Mellin transform explained

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 isoften 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 complex-valued function defined on

R ^{x} _{+=} ( 0 , inf ty) is the function l M f of complex variable s given (where it exists, see Fundamental strip below) by$$\backslash left\backslash (s)\; =\; \backslash varphi(s)=\backslash int\_0^\backslash infty\; x^\; f(x)\; \backslash ,\; dx\; =\; \backslash int\_f(x)\; x^s\; \backslash frac.$$Notice that dx/x is a Haar measure on the multiplicative group R ^{x} _{+} and x\mapsto x ^{s} is a (in general non-unitary) multiplicative character.The inverse transform is $$\backslash left\backslash (x)\; =\; f(x)=\backslash frac\; \backslash int\_^\; x^\; \backslash varphi(s)\backslash ,\; ds.$$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

The two-sided Laplace transform may be defined in terms of the Mellin transform by$$\backslash left\backslash (s)\; =\; \backslash left\backslash (s)$$and conversely we can get the Mellin transform from the two-sided Laplace transform by$$\backslash left\backslash (s)\; =\; \backslash left\backslash (s).$$

The Mellin transform may be thought of as integrating using a kernel x^s with respect to the multiplicative Haar measure, $\backslash frac$, which is invariant under dilation

x \mapsto ax , so that $\backslash frac\; =\; \backslash frac;$ the two-sided Laplace transform integrates with respect to the additive Haar measure dx , which is translation invariant, so that d(x + a) = dx .

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$$\backslash left\backslash (-s)\; =\; \backslash left\backslash (-is)=\; \backslash left\backslash (-is)\backslash \; .$$We may also reverse the process and obtain$$\backslash left\backslash (s)\; =\; \backslash left\backslash (s)\; =\; \backslash left\backslash (-is)\backslash \; .$$

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

f(x) = e ^{-x} is$$\backslash Gamma(s)\; =\; \backslash int\_0^\backslash infty\; x^e^\; dx$$where \Gamma(s) is the gamma function. \Gamma(s) is a meromorphic function with simple poles at z = 0 , -1 , -2 , ... .^[2] Therefore, \Gamma(s) is analytic for \Re(s)> 0 . Thus, letting c> 0 and z ^{-s} on the principal branch, the inverse transform gives$$e^=\; \backslash frac\; \backslash int\_^\; \backslash Gamma(s)\; z^\; \backslash ;\; ds\; .$$

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

Polynomial functions

Since $\backslash int\_0^\backslash infty\; x^a\; dx$ is not convergent for any value of

a\in R , 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, ifthen$$\backslash mathcal\; M\; f\; (s)=\; \backslash int\_0^1\; x^x^adx\; =\; \backslash int\_0^1\; x^dx\; =\; \backslash frac\; 1\; .$$

Thus

l M f (s) has a simple pole at s =- a and is thus defined for \Re (s)> - a . Similarly, ifthen$$\backslash mathcal\; M\; f\; (s)=\; \backslash int\_1^\backslash infty\; x^x^bdx\; =\; \backslash int\_1^\backslash infty\; x^dx\; =\; -\; \backslash frac\; 1\; .$$Thus l M f (s) has a simple pole at s =- b and is thus defined for \Re (s)< - b .

Exponential functions

For

p > 0 , let f(x) = e ^{-px} . Then$$\backslash mathcal\; M\; f\; (s)\; =\; \backslash int\_0^\backslash infty\; x^\; e^\backslash frac\; =\; \backslash int\_0^\backslash infty\; \backslash left(\backslash frac\; \backslash right)^e^\; \backslash frac\; =\; \backslash frac\backslash int\_0^\backslash infty\; u^e^\; \backslash frac\; =\; \backslash frac\backslash Gamma(s).$$

Zeta function

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

\zeta(s) . Let $f(x)=\backslash frac$. Then$$\backslash mathcal\; M\; f\; (s)\; =\; \backslash int\_0^\backslash infty\; x^\backslash fracdx\; =\; \backslash int\_0^\backslash infty\; x^\backslash fracdx\; =\; \backslash int\_0^\backslash infty\; x^\backslash sum\_^\backslash infty\; e^dx\; =\; \backslash sum\_^\backslash infty\; \backslash int\_0^\backslash infty\; x^e^\backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash frac\backslash Gamma(s)=\backslash Gamma(s)\backslash zeta(s)\; .$$Thus,$$\backslash zeta(s)\; =\; \backslash frac\backslash int\_0^\backslash infty\; x^\backslash frac\; dx.$$

Generalized Gaussian

For

p > 0 , let f(x) = e ^{- x ^{p}} (i.e. f is a generalized Gaussian distribution without the scaling factor.) Then$$\backslash mathcal\; M\; f\; (s)\; =\; \backslash int\_0^\backslash infty\; x^e^dx\; =\; \backslash int\_0^\backslash infty\; x^x^e^dx\; =\; \backslash int\_0^\backslash infty\; x^(x^p)^e^dx\; =\; \backslash frac\backslash int\_0^\backslash infty\; u^e^du\; =\; \backslash frac\; .$$In particular, setting s =1 recovers the following form of the gamma function$$\backslash Gamma\backslash left(1+\backslash frac\backslash right)\; =\; \backslash int\_0^\backslash infty\; e^dx.$$

Power series and Dirichlet series

Generally, assuming necessary convergence, we can connect Dirichlet series and related power series$$F(s)=\backslash sum\backslash limits\_^\backslash frac,\; \backslash quad\; f(z)=\backslash sum\backslash limits\_^a\_nz^n$$by the formal identity involving Mellin transform:^[4] $$\backslash Gamma(s)F(s)=\backslash int\_^x^f(e^)dx$$

Fundamental strip

For

\alpha,\beta\in R , let the open strip \langle\alpha,\beta\rangle be defined to be all s\in C such that s = \sigma + it with \alpha < \sigma < \beta. The fundamental strip of l{M} f(s) is defined to be the largest open strip on which it is defined. For example, for a > b the fundamental strip ofis \langle - a, - b \rangle. As seen by this example, the asymptotics of the function as x\to 0 ^{+} define the left endpoint of its fundamental strip, and the asymptotics of the function as x\to +inf ty define its right endpoint. To summarize using Big O notation, if f is O(x ^{a)} as x\to 0 ^{+} and O(x ^{b)} as x\to +inf ty, then l{M} f(s) is defined in the strip \langle - a, - b \rangle. ^[5]

An application of this can be seen in the gamma function,

\Gamma(s). Since f(x) = e ^{-x} is O(x ^{0)} as x\to 0 ^{+} and O(x ^{k} ) for all k, then \Gamma(s) = l{M} f(s) should be defined in the strip \langle 0 , +inf ty \rangle, which confirms that \Gamma(s) is analytic for \Re(s) > 0 .

Properties

The properties in this table may be found in and .

Properties of the Mellin transform

Function	Mellin transform	Fundamental strip	Comments
f(x)	\tilde{f}(s) = \{ l{M}f\}(s)=\int ^{inf ty} _{0} f(x) x ^{s} (dx)/(x)	\alpha < \Re s < \beta	Definition
x ^{\nu} f(x)	\tilde{f}(s + \nu)	\alpha - \Re \nu < \Re s < \beta - \Re \nu
f(x ^{\nu} )	(1)/(\tilde{f}\left( |\nu|) (s)/(\nu) \right)	\alpha < \nu ^{-1} \Re s < \beta	\nu\in R , \nu ≠ 0
f(x ^{-1} )	\tilde{f}( - s)	- \beta < \Re s < - \alpha
x ^{-1} f(x ^{-1} )	\tilde{f}( 1- s)	1- \beta < \Re s < 1- \alpha	Involution
\overline{f(x)}	\overline{\tilde{f}(\overline{s})}	\alpha < \Re s < \beta	Here \overline{z} denotes the complex conjugate of z .
f(\nu x)	\nu ^{-s} \tilde{f}(s)	\alpha < \Re s < \beta	\nu > 0 , Scaling
f(x) ln x	\tilde{f}'(s)	\alpha < \Re s < \beta
f'(x)	- (s -1 ) \tilde{f}(s -1 )	\alpha +1 < \Re s < \beta +1	The domain shift is conditional and requires evaluation against specific convergence behavior.
\left( (d)/(dx) \right) ^{n} f(x)	( -1 ) ^{n} (\Gamma(s))/(\Gamma(s - n)) \tilde{f}(s - n)	\alpha + n < \Re s < \beta + n
xf'(x)	- s \tilde{f}(s)	\alpha < \Re s < \beta
\left( x (d)/(dx) \right) ^{n} f(x)	( - s) ^{n} \tilde{f}(s)	\alpha < \Re s < \beta
\left( (d)/(dx) x\right) ^{n} f(x)	( 1- s) ^{n} \tilde{f}(s)	\alpha < \Re s < \beta
\int ^{x} _{0} f(y) dy	- s ^{-1} \tilde{f}(s +1 )	\alpha -1 < \Re s < min (\beta -1 , 0 )	Valid only if the integral exists.
\int ^{inf ty} _{x} f(y) dy	s ^{-1} \tilde{f}(s +1 )	max (\alpha -1 , 0 ) < \Re s < \beta -1	Valid only if the integral exists.
\int ^{inf ty} _{0} f _{1 \left( (x)/(y)} \right) f _{2(y)} (dy)/(y)	\tilde{f} _{1(s)} \tilde{f} _{2(s)}	max (\alpha _{1,\alpha} _{2)} < \Re s < min (\beta _{1,\beta} _{2)}	Multiplicative convolution
x ^{\mu} \int ^{inf ty} _{0} y ^{\nu} f _{1 \left( (x)/(y)} \right) f _{2(y)} dy	\tilde{f} _{1(s+\mu)} \tilde{f} _{2(s+\mu+\nu+1)}		Multiplicative convolution (generalized)
x ^{\mu} \int ^{inf ty} _{0} y ^{\nu} f _{1(xy)} f _{2(y)} dy	\tilde{f} _{1(s+\mu)} \tilde{f} _{2(1-s-\mu+\nu)}		Multiplicative convolution (generalized)
f _{1(x)} f _{2(x)}	(1)/(2 \pi i) \int ^{c + i inf ty} _{c - i inf ty} \tilde{f} _{1(r)\tilde{f}} _{2(s-r)dr}	\begin{aligned} \alpha _{2+c&<\Re} s<\beta _{2+c} \ \alpha _{1&<c<\beta} _{1} \end{aligned}	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

Let

f _{1(x)} and f _{2(x)} be functions with well-definedMellin transforms \tilde{f} _{1,2} (s) = l{M}\{f _{1,2} \}(s) in the fundamental strips \alpha _{1,2} <\real s<\beta _{1,2} .Let c\in R with max (\alpha _{1,1-\beta} _{2)<c<min(\beta} _{1,1-\alpha} _{2)} .If the functions x ^{c-1/2} f _{1(x)} and x ^{1/2-c} f _{2(x)} are also square-integrable over the interval ( 0 , inf ty) , then Parseval's formula holds:^[6] $$\backslash int\_0^\; f\_1(x)\backslash ,f\_2(x)\backslash ,dx\; =\; \backslash frac\; \backslash int\_^\; \backslash tilde(s)\backslash ,\backslash tilde(1-s)\backslash ,ds$$The integration on the right hand side is done along the vertical line \Re r = c thatlies entirely within the overlap of the (suitable transformed) fundamental strips.

We can replace

f _{2(x)} by f ^{s _{0-1}} _{2 (x)x} . This gives following alternative form of the theorem:Let f _{1(x)} and f _{2(x)} be functions with well-definedMellin transforms \tilde{f} _{1,2} (s) = l{M}\{f _{1,2} \}(s) in the fundamental strips \alpha _{1,2} <\real s<\beta _{1,2} .Let c\in R with \alpha _{1<c<\beta} _{1} andchoose s _{0\inC} with \alpha _{2<} \Re s _{0} - c <\beta _{2} .If the functions x ^{c-1/2} f _{1(x)} and x ^{s _{0-c-1/2}} f _{2(x)} are also square-integrable over the interval ( 0 , inf ty) , then we have^[6] $$\backslash int\_0^\; f\_1(x)\backslash ,f\_2(x)\backslash ,x^\backslash ,dx\; =\; \backslash frac\; \backslash int\_^\; \backslash tilde(s)\backslash ,\backslash tilde(s\_0-s)\backslash ,ds$$We can replace f _{2(x)} by \overline{f _{1(x)}} .This gives following theorem:Let f(x) be a function with well-defined Mellin transform \tilde{f}(s) = l{M}\{f\}(s) in the fundamental strip \alpha<\real s<\beta .Let c\in R with \alpha<c<\beta .If the function x ^{c-1/2} f(x) is also square-integrable over the interval ( 0 , inf ty) , then Plancherel's theorem holds:^[7] $$\backslash int\_0^\; |f(x)|^2\backslash ,x^dx\; =\; \backslash frac\; \backslash int\_^\; |\; \backslash tilde(c+it)\; |^2\; \backslash ,dt$$

As an isometry on L² spaces

In the study of Hilbert spaces, the Mellin transform is often posed in a slightly different way. For functions in

L ^{2(0,infty)} (see Lp space) the fundamental strip always includes \tfrac{ 1 }{ 2 } + i R , so we may define a linear operator \tilde{ l{M}} as$$\backslash tilde\backslash colon\; L^2(0,\backslash infty)\backslash to\; L^2(-\backslash infty,\backslash infty),$$$$\backslash (s)\; :=\; \backslash frac\backslash int\_0^\; x^\; f(x)\backslash ,dx.$$In other words, we have set$$\backslash (s):=\backslash tfrac\backslash (\backslash tfrac\; +\; is).$$This operator is usually denoted by just plain l{M} and called the "Mellin transform", but \tilde{ l{M}} is used here to distinguish from the definition used elsewhere in this article. The Mellin inversion theorem then shows that \tilde{ l{M}} is invertible with inverse$$\backslash tilde^\backslash colon\; L^2(-\backslash infty,\backslash infty)\; \backslash to\; L^2(0,\backslash infty),$$$$\backslash (x)\; =\; \backslash frac\backslash int\_^\; x^\; \backslash varphi(s)\backslash ,ds.$$Furthermore, this operator is an isometry, that is to say \|\tilde{ l{M}} f\| _{L ^{2(-infty,infty)}} = \|f\| _{L ^{2(0,infty)}} for all f\in L ^{2(0,infty)} (this explains why the factor of 1 /\sqrt{ 2 \pi} was used).

In probability theory

In probability theory, the Mellin transform is an essential tool in studying the distributions of products of random variables. If X is a random variable, and denotes its positive part, while is its negative part, then the Mellin transform of X is defined as$$\backslash mathcal\_X(s)\; =\; \backslash int\_0^\backslash infty\; x^s\; dF\_(x)\; +\; \backslash gamma\backslash int\_0^\backslash infty\; x^s\; dF\_(x),$$where γ is a formal indeterminate with . This transform exists for all s in some complex strip, where .

The Mellin transform

l{M} _{X(it)} of a random variable X uniquely determines its distribution function F_X. 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:$$\backslash mathcal\_(s)\; =\; \backslash mathcal\_X(s)\backslash mathcal\_Y(s)$$

Problems with Laplacian in cylindrical coordinate system

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:$$\backslash frac\; \backslash frac\; \backslash left(r\; \backslash frac\; \backslash right)\; =\; f\_\; +\; \backslash frac$$

For example, in 2-D polar coordinates the Laplacian is:$$\backslash nabla^2\; f\; =\; \backslash frac\; \backslash frac\; \backslash left(r\; \backslash frac\; \backslash right)\; +\; \backslash frac\; \backslash frac$$and in 3-D cylindrical coordinates the Laplacian is,$$\backslash nabla^2\; f\; =\; \backslash frac\; \backslash frac\; \backslash left(r\; \backslash frac\; \backslash right)\; +\; \backslash frac\; \backslash frac\; +\; \backslash frac.$$

This term can be treated with the Mellin transform,^[8] since:$$\backslash mathcal\; M\; \backslash left(r^2\; f\_\; +\; r\; f\_r,\; r\; \backslash to\; s\; \backslash right)\; =\; s^2\; \backslash mathcal\; M\; \backslash left(f,\; r\; \backslash to\; s\; \backslash right)\; =\; s^2\; F$$

For example, the 2-D Laplace equation in polar coordinates is the PDE in two variables:$$r^2\; f\_\; +\; r\; f\_r\; +\; f\_\; =\; 0$$and by multiplication:$$\backslash frac\; \backslash frac\; \backslash left(r\; \backslash frac\; \backslash right)\; +\; \backslash frac\; \backslash frac\; =\; 0$$with a Mellin transform on radius becomes the simple harmonic oscillator:$$F\_\; +\; s^2\; F\; =\; 0$$with general solution:$$F\; (s,\; \backslash theta)\; =\; C\_1(s)\; \backslash cos\; (s\backslash theta)\; +\; C\_2(s)\; \backslash sin\; (s\; \backslash theta)$$

Now let's impose for example some simple wedge boundary conditions to the original Laplace equation:$$f(r,-\backslash theta\_0)\; =\; a(r),\; \backslash quad\; f(r,\backslash theta\_0)\; =\; b(r)$$these are particularly simple for Mellin transform, becoming:$$F(s,-\backslash theta\_0)\; =\; A(s),\; \backslash quad\; F(s,\backslash theta\_0)\; =\; B(s)$$

These conditions imposed to the solution particularize it to:$$F\; (s,\; \backslash theta)\; =\; A(s)\; \backslash frac\; +\; B(s)\; \backslash frac$$

Now by the convolution theorem for Mellin transform, the solution in the Mellin domain can be inverted:$$f(r,\; \backslash theta)\; =\; \backslash frac\; \backslash int\_0^\backslash infty\; \backslash left\; (\backslash frac\; +\; \backslash frac\; \backslash right)\; x^\; \backslash ,\; dx$$where the following inverse transform relation was employed:$$\backslash mathcal\; M^\; \backslash left(\backslash frac\; ;\; s\; \backslash to\; r\; \backslash right)\; =\; \backslash frac\; 1\; \backslash frac$$where

m = (\pi)/(2 \theta _{0}) .

Applications

The Mellin Transform is widely used in computer science for the analysis of algorithms^[9] 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.^{[10] [11] [12]}

Examples

• Perron's formula describes the inverse Mellin transform applied to a Dirichlet series.
• The Mellin transform is used in analysis of the prime-counting function and occurs in discussions of the Riemann zeta function.
• Inverse Mellin transforms commonly occur in Riesz means.
• The Mellin transform can be used in audio timescale-pitch modification .

Table of selected Mellin transforms

Following list of interesting examples for the Mellin transform can be found in and :

Selected Mellin transforms

Function f(x)	Mellin transform \tilde{f}(s) = l{M}\{f\}(s)	Region of convergence	Comment
e ^{-x}	\Gamma(s)	0 < \Re s < inf ty
e ^{-x} -1	\Gamma(s)	-1 < \Re s < 0
e ^{-x} -1 + x	\Gamma(s)	-2 < \Re s < -1	And generally \Gamma(s) is the Mellin transform of^[13] e ^{-x} - \sum ^{N -1} _{n =0} (( -1 ) ^{n})/(n !) x ^{n,}
e ^{- x ^{2}}	\tfrac{ 1 }{ 2 }\Gamma(\tfrac{ 1 }{ 2 }s)	0 < \Re s < inf ty
erfc (x)	(\Gamma(\tfrac{ 1)/(2) ( 1+ s))}{\sqrt{\pi} s}	0 < \Re s < inf ty
e ^{- ( ln x) ^{2}}	\sqrt{\pi} e ^{\tfrac{1} { 4 }s ^{2}}	-inf ty < \Re s < inf ty
\delta(x - a)	a ^{s-1}	-inf ty < \Re s < inf ty	a> 0 , \delta(x) is the Dirac delta function.
u( 1- x) = \left\{ \begin{aligned} & 1 && if 0 <x< 1 &\ & 0 && if 1 <x< inf ty &\end{aligned} \right. | (1)/(s) | 0 < \Re s < inf ty | u(x) is the Heaviside step function|- | - u(x -1 ) = \left\{ \begin{aligned} & 0 && if 0 <x< 1 &\ & -1 && if 1 <x< inf ty &\end{aligned} \right. | (1)/(s) | -inf ty < \Re s < 0 ||- | u( 1- x)x ^{a} = \left\{ \begin{aligned} &x ^{a} && if 0 <x< 1 &\ & 0 && if 1 <x< inf ty &\end{aligned} \right. | (1)/(s + a) | - \Re a < \Re s < inf ty | |- | - u(x -1 )x ^{a} = \left\{ \begin{aligned} & 0 && if 0 <x< 1 &\ & - x ^{a} && if 1 <x< inf ty &\end{aligned} \right. | (1)/(s + a) | -inf ty < \Re s < - \Re a ||- | u( 1- x)x ^{a} ln x = \left\{ \begin{aligned} &x ^{a} ln x && if 0 <x< 1 &\ & 0 && if 1 <x< inf ty &\end{aligned} \right. | (1)/((s + a) ^{2}) | - \Re a < \Re s < inf ty | |- | - u(x -1 )x ^{a} ln x = \left\{ \begin{aligned} & 0 && if 0 <x< 1 &\ & - x ^{a} ln x && if 1 <x< inf ty &\end{aligned} \right. | (1)/((s + a) ^{2}) | -inf ty < \Re s < - \Re a ||- | (1)/(1+ x) | (\pi)/(\sin(\pi s)) | 0 < \Re s < 1 ||- | (1)/(1- x) | (\pi)/(\tan(\pi s)) | 0 < \Re s < 1 ||- | (1)/(1+ x ^{2}) | (\pi)/(2 \sin(\tfrac{ 1) { 2 }\pi s)} | 0 < \Re s < 2 ||- | ln ( 1+ x) | (\pi)/(s\sin(\pi s)) | -1 < \Re s < 0 ||- | \sin(x) | \sin(\tfrac{ 1 }{ 2 }\pi s) \Gamma(s) | -1 < \Re s < 1 ||- | \cos(x) | \cos(\tfrac{ 1 }{ 2 }\pi s) \Gamma(s) | 0 < \Re s < 1 ||- | e ^{ix} | e ^{i\pi} \Gamma(s) | 0 < \Re s < 1 ||- | J _{0(x)} | (2 ^{s-1})/(\pi) \sin(\pi s/ 2 ) \left[\Gamma(s/ 2 )\right] ^{2} | 0 < \Re s < \tfrac{ 3 }{ 2 } | J _{0(x)} is the Bessel function of the first kind.|- | Y _{0(x)} | - (2 ^{s-1})/(\pi) \cos(\pi s/ 2 ) \left[\Gamma(s/ 2 )\right] ^{2} | 0 < \Re s < \tfrac{ 3 }{ 2 } | Y _{0(x)} is the Bessel function of the second kind|- | K _{0(x)} | 2 ^{s-2} \left[\Gamma(s/ 2 )\right] ^{2} | 0 < \Re s < inf ty | K _{0(x)} is the modified Bessel function of the second kind|} See also Mellin inversion theorem Perron's formula Ramanujan's master theorem References Book: Lokenath Debnath. Dambaru Bhatta. Integral Transforms and Their Applications. 19 April 2016. CRC Press. 978-1-4200-1091-6. Book: Galambos . Janos . Simonelli . Italo . 2004 . Products of random variables: applications to problems of physics and to arithmetical functions . Marcel Dekker, Inc. . 0-8247-5402-6 . registration . Book: Paris . R. B. . Kaminski . D. . Asymptotics and Mellin-Barnes Integrals . registration . Cambridge University Press . 2001 . 9780521790017 . Book: A. D. . Polyanin . A. V. . Manzhirov . Handbook of Integral Equations . CRC Press . Boca Raton . 1998 . 0-8493-2876-4 . Book: Bracewell . Ronald N. . 2000 . The Fourier Transform and Its Applications . 3rd. Book: Erdélyi . Arthur . 1954 . Tables of Integral Transforms . 1 . McGraw-Hill. Book: Titchmarsh . E.C. . 1948 . Introduction to the Theory of Fourier Integrals . 2nd. P. . Flajolet . X. . Gourdon . P. . Dumas . Mellin transforms and asymptotics: Harmonic sums . Theoretical Computer Science . 144 . 1–2 . 3–58 . 1995 . 10.1016/0304-3975(95)00002-e. Tables of Integral Transforms at EqWorld: The World of Mathematical Equations. Some Applications of the Mellin Transform in Statistics (paper) External links Philippe Flajolet, Xavier Gourdon, Philippe Dumas, Mellin Transforms and Asymptotics: Harmonic sums. Antonio Gonzáles, Marko Riedel Celebrando un clásico, newsgroup es.ciencia.matematicas Juan Sacerdoti, Funciones Eulerianas (in Spanish). Mellin Transform Methods, Digital Library of Mathematical Functions, 2011-08-29, National Institute of Standards and Technology Antonio De Sena and Davide Rocchesso, A FAST MELLIN TRANSFORM WITH APPLICATIONS IN DAFX

Notes and References

1. Mellin. Hj.. Zur Theorie zweier allgemeinen Klassen bestimmter Integrale. Acta Societatis Scientiarum Fennicæ. XXII, N:o 2. 1–75.
2. Book: E.T. . Whittaker . E. T. Whittaker. G.N. . Watson. G. N. Watson . . 1996 . Cambridge University Press.
3. G. H. . Hardy. G. H. Hardy . J. E. . Littlewood. J. E. Littlewood . Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes . . 41 . 1 . 1916 . 119–196 . 10.1007/BF02422942 . free . (See notes therein for further references to Cahen's and Mellin's work, including Cahen's thesis.)
4. Aurel . Wintner. Aurel Wintner . On Riemann's Reduction of Dirichlet Series to Power Series . . 69 . 4 . 1947 . 769–789 . 10.2307/2371798. free .
5. P. . Flajolet . X. . Gourdon . P. . Dumas . Mellin transforms and asymptotics: Harmonic sums . Theoretical Computer Science . 144 . 1–2 . 3–58 . 1995 . 10.1016/0304-3975(95)00002-e.
6. .
7. .
8. Bhimsen, Shivamoggi, Chapter 6: The Mellin Transform, par. 4.3: Distribution of a Potential in a Wedge, pp. 267–8
9. 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.
10. A. Liam Fitzpatrick, Jared Kaplan, Joao Penedones, Suvrat Raju, Balt C. van Rees. "A Natural Language for AdS/CFT Correlators".
11. A. Liam Fitzpatrick, Jared Kaplan. "Unitarity and the Holographic S-Matrix"
12. A. Liam Fitzpatrick. "AdS/CFT and the Holographic S-Matrix", video lecture.
13. Jacqueline Bertrand, Pierre Bertrand, Jean-Philippe Ovarlez. The Mellin Transform. The Transforms and Applications Handbook, 1995, 978-1420066524. ffhal-03152634f