The Experts below are selected from a list of 8484 Experts worldwide ranked by ideXlab platform
Judd S. Gardner - One of the best experts on this subject based on the ideXlab platform.
-
Approximate Expansion of a Narrow Gaussian Beam in Spherical Vector Wave Functions
IEEE Transactions on Antennas and Propagation, 2007Co-Authors: Judd S. GardnerAbstract:The expansion of electromagnetic sources is fundamental to the analysis of field propagation and scattering. Of these electromagnetic sources, one of the most commonly used optical sources is the narrow Gaussian beam. It is, therefore, useful to find an expansion of the narrow Gaussian beam that could be used in applications such as scattering and propagation studies in free-space and materials. The approximate method of expansion here is based on an exact expansion that was obtained for a vector plane wave in terms of spherical vector wave Functions. Since the simpler vector plane wave representation that previously enabled an exact solution is replaced here by the more complicated narrow Gaussian beam representation, an approximation of the associated Legendre Function will be applied in order to obtain an approximate expansion of a narrow Gaussian beam in terms of spherical vector wave Functions. Although the expansion is taken about the source point, field points close to the beam axis located at distances near and far from the source are found to have good accuracy.
-
Uniform Asymptotic Expansion of the Associated Legendre Function to Leading Term for Complex Degree and Integral Order
IEEE Transactions on Antennas and Propagation, 2007Co-Authors: Judd S. GardnerAbstract:The associated Legendre Function arises naturally in the study of spherical waves. Since in practical applications it is most often symbolically represented by Pn m(xi) for m les n and Pn m(xi) equiv 0 for m > n where m is the integer order and n is the integer degree, this form will be employed to develop the uniform asymptotic expansion. The considerable extent to which this Function appears in literature substantiates its importance in engineering and science, and particularly to spherical harmonics. In his book, "Partial Differential Equations in Physics" Sommerfeld covers a variety of subjects including spherical harmonics, and gives a detailed account of obtaining an expansion of the associated Legendre Function, Pn m(cos(thetas)), by the method of steepest descents over the interval 0 les thetas les pi. The results he obtains are quite accurate for n Gt m except as thetas approaches the critical points, thetas rarr 0 or thetas rarr pi. Beginning with the same integral representation of the associated Legendre Function with integer order and degree that Sommerfeld employed, a uniform asymptotic expansion is found that is applicable to the neighborhoods of thetas = 0 and thetas = pi and that becomes increasingly more accurate as n increases beyond m. Furthermore, the accuracy of the resulting uniform asymptotic expansion remains for real degree and complex degree as well. The results are plotted in order to assess the accuracy and the domain of validity of the uniform asymptotic expansion. The results of the uniform asymptotic expansion are also compared to the available approximation of the associated Legendre Function given in terms of Bessel Functions for small values of thetas.
Toshio Fukushima - One of the best experts on this subject based on the ideXlab platform.
-
Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere
Journal of Geodesy, 2018Co-Authors: Toshio FukushimaAbstract:In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary Function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the $$4 \pi $$ 4 π fully normalized associated Legendre Function in terms of the rectangle values of the Wigner d Function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as $$2^{30}\,{\approx }\,10^9$$ 2 30 ≈ 10 9 . The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.
-
numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers ii first second and third order derivatives
Journal of Geodesy, 2012Co-Authors: Toshio FukushimaAbstract:We confirm that the first-, second-, and third-order derivatives of fully-normalized Legendre polynomial (LP) and associated Legendre Function (ALF) of arbitrary degree and order can be correctly evaluated by means of non-singular fixed-degree formulas (Bosch in Phys Chem Earth 25:655–659, 2000) in the ordinary IEEE754 arithmetic when the values of fully-normalized LP and ALF are obtained without underflow problems, for e.g., using the extended range arithmetic we recently developed (Fukushima in J Geod 86:271–285, 2012). Also, we notice the same correctness for the popular but singular fixed-order formulas unless (1) the order of differentiation is greater than the order of harmonics and (2) the point of evaluation is close to the poles. The new formulation using the fixed-order formulas runs at a negligible extra computational time, i.e., 3–5 % increase in computational time per single ALF when compared with the standard algorithm without the exponent extension. This enables a practical computation of low-order derivatives of spherical harmonics of arbitrary degree and order.
-
Recursive computation of finite difference of associated Legendre Functions
Journal of Geodesy, 2012Co-Authors: Toshio FukushimaAbstract:The existing methods to compute the definite integral of associated Legendre Function (ALF) with respect to the argument suffer from a loss of significant figures independently of the latitude. This is caused by the subtraction of similar quantities in the additional term of their recurrence formulas, especially the finite difference of their values between two endpoints of the integration interval. In order to resolve the problem, we develop a recursive algorithm to compute their finite difference. Also, we modify the algorithm to evaluate their definite integrals assuming that their values at one endpoint are known. We numerically confirm a significant increase in computing precision of the integral by the new method. When the interval is one arc minute, for example, the gain amounts to 2–4 digits for the degree of harmonics in the range 2 ≤ n ≤ 2,048. This improvement in precision is achieved at a negligible increase in CPU time, say less than 5%.
Demni Nizar - One of the best experts on this subject based on the ideXlab platform.
-
Explicit expressions of the Hua-Pickrell semi-group
HAL CCSD, 2021Co-Authors: Arista Jonas, Demni NizarAbstract:A paraitre dans Theory of Probability and its ApplicationsIn this paper, we study the one-dimensional Hua-Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semi-group density through the associated Legendre Function in the cut. Next, we focus on the general (not necessarily stationary) case for which we prove an intertwining relation between Hua-Pickrell diffusions corresponding to different sets of parameters. Using Cauchy Beta integral on the one hand and Girsanov's Theorem on the other hand, we discuss the connection between the stationary and general cases. Afterwards, we prove our main result providing novel integral representations of the Hua-Pickrell semi-group density, answering a question raised by Alili, Matsumoto and Shiraishi (S\'eminaire de Probabilit\'es, 35, 2001). To this end, we appeal to the semi-group density of the Maass Laplacian and extend it to purely-imaginary values of the magnetic field. In the last section, we use the Karlin-McGregor formula to derive an expression of the semi-group density of the multi-dimensional Hua-Pickrell particle system introduced by T. Assiotis
-
Explicit expressions of the Hua-Pickrell semi-group
2020Co-Authors: Arista Jonas, Demni NizarAbstract:In this paper, we study the one-dimensional Hua-Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semi-group density through the associated Legendre Function in the cut. Next, we focus on the general (not necessarily stationary) case for which we prove an intertwining relation between Hua-Pickrell diffusions corresponding to different sets of parameters. Using Cauchy Beta integral on the one hand and Girsanov's Theorem on the other hand, we discuss the connection between the stationary and general cases. Afterwards, we prove our main result providing novel integral representations of the Hua-Pickrell semi-group density, answering a question raised by Alili, Matsumoto and Shiraishi (S\'eminaire de Probabilit\'es, 35, 2001). To this end, we appeal to the semi-group density of the Maass Laplacian and extend it to purely-imaginary values of the magnetic field. In the last section, we use the Karlin-McGregor formula to derive an expression of the semi-group density of the multi-dimensional Hua-Pickrell particle system introduced by T. Assiotis.Comment: 19 pages, the final expression of the semi-group density is obtaine
Radoslaw Szmytkowski - One of the best experts on this subject based on the ideXlab platform.
-
alternative approach to the solution of the momentum space schrodinger equation for bound states of the n dimensional coulomb problem
Annalen der Physik, 2012Co-Authors: Radoslaw SzmytkowskiAbstract:The Schrodinger–Coulomb problem in R N , N � 2 ,i s considered in the momentum representation. The adjoint Sturmian eigenvalue problem is discussed first. The resulting radial integral equation is solved with the aid of a symmetric Poisson-type series expansion of the Legendre Function of the second kind into products of the Gegenbauer polynomials, established in the 1950’s by Ossicini. A relationship between solutions to the Sturmian problem and to an energy eigenvalue problem is then exploited to find the Coulomb bound-state energy levels in R N ,t ogether with explicit representations of the associated momentum-space wave Functions.
-
solution of the momentum space schrodinger equation for bound states of the n dimensional coulomb problem revisited
arXiv: Quantum Physics, 2011Co-Authors: Radoslaw SzmytkowskiAbstract:The Schr\"odinger-Coulomb Sturmian problem in $\mathbb{R}^{N}$, $N\geqslant2$, is considered in the momentum representation. An integral formula for the Gegenbauer polynomials, found recently by Cohl [arXiv:1105.2735], is used to separate out angular variables and reduce an integral Sturmian eigenvalue equation in $\mathbb{R}^{N}$ to a Fredholm one on $\mathbb{R}_{+}$. A kernel of the latter equation contains the Legendre Function of the second kind. A symmetric Poisson-type series expansion of that Function into products of the Gegenbauer polynomials, established by Ossicini [Boll. Un. Mat. Ital. 7 (1952) 315], is then used to determine the Schr\"odinger-Coulomb Sturmian eigenvalues and associated momentum-space eigenFunctions. Finally, a relationship existing between solutions to the Sturmian problem and solutions to a (physically more interesting) energy eigenvalue problem is exploited to find the Schr\"odinger-Coulomb bound-state energy levels in $\mathbb{R}^{N}$, together with explicit representations of the associated normalized momentum-space Schr\"odinger-Coulomb Hamiltonian eigenFunctions.
-
closed form of the generalized green s Function for the helmholtz operator on the two dimensional unit sphere
Journal of Mathematical Physics, 2006Co-Authors: Radoslaw SzmytkowskiAbstract:The closed representation of the generalized (known also as reduced or modified) Green’s Function for the Helmholtz partial differential operator on the surface of the two-dimensional unit sphere is derived. In its compact form, the derived formula contains a Legendre polynomial and a derivative of the Legendre Function of the first kind with respect to its index. An explicit expression for that derivative is found and used to obtain an expanded (and potentially more suitable in applications) form of the generalized Green’s Function for the operator in question. The related problem of constructing the closed form of the generalized Green’s Function for the Legendre ordinary differential operator on the segment −1
J S Gardner - One of the best experts on this subject based on the ideXlab platform.
-
asymptotic expansion of the associated Legendre Function over the interval 0 leq theta leq pi
IEEE Transactions on Antennas and Propagation, 2011Co-Authors: J S GardnerAbstract:The associated Legendre Function is generally defined by an integral or differential equation. However, a closed-form representation of the associated Legendre Function of order 0 or 1 could be beneficial to model the propagation and scattering of waves in spherical coordinates. To this end, an associated Legendre approximation with accuracy over the entire defining interval is presented that is particularly conducive to integral evaluation using stationary phase or steepest descents.