arXiv subjects

Riccardo Borghi

Publications and source records attributed to Riccardo Borghi.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic Convergence of Weniger's $δ$-Transformation for a Class of Superfactorially Divergent Stieltjes Series

The resummation of superfactorially divergent series represents a significant computational challenge in mathematical physics. In the present paper the resummation of a specific class of Stieltjes series characterized by a moment sequence growing as $(2n)!$ will be addressed. Despite the fact that Carleman's condition is satisfied for these series, the convergence rate of Padé approximants is severely hindered by the logarithmic divergence of the associated Carleman series. Weniger's $δ$ transformation is proposed as a highly efficient alternative resummation tool. By employing recently established results on the converging factors of superfactorially divergent Stieltjes series, an exact integral representation for the truncation error is obtained. This representation enables the rigorous derivation of the leading-order asymptotic behavior of the transformation error, as well as the estimation of the related convergence rate, for real positive arguments. Numerical experiments strongly support the theoretical findings, suggesting that the $δ$ transformation offers a robust and computationally efficient framework for decoding this class of wildly divergent expansions

math.GM↗

arXiv2025

On the Twistability of Partially Coherent, Schell-model Sources

In this paper, the problem of assessing the twistability of a given bona fide cross-spectral density is tackled for the class of Schell-model sources, whose shift-invariant degree of coherence is represented by a real and symmetric function, {denoted as} $μ(-\bfr)=μ(\bfr)$. By employing an abstract operatorial language, the problem of determining the highly degenerate spectrum of a twisted operator $\hat W_u$ is addressed through a modal analysis based on {the} complete knowledge of the spectrum of the {\em sole} twist operator $\hat T_u$, as found by R. Simon and N. Mukunda. [J. Opt. Soc. Am. A \textbf{15,} 1361 (1998)]. To this end, the evaluation of the complete tensor of the matrix elements $\bra n',\ell'|\hat W_u|n,\ell\ket$ is carried out within the framework of the so-called {\em extended Wigner distribution function}, a concept recently introduced by M. {VanValkenburgh} [J. Mod. Opt. \textbf{55,} 3537 - 3549 (2008)]. As a nontrivial application of the algorithm developed here, the analytical determination of the spectrum of saturated twisted astigmatic Gaussian Schell-model sources is also presented.

physics.optics↗

arXiv2024

Factorial Series Representation of Stieltjes Series Converging Factors

The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series, is nowadays beyond dispute. Weniger's transformation, in particular, is able to accomplish spectacular results when used to overcome resummation problems, often outperforming better known resummation techniques, the most known being Padé approximants. However, our understanding of its theoretical features is still far from being satisfactory and particularly bad as far as the decoding of factorially divergent series is concerned. Stieltjes series represent a class of power series of fundamental interest in mathematical physics. In the present paper, it is shown how the Stieltjes series converging factor of any order is expressible as an inverse factorial series, whose terms can be analytically retrieved through a simple recursive algorithm. A few examples of applications of our algorithm are presented, in order to show its effectiveness and implementation ease. We believe the results presented here could constitute an important, preliminary step for the development of a general convergence theory of Weniger's transformation on Stieltjes series. A rather ambitious project, but worthy of being pursued in the future.

math.NA↗

arXiv2024

Dressing the Cusp: How Sharp-Edge Diffraction Theory Solves a Basic Issue in Catastrophe Optics

The description of light diffraction using catastrophe optics is one of the most intriguing theoretical invention in the field of classical optics of the last four decades. Its practical implementation has faced some resistance over the years, mainly due to the difficulty of mathematically decorating the different, topologically speaking, types of optical singularities (caustics) that concur to build up the skeleton on which diffraction patterns stem. Such a fundamental {\em dressing problem} has been solved in the past only for the so-called {\em fold}, which lies at the bottom of the hierarchy of structurally stable caustics. Climbing this hierarchy implies considerably more challenging mathematical problems to be solved. An ancient mathematical theorem is here employed to find the complete solution of the dressing problem for the {\em cusp}, which is placed, in the stable caustic hierarchy, immediately after the fold. The other ingredient used for achieving such an important theoretical result is the paraxial version of the boundary diffraction wave theory, whose tight connection with catastrophe optics has recently been emphasized in [R. Borghi, Opt. Lett. {\bf 41,} 3114 - 3117 (2016)]. A significant example of the developed algorithm aimed at demonstrating its effectiveness and ease of implementation, is also presented.

physics.optics↗

arXiv2024

Paraxial Sharp-Edge Diffraction of Vortex Beams by Elliptic Apertures

A semi-analytical computational algorithm to model the wavefield generated by paraxial diffraction of a class of Laguerre-Gauss beams by sharp-edge elliptic apertures is here developed. Thanks to such a powerful computational tool, some basic aspects of an intriguing and still unexplored singular optics scenario can be studied, within a geometry as simple as possible, with arbitrarily high accuracies.

physics.optics↗

arXiv2024

On the Bessel solution of Kepler's equation

Since its introduction in 1650, Kepler's equation has never ceased to fascinate mathematicians, scientists, and engineers. Over the course of five centuries, a large number of different solution strategies have been devised and implemented. Among them, the one originally proposed by J. L. Lagrange and later by F. W. Bessel still continues to be a source of mathematical treasures. Here, the Bessel solution of the elliptic Kepler equation is explored from a new perspective offered by the theory of the Stieltjes series. In particular, it has been proven that a complex Kapteyn series obtained directly by the Bessel expansion is a Stieltjes series. This mathematical result, to the best of our knowledge, is a new integral representation of the KE solution. Some considerations on possible extensions of our results to more general classes of the Kapteyn series are also presented.

math-ph↗

arXiv2023

"Analytical Continuation'' of Flattened Gaussian Beams

A purely analytical extension of the flattened Gaussian beams [Opt. Commun. \textbf{107,} 335 (1994)] to any values of the beam order, is here proposed. Due to it, the paraxial propagation problem of axially symmetric, coherent flat-top beams through arbitrary $ABCD$ optical systems can definitely be solved in closed form via a particular bivariate confluent hypergeometric function.

physics.optics↗

arXiv2022

Huygens' cycloidal pendulum: an elementary derivation

A pedagogical derivation of the Huygens cycloidal pendulum, suitable for high-school students, is here presented. Our derivation rests only on simple algebraic and geometrical tricks, without the need of any Calculus concept.

physics.class-ph↗

arXiv2022

Paraxial Sharp-Edge Diffraction: A General Computational Approach

A general reformulation of classical sharp-edge diffraction theory is proposed within paraxial approximation. The, not so much known, Poincaré vector potential construction is employed directly inside Fresnel's 2D integral in order for it to be converted into a single 1D contour integral over the aperture boundary. Differently from the recently developed paraxial revisitation of BDW's theory, such approach can be applied to arbitrary wavefield distributions impinging onto arbitrarily shaped sharp-edge planar apertures. A couple of interesting examples of application of the proposed method is presented.

physics.optics↗

arXiv2021

Solving Kepler's equation via nonlinear sequence transformations

Since more than three centuries Kepler's equation continues to represents an important benchmark for testing new computational techniques. In the present paper, the classical Kapteyn series solution of Kepler's equation originally conceived by Lagrange and Bessel will be revisited from a different perspective, offered by the relatively new and still largely unexplored framework of the so-called nonlinear sequence transformations. The main scope of the paper is to provide numerical evidences supporting the fact that the Kapteyn series solution of Kepler's equation could be a Stieltjes series. To support such a conjecture, two types of Levin-type sequence transformations, namely Levin $d$- and Weniger $δ$-transformations, will be employed to sum up several wildly divergent series derived by the Debye representation of Bessel functions. As an interesting byproduct of this analysis, an effective recursive algorithm to generate arbitrarily higher-order Debye's polynomials will be developed. Such a "Stieltjeness" conjecture will also be numerically validated by directly employing the Levin-type transformations to accelerate the complex Kapteyn series solution of the Kepler equation. Both $d$- and $δ$- transformations display exponential convergence, whose rate will be numerically estimated. A few conclusive words, together with some hints for future extensions in the direction of more general class of Kapteyn series, eventually close the paper.

math.CA↗

arXiv2020

From Kepler's laws to Newton's law: a didactical proof

An elementary derivation of the Newton "inverse square law" from the three Kepler laws is proposed. Our proof, thought essentially for first-year undergraduates, basically rests on Euclidean geometry. It could then be offered even to high-school students possessing only the first basics of Calculus.

physics.class-ph↗

arXiv2019

Exact paraxial diffraction theory for polygonal apertures under Gaussian illumination

Paraxial diffraction of monochromatic Gaussian beams by arbitrarily shaped polygonal apertures is analytically explored within the boundary diffraction wave theory framework. Exact closed-form expressions of the diffracted wavefield are obtained, as well as an interesting connection between classical optics and probability theory.

physics.optics↗

arXiv2018

Gaussian sharp-edge diffraction: a paraxial revisitation of Miyamoto-Wolf's theory

A "genuinely" paraxial version of Miyamoto-Wolf's theory aimed at dealing with sharp-edge diffraction under Gaussian beam illumination is presented. The theoretical analysis is carried out in such a way the well known Young-Maggi-Rubinowicz boundary diffraction wave theory can be extended to deal with Gaussian beams in an apparently straightforward way. The key for achieving such an extension is the introduction of suitable "complex angles" within the integral representations of the geometrical and BDW components of the total diffracted wavefield. Surprisingly enough, such a simple (although not rigorously justified) mathematical generalization seems to work well within the complex Gaussian realm. The resulting integrals provide meaningful quantities that, once suitably combined, give rise to predictions which are in perfect agreement with results already obtained in the past. An interesting and still open theoretical question about how to evaluate "Gaussian geometrical shadows" for arbitrarily shaped apertures is also discussed.

physics.optics↗

arXiv2014

Convergence Analysis of the Summation of the Euler Series by Padé Approximants and the Delta Transformation

Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series $\mathcal{E}(z) \sim \sum_{n=0}^{\infty} (-1)^n n! z^n$ is a very important model for the ubiquitous factorially divergent perturbation expansions in physics. In this article, we analyze the summation of the Euler series by Padé approximants and the delta transformation [E. J. Weniger, Comput. Phys. Rep. Vol.10, 189 (1989), Eq. (8.4-4)] which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a new factorial series representation of the truncation error of the Euler series [R. Borghi, Appl. Num. Math. Vol.60, 1242 (2010)]. We derive explicit expressions for the transformation errors of Padé approximants and of the delta transformation. A subsequent asymptotic analysis proves \emph{rigorously} the convergence of both Padé and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Padé. This is in agreement with previous numerical results.

math-ph↗

arXiv2013

Uniform approximation of paraxial flat-topped beams

A uniform asymptotic theory of the free-space paraxial propagation of coherent flattened Gaussian beams is proposed in the limit of nonsmall Fresnel numbers. The pivotal role played by the error function in the mathematical description of the related wavefield is stressed.

physics.optics↗

arXiv2013

Simple pendulum dynamics: revisiting the Fourier-based approach to the solution

The Fourier-based analysis customarily employed to analyze the dynamics of a simple pendulum is here revisited to propose an elementary iterative scheme aimed at generating a sequence of analytical approximants of the exact law of motion. Each approximant is expressed by a Fourier sum whose coefficients are given by suitable linear combinations of Bessel functions, which are expected to be more accessible, especially at an undergraduate level, with respect to Jacobian elliptic functions. The first three approximants are explicitely obtained and compared with the exact solution for typical initial angular positions of the pendulum. In particular, it is shown that, at the lowest approximation level, the law of motion of the pendulum turns out to be adequately described, up to oscillation amplitudes of $π/2$, by a sinusoidal temporal behaviour with a frequency proportional to the square root of the so-called "besinc" function, well known in physical optics.

physics.class-ph↗

arXiv2012

Trajectory of a body in a resistant medium: an elementary derivation

A didactical exposition of the classical problem of the trajectory determination of a body, subject to the gravity in a resistant medium, is proposed. Our revisitation is aimed at showing a derivation of the problem solution which should be as simple as possible from a technical point of view, in order to be grasped even by first-year undergraduates. A central role in our analysis is played by the so-called "chain rule" for derivatives, which is systematically used to remove the temporal variable from Newton's law to derive the differential equation of the Cartesian representation of the trajectory, with a considerable reduction of the overall mathematical complexity. In particular, for a resistant medium exerting a force quadratic with respect to the velocity our approach leads, in an elementary way, to the differential equation of the trajectory, which is subsequently solved by series expansion. A comparison of the polynomial approximants obtained by truncating such series with the solution recently proposed through a homotopy analysis is also presented.

physics.class-ph↗

arXiv2010

Asymptotic and factorial expansions of Euler series truncation errors via exponential polynomials

A detailed analysis of the remainder obtained by truncating the Euler series up to the $n$th-order term is presented. In particular, by using an approach recently proposed by Weniger, asymptotic expansions of the remainder, both in inverse powers and in inverse rising factorials of $n$, are obtained. It is found that the corresponding expanding coefficients are expressed, in closed form, in terms of exponential polynomials, well known in combinatorics, and in terms of associated Laguerre polynomials, respectively. A study of the divergence and/or of the convergence of the above expansions is also carried out for positive values of the Euler series argument.

physics.comp-ph↗