Mellin Transform Research Articles

Averaged Integral Transforms

An integral transform $\bar f(s) = \int f(t)\varphi (s,t)dt$ can be averaged over the convex hull of $\{ s_1 , \cdots ,s_k \} $ to produce an analogous function $\bar F\{ s_1 , \cdots ,s_k \} $ of several real or complex variables. The question arises whether it is legitimate to take the average under the integral sign, so that $\bar F\{ s_1 , \cdots ,s_k \} = \int f(t)\Phi (s_1 , \cdots ,s_k ;t)dt$, where $\Phi $ is the corresponding average of $\varphi $. Conditions for the validity of this equation are of interest in the theory of special functions because the kernel $\Phi $ may be a Bessel function, elliptic integral, or other hypergeometric function when $\varphi $ is the kernel of a Laplace, Fourier, or Stieltjes transformation. Sufficient conditions of validity are established in the case of these three transformations and the inverse Laplace, Fourier, and Mellin transformations. The averages have some but not all of the operational properties of the corresponding ordinary transforms. Some examples are given involving various special functions.

Saturation Theory in Connection with Mellin Transform Methods

If X denotes one of the spaces $L^p $, $1 \leqq p < \infty $, or C, where $L^p $ is the set of all f with $[\int _0^\infty | {f(r)} |^p r^{ - 1} dr]^{{1 / p}} < \infty $ and C the set of all f bounded and continuous on $(0,\infty )$ with $\lim _{a \to 1} [\sup _{0 < r < \infty } |f(ar) - f(r)|] = 0$, then $f \in X_{\sigma _1 ,\sigma _2 } $ provided $r^\sigma f(r) \in X$ for every $\sigma \in (\sigma _1 ,\sigma _2 )$. For $f \in X_{\sigma _1 ,\sigma _2 } $ a general approximation process $I_\vartheta (f;r) = \int _0^\infty f({r / \rho })k_\vartheta (\rho )\rho ^{ - 1} d\rho $ of Mellin convolution type is considered, the kernel $\{ k_\vartheta (r)\} $, $\varphi > 0$,satisfying suitable conditions such that it constitutes an approximate identity for $\vartheta \to 0 + $, i.e., \[ \mathop {\lim }\limits_{\vartheta \to 0 + } \left\| {r^\sigma [I_\vartheta (f;r) - f(r)]} \right\|_x = 0\quad {\text{for each }}\sigma \in (\sigma _1 ,\sigma _2 ), f\in X_{\sigma _1,\sigma _2 }.\] The main purpose of this paper is the study of saturation phenomena of the process $I_\vartheta (f;r)$. Using Butzer’s integral transform method, a complete saturation theorem is obtained under suitable conditions upon the Mellin transform of $k_\vartheta (r)$. As a significant application of the general results, the boundary behavior of the solution $u(r,\vartheta )$ of Dirichlet’s problem for the wedge $W = \{ (r,\vartheta )\mid 0 < r < \infty ,0 < \vartheta < \vartheta _0 \} $, $0 < \vartheta _0 < 2\pi $, is considered in detail. This problem is first raised in a strong sense (essentially as an abstract Cauchy problem) which allows a rigorous treatment of its solution via the classical Mellin transform method. In particular, boundary values $f_1 ,f_2 \in X_{ - \sigma _0 ,\sigma _0 } = {\pi / {\vartheta _0 }}$, are attained in the sense that for each $\sigma \in ( - \sigma _0 ,\sigma _0 )$, \[\mathop {\lim }\limits_{\vartheta \to 0 + } \left\| {r^\sigma \left[ {u(r,\vartheta ) - f_1 (r)} \right]} \right\|_X = 0,\quad \mathop {\lim }\limits_{\vartheta \to \vartheta _0 - } \left\| {r^\sigma \left[ {u(r,\vartheta ) - f_2 (r)} \right]} \right\|_X = 0.\] For, for example, symmetric boundary values $f_1 = f_2 = f$, the general saturation theorem then gives that the above quantities cannot tend to zero too rapidly; thus $\| {r^\sigma [u(r,\vartheta ) - f(r)]} \|_X = o(\vartheta )$, $\vartheta \to 0 + $, implies $f = 0$.

Relativistic Spin-Zero Wave Equation

We have studied the solutions of a wave equation which describes a spin-zero particle in the Coulomb field of a nucleus. An interesting feature of this equation is that the kernel is not of the Fredholm type. The behavior of the momentum space wavefunction for large momentum is not determined solely by the angular momentum state but, as in the cases of the Dirac and Klein-Gordon equations, it depends on the electric charge as well. Our analysis of the asymptotic properties is based on a Mellin transformation of the momentum space equation. This leads to a singular integral equation with a Cauchy-type kernel which may be treated by standard methods. The equation is shown to have unique solutions.

On the summation formula of Voronoi

A formula involving sums of the form _ d(n)f(n) and _ d(n)g(n) is derived, where d(n) is the number of divisors of n, andf(x), g(x) are Hankel transforms of each other. Many forms of such a formula, generally known as Voronoi's summation formula, are known, but we give a more symmetrical formula. Also, the reciprocal relation between f(x) and g(x) is expressed in terms of an elementary kernel, the cosine kernel, by introducing a function of the class L2(0, cc). We use L2-theory of Mellin and Fourier-Watson transformations. Introduction. In 1904 Voronoi [10] published the following general formula: If 7(n) is an arithmetic function andf(x) is continuous and has a finite number of maxima and minima in a a n2~_,na a = a One of the better known special cases of this formula is when 7(n)=d(n), the number of divisors of n, and a(x) = (2/i7)Ko(4i7x1/2) Yo(4i7x1/2), 8(x) = log x + 2y, y being Euler's constant and Y0, Ko denote Bessel functions of second and third kinds respectively, of order zero. This special case is generally known as Voronoi's summation formula. Later, this formula received considerable attention as a result of which many modifications were put forth by A. L. Dixon and W. L. Ferrar [2], J. R. Wilton [13], A. P. Guinand [3] and others. Most of the authors used complex analysis and in all the new forms of the Voronoi formula, the kernel used was a combination of the Bessel functions Yo(x) and Ko(x). Our object in this paper is to obtain a more symmetric and simplified form of Voronoi's formula, which holds under simple conditions. We state below the main result. First, a definition, due to Miller [6] and Guinand [4]. Received by the editors August 29, 1968 and, in revised form, February 18, 1969. AMS 1970 subject classifications. Primary 1OA20; Secondary 1OA20.

Approximate Formula for Rainbow Scattering Angle

An integral expression of the classical deflection function by the use of Mellin transformation has been presented, which is convenient for the differentitation with respect to the impact parameter b to give the relation between the rainbow scattering angle θ R and the reduced kinetic energy K of the incident particle. A simple and useful approximate formula, -θ R⋅K=0.29 γ+1.75, valid for the value of K not less than five, has been derived for the three-parameter(12, 6, 4)potential.

On the distribution of VOTAW's likelihood ratio criterion L for testing the bipolarity of a covariance matrix

AbstractIn this paper the exact distribution function of Votaw's likelihood ratio criterion L has been obtained in the form of an infinite series by applying Mellin's transformation on the moments and then by using the asymptotic expansion for log Γ(x + h). The author has actually calculated the values of the constants for some particular cases and has shown that the first four or five terms give a high degree of accuracy. An expression for the Integral probability of the distribution has also been obtained.

A Property of Mellin Transforms

Previous article Next article A Property of Mellin TransformsW. J. HarringtonW. J. Harringtonhttps://doi.org/10.1137/1009074PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] H. S. Carslaw and , J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, 1959 0095.30201 Google Scholar[2] C. J. Tranter, The use of the Mellin transform in finding the stress distribution in an infinite wedge, Quart. J. Mech. Appl. Math., 1 (1948), 125–130 MR0025666 0033.31203 CrossrefISIGoogle Scholar[3] Eli Sternberg and , W. T. Koiter, The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity, J. Appl. Mech., 25 (1958), 575–581 MR0102947 0084.40601 Google Scholar[4] A. Erdélyi, , W. Magnus, , F. Oberhettinger and , F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954xx+391, Bateman Manuscript Project MR0061695 0055.36401 Google Scholar[5] Henri Cartan, Elementary theory of analytic functions of one or several complex variables, Éditions Scientifiques Hermann, Paris, 1963, 228– MR0154968 0121.30501 Google Scholar[6] Balth van der Pol and , H. Bremmer, Operational calculus based on the two-sided Laplace integral, Cambridge, at the University Press, 1955xiii+415, London MR0072988 0066.09101 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Generating functions of Pollaczek polynomials: a revisit1 July 2019 | Integral Transforms and Special Functions, Vol. 30, No. 11 Cross Ref Bibliography21 January 2010 Cross Ref The Mellin Transform Cross Ref Mellin Transforms Cross Ref Mellin Transforms Cross Ref Naylor Transforms of Mellin TypeW. J. Harrington and K. A. Patel17 February 2012 | SIAM Journal on Mathematical Analysis, Vol. 4, No. 1AbstractPDF (465 KB)Stress boundary-value problems for infinite wedgesJournal of Elasticity, Vol. 1, No. 1 Cross Ref Volume 9, Issue 3| 1967SIAM Review History Submitted:10 October 1966Published online:18 July 2006 InformationCopyright © 1967 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1009074Article page range:pp. 542-547ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

On the Generalized Mellin Transform of a Complex Random Variable and Its Applications

On the Generalized Mellin Transform of a Complex Random Variable and Its Applications

The impact of a rigid circular cylinder on an elastic solid

A method is described for approximating to any degree of accuracy the solution of the following problem: An elastic body which is bounded by a plane on one side, but extends to infinity otherwise, is hit by a circular disk of given mass, radius, and initial speed perpendicular to the plane boundary. The whole surface of the disk enters into contact with the elastic body at the same time and stays in contact at all its points from then on. The disk is assumed to be rigid, i.e. it does not allow the particles of the elastic body in the contact area to move relative to each other in a direction perpendicular to the plane boundary. For the relative motion of these particles parallel to the face of the disk several conditions are considered, representing perfect lubrication, various degrees of viscous friction and perfect adherence. With the help of various Mellin transformations a method is indicated which leads to an expansion of the motion in powers of the Laplace transform variable. The case of perfect adherence needs some special consideration, and a simple approximation for the static problem is found. The case of perfect lubrication is then treated in more detail by a different method which replaces the condition of constant normal displacement in the contact area by an equivalent number of requirements for certain averages over the normal displacement in the contact area. The condition of rigidity for the disk is not exactly satisfied, but it is possible to judge the accuracy of the approximation (with the help of asymptotic expansions in the Laplace transform variable) at the initial time, when discrepancies are largest. The concept of ‘mode of vibration’ is introduced. Any transient in the coupled system of elastic body and rigid disk can be described as superposition of modes, each of which is an exponentially damped harmonic oscillation of the coupled system with a frequency and damping constant independent of the particular transient. The motion of the impinging disk is then seen to be dominated mostly by the lowest mode, provided the mass of the disk is not too small. The displacement perpendicular to the boundary outside of the contact area has been calculated. This calculation is not more difficult than the corresponding one in the case of a point-like source at the plane boundary of an elastic solid. Numerical computations were carried out for the case of perfect lubrication with the help of the Elecom digital computer in order to determine the first two modes and their contributions to the motion of the disk. As long as Poisson’s ratio for the elastic solid exceeds 1/4, the results do not depend strongly on the value of Poisson’s ratio. The ratio of the areal mass densities of the disk to the elastic solid is the main parameter of the theory. The shear wave velocity of the elastic solid determines the time scale of the motion.

Applications of Mellin Transforms to Some Problems of Statistical Mechanics

Mellia transforms are used to obtain an asymptotic form of an integral contsining a power series, in which tbe term by term integration cannot be allowed. Examples are given concerning problems of statistical mechanics, e.g., the partition funuction of an electron gas, the connelation energy of a free electron gas, and the partition function of a hard sphere gas at zero-temperature limit. (auth)

Application of Mellin and Hankel Transforms to Networks with Time-Varying Parameters

Integral transform techniques for solving linear integro-differential equations can provide insight and flexibility in solving physical problems, especially network problems. The type of differential equation which describes the physical system will dictate the transform that should be applied to simplify the solution and this paper deals with two transforms, namely, the Mellin transform and the Hankel transform. The Laplace transform can be used to solve linear constant coefficient differential equations or networks which are represented by this type of equation. A familiarity with this transform is assumed and is not covered in this paper. Mellin transforms may be applied to networks which yield the Euler-Cauchy differential equation. This transform will simplify the solution of such an equation. A transform table, similar to that type used in Laplace transform theory, is developed and applied to network problems. Hankel transforms may be applied to networks which yield the Bessel differential equation or variations of this equation. Unlike the Laplace and Mellin transforms, the Hankel transform is symmetric and the transformed variable is a real, rather than a complex variable. A transform table of both operations and functions is developed anti applied to network problems as before. Three methods can be used to establish the table of transform pairs. They can be described as: performing the integral operation, applying the table of operations on known transform pairs, and deriving the Hankel transform from the Laplace transform. With both transforms, the applications are made to problems in analysis, instrumentation, and synthesis.

Stress Distributions within Solids of Revolution

AbstractIn this paper the problem of axially symmetric stress distribution within semi‐infinite transversely isotropic solids bounded by one or two cones has been considered and a formal solution is obtained with the help of Mellin Transform. As an example, the problem of stress distribution within semi‐infinite solids under a uniform normal loading applied over a circular area, is worked out in detail and the results in the special case of isotropy, agree with those previously known.

On the Exact Distribution of a Test in Multivariate Analysis

SUMMARY By rearranging the factors of the complicated r th moment of the distribution of a statistic in multivariate analysis and using Mellin’s Transformation it is possible to find the exact distribution of the statistic. This cannot be found easily by other methods.

Some Applications of the Mellin Transform in Statistics

It is well known that the Fourier transform is a powerful analytical tool in studying the distribution of sums of independent random variables. In this paper it is pointed out that the Mellin transform is a natural analytical tool to use in studying the distribution of products and quotients of independent random variables. Formulae are given for determining the probability density functions of the product and the quotient $\frac{\xi}{\eta}$, where $\xi$ and $\eta$ are independent positive random variables with p.d.f.'s $f(x)$ and $g(y)$, in terms of the Mellin transforms $F(s) = \int_0^\infty f(x) x^{s-1} dx$ and $G(s) = \int_0^\infty g(y)y^{s-1} dy$. An extension of the transform technique to random variables which are not everywhere positive is given. A number of examples including Student's $t$-distribution and Snedecor's $F$-distribution are worked out by the technique of this paper.

Applications of Mellin's Transformation to Integral Equations

Applications of Mellin's Transformation to Integral Equations