Theorems connecting Stieltjes transform and Hankel transform

2020 ◽  
Vol 14 (2) ◽  
pp. 605-613
Author(s):  
Virendra Kumar
Keyword(s):  
Hankel Transform ◽  
Stieltjes Transform

scholarly journals The H-function transform

1970 ◽  
Vol 11 (2) ◽  
pp. 142-148 ◽  
Author(s):  
K. C. Gupta ◽  
P. K. Mittal
Keyword(s):  
Laplace Transform ◽  
Inversion Formula ◽  
Integral Transform ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Mathematics And Physics

Here we introduce a new integral transform whose kernel is the H-function. Since most of the important functions occurring in Applied Mathematics and Physics are special cases of the H-function, various integral transforms involving these functions as kernels follow as special cases of our transform. We mention some of them here and observe that a study of this transform gives general and useful results which serve as key formulae for several important integral transforms viz. Laplace transform, Hankel transform. Stieltjes transform and the various generalizations of these transforms. In the end we establish an inversion formula for the new transform and point out its special cases which are generalizations of results found recently.

Titchmarsh's theorem of Hankel transform

2019 ◽  
pp. 11-24
Author(s):  
Mohamed-Ahmed Boudref
Keyword(s):  
Number Theory ◽  
Data Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Analytic Number Theory ◽  
Hankel Transform ◽  
Partial Differential ◽  
Analytic Number ◽  
Bessel Transform

Hankel transform (or Fourier-Bessel transform) is a fundamental tool in many areas of mathematics and engineering, including analysis, partial differential equations, probability, analytic number theory, data analysis, etc. In this article, we prove an analog of Titchmarsh's theorem for the Hankel transform of functions satisfying the Hankel-Lipschitz condition.

The cost of a general stochastic epidemic

1972 ◽  
Vol 9 (02) ◽  
pp. 257-269 ◽  
Author(s):  
J. Gani ◽  
D. Jerwood
Keyword(s):  
Recursive Equation ◽  
Stieltjes Transform ◽  
Stochastic Epidemic ◽  
The Mean ◽  
The Cost ◽  
Mean Cost ◽  
General Stochastic

This paper is concerned with the cost Cis = aWis + bTis (a, b > 0) of a general stochastic epidemic starting with i infectives and s susceptibles; Tis denotes the duration of the epidemic, and Wis the area under the infective curve. The joint Laplace-Stieltjes transform of (Wis, Tis ) is studied, and a recursive equation derived for it. The duration Tis and its mean Nis are considered in some detail, as are also Wis and its mean Mis . Using the results obtained, bounds are found for the mean cost of the epidemic.

scholarly journals Interior-stress fields produced by a general axisymmetric punch

Friction ◽  
2021 ◽  
Author(s):  
Longxiang Yang ◽  
Zhanjiang Wang ◽  
Weiji Liu ◽  
Guocheng Zhang ◽  
Bei Peng
Keyword(s):  
Hankel Transform ◽  
Elastic Half Space ◽  
Stress Fields ◽  
Dual Integral Equations ◽  
Total Load ◽  
Interior Stress ◽  
Von Mises ◽  
Von Mises Stresses ◽  
Relationship Of ◽  
Analytical Expressions

AbstractThis work is a supplement to the work of Sneddon on axisymmetric Boussinesq problem in 1965 in which the distributions of interior-stress fields are derived here for a punch with general profile. A novel set of mathematical procedures is introduced to process the basic elastic solutions (obtained by the method of Hankel transform, which was pioneered by Sneddon) and the solution of the dual integral equations. These processes then enable us to not only derive the general relationship of indentation depth D and total load P that acts on the punch but also explicitly obtain the general analytical expressions of the stress fields beneath the surface of an isotropic elastic half-space. The usually known cases of punch profiles are reconsidered according to the general formulas derived in this study, and the deduced results are verified by comparing them with the classical results. Finally, these general formulas are also applied to evaluate the von Mises stresses for several new punch profiles.

A METHOD FOR THE DIRECT INTERPRETATION OF ELECTRICAL SOUNDINGS MADE OVER A FAULT OR DIKE

Geophysics ◽  
1973 ◽  
Vol 38 (4) ◽  
pp. 762-770 ◽  
Author(s):  
Terry Lee ◽  
Ronald Green
Keyword(s):  
Bessel Functions ◽  
Direct Analysis ◽  
Hankel Transform ◽  
Approximation Technique ◽  
Polarization Data ◽  
Resistivity Data ◽  
Vertical Fault ◽  
Direct Interpretation ◽  
Electrical Soundings ◽  
Infinite Integral

The potential function for a point electrode in the vicinity of a vertical fault or dike may be expressed as an infinite integral involving Bessel functions. Beginning with such an expression, two methods are presented for the direct analysis of resistivity data measured both normal and parallel to dikes or faults. The first method is based on the asymptotic expansion of the Hankel transform of the field data and is suitable for surveys done parallel to the strike of the dike or fault. The second method is based on a successive approximation technique which starts from an initial approximate solution and iterates until a solution with prescribed accuracy is found. Both methods are suitable for programming on a digital computer and some illustrative numerical results are presented. These examples show the limitations of the methods. In addition, the application of resistivity data to the interpretation of induced‐polarization data is pointed out.

An Inventory With Constant Demand and Poisson Restocking

1987 ◽  
Vol 1 (2) ◽  
pp. 203-210 ◽  
Author(s):  
Laurence A. Baxter ◽  
Eui Yong Lee
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Distribution Function ◽  
Poisson Process ◽  
Stationary Case ◽  
Stieltjes Transform ◽  
Partial Differential

An inventory whose stock decreases linearly with time is considered. The inventory may be replenished at the instants at which a deliveryman arrives provided that the level of the inventory does not exceed a certain threshold; deliveries are made according to a Poisson process. A partial differential equation for the distribution function of the level of the inventory is solved to yield a formula for the corresponding Laplace–Stieltjes transform. The evaluation of the transform is discussed and explicit results are obtained for the stationary case.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

Some analyses on the control of queues using level crossings of regenerative processes

1980 ◽  
Vol 17 (3) ◽  
pp. 814-821 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Density Functions ◽  
Waiting Time Distribution ◽  
Regenerative Processes ◽  
Stieltjes Transform ◽  
Expected Number ◽  
Control Of Queues ◽  
Special Case

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

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