Quadrature relations for finite-limit, principal-value integrals

1992 ◽  
Vol 70 (8) ◽  
pp. 656-666 ◽  
Author(s):  
K. T. R. Davies ◽  
M. L. Glasser ◽  
R. W. Davies
Keyword(s):  
Arbitrary Order ◽  
Finite Limit ◽  
Limit Case ◽  
Double Integral ◽  
Divergent Integral ◽  
Step Functions ◽  
Principal Value ◽  
Indefinite Integral ◽  
The Difference ◽  
Double Integrals

An intuitive expression is obtained for a general finite-limit, prinicipal-value (PV) integral containing a pole of arbitrary order. Such an integral is simply the difference between its "end-point quadratures," the quadrature being the result of the indefinite integral. This PV expression is fairly obvious for simple poles but requires careful justification for higher order poles. Moreover, quadratures are useful in evaluating related integrals that contain both poles and other singularities (e.g., step functions or logarithmic divergences). Then, for a certain type of finite-limit divergent integral, the PV is interpreted as its "convergent part." Also, for cases in which there are two PV's in a double integral, it is shown that the famous Poincaré–Bertrand theorem applies to finite-limit as well as infinite-limit integrals. Finally, an interesting quadrature relation is derived for such double integrals, and the validity of the Poincaré–Bertrand theorem is explicitly demonstrated for a simple finite-limit case.

scholarly journals Tauberian conditions under which convergence follows from Cesàro summability of double integrals over R2

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3425-3440
Author(s):  
Gökşen Fındık ◽  
İbrahim Çanak
Keyword(s):  
Continuous Function ◽  
The Other ◽  
Cesàro Summability ◽  
Double Integral ◽  
Tauberian Conditions ◽  
Summability Methods ◽  
Cesaro Summability ◽  
The Difference ◽  
Complex Valued ◽  
Double Integrals

For a real- or complex-valued continuous function f over R2+:= [0,1) x [0,1), we denote its integral over [0,u] x [0,v] by s(u,v) and its (C,1, 1) mean, the average of s(u,v) over [0,u] x [0,v], by ?(u,v). The other means (C,1,0) and (C; 0; 1) are defined analogously. We introduce the concepts of backward differences and the Kronecker identities in different senses for double integrals over R2+. We give onesided and two-sided Tauberian conditions based on the difference between double integral of s(u, v) and its means in different senses for Ces?ro summability methods of double integrals over [0,u] x [0,v] under which convergence of s(u,v) follows from integrability of s(u,v) in different senses.

Using double integrals to solve single integrals

2016 ◽  
Vol 100 (548) ◽  
pp. 257-265
Author(s):  
G. J. O. Jameson
Keyword(s):  
Finite Limit ◽  
Double Integral ◽  
Image Position ◽  
Double Integrals

Consider the integralwhere b > a > 0. First, let us clarify why it even exists. Of course, convergence at infinity is ensured by the exponential terms, but the integrals of and e–ax/x and e–bx/x, taken separately, are divergent at 0, since these integrands equate asymptotically to 1/x as x → 0. However,so (e–ax – e–bx)/x tends to the finite limit b – a as x → 0 and there is no problem integrating it on intervals of the form [0, r].A neat way to evaluate I1 starts by expressing the integrand itself as an integral:(1)Inserting this into I1 converts it into a double integral.

The Effect of the Steady Flow Potential on the Motions of a Moving Ship in Waves

2000 ◽  
Vol 44 (01) ◽  
pp. 14-32
Author(s):  
Ming-Chung Fang
Keyword(s):  
Steady Flow ◽  
Present Method ◽  
Three Dimensional ◽  
Source Distribution ◽  
Series Expansions ◽  
Ship Motions ◽  
Double Integral ◽  
Flow Potential ◽  
Principal Value ◽  
Good Agreement

A three-dimensional method to analyze the motions of a ship running in waves is presented, including the effects of the steady-flow potential. Basically, the general formulations are based on the source distribution technique by which the ship hull surface is regarded as the assembly of many panels. The present study includes three algorithms for treating the corresponding Green function:the Hess & Smith algorithm for the part of simple source I/r,the complex plane contour integral of the Shen & Farell algorithm for the double integral of steady flow, andthe series expansions of the Telste & Noblesse algorithm for the Cauchy principal value integral of unsteady flow. The study reveals that the effect of steady flow on ship motions is generally small, but it still cannot be neglected in some cases, especially for the ship running in oblique waves. The effect also depends on the fore-aft configuration of the ship. The results predicted by the present method are found to be in fairly good agreement with existing experiments and other theories.

scholarly journals Global Behavior of the Difference Equationxn+1=(p+xn-1)/(qxn+xn-1)

2010 ◽  
Vol 2010 ◽  
pp. 1-6
Author(s):  
Taixiang Sun ◽  
Hongjian Xi ◽  
Hui Wu ◽  
Caihong Han
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Initial Conditions ◽  
Global Behavior ◽  
Finite Limit ◽  
The Difference

We study the following difference equationxn+1=(p+xn-1)/(qxn+xn-1),n=0,1,…,wherep,q∈(0,+∞)and the initial conditionsx-1,x0∈(0,+∞). We show that every positive solution of the above equation either converges to a finite limit or to a two cycle, which confirms that the Conjecture 6.10.4 proposed by Kulenović and Ladas (2002) is true.

FEEDBACK SYNCHRONIZATION OF CHAOTIC SYSTEMS

2003 ◽  
Vol 13 (01) ◽  
pp. 177-191 ◽  
Author(s):  
C. SARASOLA ◽  
F. J. TORREALDEA ◽  
A. D'ANJOU ◽  
A. MOUJAHID ◽  
M. GRAÑA
Keyword(s):  
Chaotic Systems ◽  
Structural Parameter ◽  
Behavioral Characteristics ◽  
Finite Limit ◽  
Complete Synchronization ◽  
Coupling Interaction ◽  
Adaptive Laws ◽  
Interaction Term ◽  
The Difference ◽  
Variable Errors

Feedback coupling through an interaction term proportional to the difference in the value of some behavioral characteristics of two systems is a very common structural setting that leads to synchronization of the behavior of both systems. The degree of synchronization attained depends on the strength of the interaction term and on the mutual interdependency of the structures of both systems. In this paper, we show that two chaotic systems linked through a feedback coupling interaction term of gain parameter k reach a synchronized regime characterized by a vector of variable errors which tends towards zero with parameter k while the interaction term tends towards a finite nonzero permanent regime. This means that maintaining a certain degree of synchronization has a cost. In the limit, complete synchronization occurs at a finite limit cost. We show that feedback coupling in itself brings about conditions permitting that systems with a degree of structural parameter flexibility evolve close towards each other structures in order to facilitate the maintenance of the synchronized regime. In this paper, we deduce parameter adaptive laws for any family of homochaotic systems provided they are previously forced to work, via feedback coupling, within an appropriate degree of synchronization. The laws are global in the space of parameters and lead eventually to identical synchronization at no interaction cost. We illustrate this point with homochaotic systems from the Lorenz, Rössler and Chua families.

Double integral involving G-function of two variables

2020 ◽  
pp. 333-338
Author(s):  
Shukla Vinay Kumar
Keyword(s):  
Integral Equation ◽  
Population Density ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Research Paper ◽  
Statistical Distribution ◽  
Double Integral ◽  
Expansion Formulae ◽  
Function Of Two Variables ◽  
Double Integrals

In the study of certain boundary value problems integrals are useful with their connections. To obtain expansion formulae it also helps. In the study of integral equation, probability and statistical distribution, integrals are also used. To measure population density within a certain area, we can also use integrals. With integrals we can analyzed anything that changes in time. The object of this research paper is to establish a double integrals involving G-Function of two variables.

Evaluation of the Wave-Resistance Green Function: Part 2—The Single Integral on the Centerplane

1987 ◽  
Vol 31 (03) ◽  
pp. 145-150 ◽  
Author(s):  
J. N. Newman
Keyword(s):  
Green Function ◽  
Arbitrary Order ◽  
Asymptotic Series ◽  
Neumann Series ◽  
Absolute Accuracy ◽  
Double Integral ◽  
Vertical Coordinate ◽  
Recursive Scheme ◽  
Thin Region ◽  
Single Integral

Effective series expansions are derived for the evaluation of the single integral in the potential of a submerged source which moves with constant velocity, when the source and field point are in the same longitudinal centerplane. In conjunction with the polynomial approximations for the double integral component which have been derived in Part 1 of this work, the present results facilitate the computation of the source potential or Green function. Three complementary domains of the centerplane are considered, with different expansions developed for use in each domain. The principal expansion is based on a Neumann series which is effective for small or moderate distances from the origin, except in a thin region near the free surface. To deal with the latter domain an asymptotic expansion is derived in ascending powers of the vertical coordinate. Both of these expansions are refined by subtracting a simpler component with the same behavior at the origin, and relating this component to Dawson's integral. Special algorithms for the evaluation of the latter function are presented in the Appendix. The third and final expansion, based upon the method of steepest descents, is effective at large distances from the origin. This asymptotic series is derived by a systematic recursive scheme to permit an arbitrary order of the approximation. Used in conjunction with the first two expansions, this permits the single integral to be evaluated with an absolute accuracy of six decimals throughout the centerplane.

scholarly journals Efektifitas Validasi Bea Perolehan Hak atas Tanah dan Bangunan terhadap Ketidaksesuaian Nilai Objek Pajak dalam Akta Jual Beli dengan Harga Sebenarnya

2021 ◽  
Vol 4 (2) ◽  
pp. 875-885
Author(s):  
Dejan Gemelar Raja Guk-Guk ◽  
Isnaini Isnaini ◽  
M. Citra Ramadhan
Keyword(s):  
Validation Process ◽  
Online System ◽  
Field Verification ◽  
Principal Value ◽  
The Difference ◽  
Transaction Value

This study aims to determine the effectiveness of the validation of Land and Building Rights Acquisition Duty (BPHTB) against tax object value discrepancies, the BPHTB validation process carried out by the Medan City Regional Revenue Service (BPPRD) and how to determine the Tax Object Principal Value (NPOP) in accordance with the actual value of the tax object. This type of research is a descriptive normative research. The effectiveness of BPHTB validation against the discrepancy in the value of tax objects is seen based on the BPHTB revenue data mentioned above, it is known that the number of targets and realization of BPHTB taxes seen from 2017 to 2019 is quite effective. The BPHTB validation process carried out by the Medan City BPPRD in general is quite satisfactory, plus currently the BPHTB SSPD validation process is carried out with an online system, so that the BPHTB SSPD validation process becomes easier and faster. How to determine the Principal Value of the Tax Object (NPOP) by determining the transaction value in the sale and purchase transaction. The use of the transaction value used as the basis for calculating BPHTB often creates problems. The difference in the transaction value agreed by the parties and stated in the deed with the transaction value used as the basis for calculating BPHTB according to BPPRD research, in this case there is uncertainty about which value is correct, so determining the NPOP in accordance with the actual value of the tax object becomes difficult. except by conducting field verification on the said BPHTB object.

scholarly journals Weighted generalization of some inequalities for double integrals

2021 ◽  
Vol 110 (124) ◽  
pp. 71-79
Author(s):  
Mehmet Sarikaya ◽  
Hüseyin Budak
Keyword(s):  
Convex Functions ◽  
Integral Inequalities ◽  
Double Integral ◽  
Double Integrals

We give some weighted double integral inequalities of Hermite-Hadamard type for co-ordinated convex functions in a rectangle from R2. The inequalities obtained provide generalizations of some result given in earlier works.

PROPAGATION OF A PULSE IN A FLUID SPHERE

Geophysics ◽  
1964 ◽  
Vol 29 (2) ◽  
pp. 259-287 ◽  
Author(s):  
Z. Alterman ◽  
P. Kornfeld
Keyword(s):  
Geometrical Optics ◽  
Step Function ◽  
Surface Velocity ◽  
Fluid Sphere ◽  
Function Type ◽  
Arrival Times ◽  
Step Functions ◽  
The Difference ◽  
Deep Focus ◽  
Theoretic Solution

The exact solution obtained in a previous paper for the motion of a uniform compressible fluid sphere due to a pressure pulse from a point source situated below the surface is applied to a source at a distance of one‐eighth of the radius below the surface. Taking the sphere as a simplified model of the earth, this corresponds to a source at a depth of about 800 km, which is not far from the depth of a deep‐focus earthquake. The time variation of pressure due to the source is represented by the difference between two step functions with rounded shoulders. The surface velocity due to sources of different durations has been evaluated for eight angular distances. The solution exhibits step function type and “logarithmic” reftected pulses, which one would anticipate from the “steepest descents” analysis of Jeffreys and Lapwood. In addition, the solution reveals single diffracted pulses and groups of diffracted pulses which have no counterpart in ray theory. When geometrical optics allows a ray to appear only after a minimum range [Formula: see text] from the epicenter, the complete wave‐theoretical solution shows that these pulses show up earlier in the forbidden zones. Similarly, in the case where the geometrical optics predicts that a certain ray should appear only for ranges [Formula: see text], and should not appear for [Formula: see text], the wave‐theoretic solution shows that such a ray does appear, by diffraction, for some range of [Formula: see text]. Arrival times of the diffracted pulses of the first group increase with increasing θ, while for the second group they decrease with θ.

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