming necessary convergence, we can connect Dirichlet series and related power series

by the formal identity involving Mellin transform:[4]

Fundamental strip[edit]

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 [5]

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


The properties in this table may be found in Bracewell (2000) and Erdélyi (1954).

Properties of the Mellin transform
Function Mellin transform Fundamental strip Comments
Here denotes the complex conjugate of .
, Scaling
Valid only if the integral exists.
Valid only if the integral exists.
Multiplicative convolution
Multiplicative convolution (generalized)
Multiplicative convolution (generalized)
Multiplication. Only valid if integral exists. See Parseval's theorem below for conditions which ensure the existence of the integral.

Parseval's theorem and Plancherel's theorem[edit]

Let and be functions with well-defined Mellin transforms in the fundamental strips . Let with . If the functions and are also square-integrable over the interval , then Parseval's formula holds: [6]

The integration on the right hand side is done along the vertical line that lies entirely within the overlap of the (suitable transformed) fundamental strips.

We can replace by . This gives following alternative form of the theorem: Let and be functions with well-defined Mellin transforms in the fundamental strips . Let with and choose with . If the functions and are also square-integrable over the interval , then we have [6]

We can replace by . This gives following theorem: Let be a function with well-defined Mellin transform in the fundamental strip . Let with . If the function is also square-integrable over the interval , then Plancherel's theorem holds: [7]

As an isometry on L2 spaces[edit]

In the study of Hilbert spaces, the Mellin transform is often posed in a slightly different way. For functions in (see Lp space) the fundamental strip always includes , so we may define a linear operator as

In other words, we have set

This operator is usually denoted by just plain and called the "Mellin transform", but is used here to distinguish from the definition used elsewhere in this article. The Mellin inversion theorem then shows that is invertible with inverse

Furthermore, this operator is an isometry, that is to say for all (this explains why the factor of was used).

In probability theory[edit]

In probability theory, the Mellin transform is an essential tool in studying the distributions of products of random variables.[8] If X is a random variable, and X+ = max{X,0} denotes its positive part, while X − = max{−X,0} is its negative part, then the Mellin transform of X is defined as[9]

where γ is a formal indeterminate with γ2 = 1. This transform exists for all s in some complex strip D = {s : a ≤ Re(s) ≤ b} , where a ≤ 0 ≤ b.[9]

The Mellin transform of a random variable X uniquely determines its distribution function FX.[9] 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 product is equal to the product of the Mellin transforms of X and Y:[10]

Problems with Laplacian in cylindrical coordinate system[edit]

In the Laplacian in cylindrical coordinates in a generic dimension (orthogonal coordinates with one angle and one radius, and the remaining lengths) there is always a term:

For example, in 2-D polar coordinates the Laplacian is:

and in 3-D cylindrical coordinates the Laplacian is,

This term can be treated with the Mellin transform,[11] since:

For example, the 2-D Laplace equation in polar coordinates is the PDE in two variables:

and by multiplication:

with a Mellin transform on radius becomes the simple harmonic oscillator:

with general solution:

Now let's impose for example some simple wedge boundary conditions to the original Laplace equation:

these are particularly simple for Mellin transform, becoming:

These conditions imposed to the solution particularize it to:

Now by the convolution theorem for Mellin transform, the solution in the Mellin domain can be inverted:

where the following inverse transform relation was employed:

where .


The Mellin Transform is widely used in computer science for the analysis of algorithms[12] 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 for purely imaginary inputs. 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 magnitude of the Fourier transform of the original function.

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.

In quantum mechanics and especially quantum field theory, Fourier space is enormously useful and used extensively because momentum and position are Fourier transforms of each other (for instance, Feynman diagrams are much more easily computed in momentum space). In 2011, A. Liam Fitzpatrick, Jared Kaplan, João Penedones, Suvrat Raju, and Balt C. van Rees showed that Mellin space serves an analogous role in the context of the AdS/CFT correspondence.[13][14][15]


Table of selected Mellin transforms[edit]

Following list of interesting examples for the Mellin transform can be found in Bracewell (2000) and Erdélyi (1954):

Selected Mellin transforms
Function Mellin transform Region of convergence Comment
And generally is the Mellin transform of[16]
is the Dirac delta function.
is the Heaviside step function
is the Bessel function of the first kind.
is the Bessel function of the second kind
is the modified Bessel function of the second kind

See also[edit]


  1. ^ Mellin, Hj. "Zur Theorie zweier allgemeinen Klassen bestimmter Integrale". Acta Societatis Scientiarum Fennicæ. XXII, N:o 2: 1–75.
  2. ^ Whittaker, E.T.; Watson, G.N. (1996). A Course of Modern Analysis. Cambridge University Press.
  3. ^ Hardy, G. H.; Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes". Acta Mathematica. 41 (1): 119–196. doi:10.1007/BF02422942. (See notes therein for further references to Cahen's and Mellin's work, including Cahen's thesis.)
  4. ^ Wintner, Aurel (1947). "On Riemann's Reduction of Dirichlet Series to Power Series". American Journal of Mathematics. 69 (4): 769–789. doi:10.2307/2371798.
  5. ^ Flajolet, P.; Gourdon, X.; Dumas, P. (1995). "Mellin transforms and asymptotics: Harmonic sums" (PDF). Theoretical Computer Science. 144 (1–2): 3–58. doi:10.1016/0304-3975(95)00002-e.
  6. ^ a b Titchmarsh (1948, p. 95).
  7. ^ Titchmarsh (1948, p. 94).
  8. ^ Galambos & Simonelli (2004, p. 15)
  9. ^ a b c Galambos & Simonelli (2004, p. 16)
  10. ^ Galambos & Simonelli (2004, p. 23)
  11. ^ Bhimsen, Shivamoggi, Chapter 6: The Mellin Transform, par. 4.3: Distribution of a Potential in a Wedge, pp. 267–8
  12. ^ Philippe Flajolet and Robert Sedgewick. The Average Case Analysis of Algorithms: Mellin Transform Asymptotics. Research Report 2956. 93 pages. Institut National de Recherche en Informatique et en Automatique (INRIA), 1996.
  13. ^ A. Liam Fitzpatrick, Jared Kaplan, Joao Penedones, Suvrat Raju, Balt C. van Rees. "A Natural Language for AdS/CFT Correlators".
  14. ^ A. Liam Fitzpatrick, Jared Kaplan. "Unitarity and the Holographic S-Matrix"
  15. ^ A. Liam Fitzpatrick. "AdS/CFT and the Holographic S-Matrix", video lecture.
  16. ^ Jacqueline Bertrand, Pierre Bertrand, Jean-Philippe Ovarlez. The Mellin Transform. The Transforms and Applications Handbook, 1995, 978-1420066524. ffhal-03152634f


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