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Journal ArticleDOI

On Certain Integrals of Lipschitz-Hankel Type Involving Products of Bessel Functions

19 Apr 1955-Philosophical Transactions of the Royal Society A (The Royal Society)-Vol. 247, Iss: 935, pp 529-551
TL;DR: In this article, the evaluation and tabulation of integrals of the type (* 00 I(p, v; A) = J J fa t) ) e~cttxdt.
Abstract: This paper is concerned with the evaluation and tabulation of certain integrals of the type (* 00 I(p, v; A) = J J fa t) ) e~cttxdt. In part I of this paper, a formula is derived for the integrals in terms of an integral of a hypergeometric function. This new integral is evaluated in the particular cases which are of most frequent use in mathematical physics. By means of these results, approximate expansions are obtained for cases in which the ratio b/a is small or in which b~a and is small. In part II, recurrence relations are developed between integrals with integral values of the parameters pt, v and A. Tables are given by means of which 7(0, 0; 1), 7(0, 1; 1), 7(1, 0; 1), 7(1,1; 1), 7(0, 0 ;0), 7(1, 0;90), 7(0, 1; 0), 7(1, 1; 0), 7(0,1; - 1 ), 7(1,0; - 1 ) and 7(1,1; - 1 ) may be evaluated for 0
Citations
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Journal ArticleDOI
TL;DR: In this paper, the real variable is replaced by a complex variable, and the factorial and related functions of the complex variable are used to solve linear differential equations of the second order.
Abstract: 1. The real variable 2. Scalars and vectors 3. Tensors 4. Matrices 5. Multiple integrals 6. Potential theory 7. Operational methods 8. Physical applications of the operational method 9. Numerical methods 10. Calculus of variations 11. Functions of a complex variable 12. Contour integration and Bromwich's integral 13. Contour integration 14. Fourier's theorem 15. The factorial and related functions 16. Solution of linear differential equations of the second order 17. Asymptotic expansions 18. The equations of potential, waves and heat conduction 19. Waves in one dimension and waves with spherical symmetry 20. Conduction of heat in one and three dimensions 21. Bessel functions 22. Applications of Bessel functions 23. The confluent hypergeometric function 24. Legendre functions and associated functions 25. Elliptic functions Notes Appendix on notation Index.

771 citations

Journal ArticleDOI
Abstract: Fluxmetric (ballistic) and magnetometric demagnetizing factors N/sub f/ and N/sub m/ for cylinders as functions of susceptibility chi and the ratio gamma of length to diameter have been evaluated. Using a one-dimensional model when gamma >or=10, N/sub f/ was calculated for -1 >

472 citations


Cites background from "On Certain Integrals of Lipschitz-H..."

  • ...We have arbitrarily set zi = 0 for ease of notation and have included the factor z,/ 1 zj I when needed to account for the sign reversal that occurs when zj 84 ] and reduce to...

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Journal ArticleDOI
TL;DR: Soil-structure interaction is an interdisciplinary field of endeavor which lies at the intersection of soil and structural mechanics, soil-and structural dynamics, earthquake engineering, geophysics and geomechanics, material science, computational and numerical methods, and diverse other technical disciplines as discussed by the authors.

302 citations

Journal ArticleDOI
TL;DR: It is found that the variance of RSSI value changes along with distance by analyzing a large number of experimental data, and the log-normal shadowing model with dynamic variance (LNSM-DV) is established, which might be adjusted dynamically according to the change of environment and be self-adaptable.
Abstract: The relationship between RSSI (Received Signal Strength Indication) values and distance is the foundation and the key of ranging and positioning technologies in wireless sensor networks. Log-normal shadowing model (LNSM), as a more general signal propagation model, can better describe the relationship between the RSSI value and distance, but the parameter of variance in LNSM is depended on experiences without self-adaptability. In this paper, it is found that the variance of RSSI value changes along with distance regu- larly by analyzing a large number of experimental data. Based on the result of analysis, we proposed the relationship function of the variance of RSSI and distance, and established the log-normal shadowing model with dynamic variance (LNSM-DV). At the same time, the method of least squares(LS) was selected to es- timate the coefficients in that model, thus LNSM-DV might be adjusted dynamically according to the change of environment and be self-adaptable. The experimental results show that LNSM-DV can further reduce er- ror, and have strong self-adaptability to various environments compared with the LNSM.

294 citations


Additional excerpts

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Journal ArticleDOI
TL;DR: In this paper, the authors compared the observed subsidence and hypocentral distributions with the predicted displacement and stress fields, and found that the relationship between average reservoir pressure drop and subsidence is remarkably linear, lending support to the linear poroelastic model.
Abstract: o Hundreds of shallow, small to moderate earthquakes have occurred near the Lacq deep gas field in southwestern France since 1969. These earthquakes are clearly separated from tectonic seismicity occurring in the Pyrenees, 25 km to the southwest. The induced seismicity began when the reservoir pressure had declined by • 30 MPa. Repeated levehng over the field shows localized subsidence reaching a maximum of 60 mm in 1989. Segall (1989) suggested that poroelastic stressing, associated with volumetric contraction of the reservoir rocks, is responsible for induced seismicity associated with fluid extraction. To test this model, we compare the observed subsidence and hypocentral distributions with the predicted displacement and stress fields. We find that the relationship between average reservoir pressure drop and subsidence is remarkably linear, lending support to the linear poroelastic model. Displacements and stresses are computed based on a priori knowledge of the reservoir geometry, materiM properties, and reservoir pressure changes. The computed vertical displacements are found to be in excellent agreement with the subsidence observed from leveling. Stress perturbations accompanying gas extraction, computed using the same parameters, are found to be • 0.2 MPa or less. Changes in Coulomb failure stress are computed assuming that shp occurs on optimally oriented planes. The predicted failure zones correlate very well with the spatial distribution of earthquakes if the perturbing stresses are small in comparison to the ambient regional deviatoric stresses and if the minimum regional compresslye stress axis is vertical. Accurate determination of focal mechanisms of the induced events would allow a more rigorous test of the poroelastic model and could lead to important inferences about the crustal stress state.

257 citations

References
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Book
03 Jan 1956
TL;DR: In this article, the authors present an account of those parts of mathematics that are most frequently needed in physics, including methods which have applications in at least two branches of physics and conditions under which theorems hold.
Abstract: This well-known text and reference contains an account of those parts of mathematics that are most frequently needed in physics. As a working rule, it includes methods which have applications in at least two branches of physics. The authors have aimed at a high standard of rigour and have not accepted the often-quoted opinion that 'any argument is good enough if it is intended to be used by scientists'. At the same time, they have not attempted to achieve greater generality than is required for the physical applications: this often leads to considerable simplification of the mathematics. Particular attention is also paid to the conditions under which theorems hold. Examples of the practical use of the methods developed are given in the text: these are taken from a wide range of physics, including dynamics, hydrodynamics, elasticity, electromagnetism, heat conduction, wave motion and quantum theory. Exercises accompany each chapter.

872 citations

Journal ArticleDOI
TL;DR: In this paper, the real variable is replaced by a complex variable, and the factorial and related functions of the complex variable are used to solve linear differential equations of the second order.
Abstract: 1. The real variable 2. Scalars and vectors 3. Tensors 4. Matrices 5. Multiple integrals 6. Potential theory 7. Operational methods 8. Physical applications of the operational method 9. Numerical methods 10. Calculus of variations 11. Functions of a complex variable 12. Contour integration and Bromwich's integral 13. Contour integration 14. Fourier's theorem 15. The factorial and related functions 16. Solution of linear differential equations of the second order 17. Asymptotic expansions 18. The equations of potential, waves and heat conduction 19. Waves in one dimension and waves with spherical symmetry 20. Conduction of heat in one and three dimensions 21. Bessel functions 22. Applications of Bessel functions 23. The confluent hypergeometric function 24. Legendre functions and associated functions 25. Elliptic functions Notes Appendix on notation Index.

771 citations

Book
01 Dec 1932
TL;DR: In this paper, the classical integral theorems of green and stokes are applied to two-dimensional problems and to non-linear problems, such as linear equations in three variables.
Abstract: Preface Introduction 1. The Classical Equations 2. Applications of the integral theorems of green and stokes 3. Two-dimensional problems 4. Conformal Representation 5. Equations in three variables 6. Polar Co-ordinates 7. Cylindrical Co-ordinates 8. Ellipsoidal Co-ordinates 9. Paraboloidal Co-ordinates 10. Toroidal Co-ordinates 11. Diffraction problems 12. Non-linear equations

632 citations

Journal ArticleDOI
01 Apr 1945

90 citations