Bessel function

Summary

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the order of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha

Official source

https://en.wikipedia.org/wiki/Bessel_function

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This course complements the Analysis and Linear Algebra courses by providing further mathematical background and practice required for 3rd year physics courses, in particular electrodynamics and quantum mechanics.

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A central limit theorem for integrals of random waves

Matthew De Courcy-Ireland, Marius Christopher Lemm

We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of arbitrary dimension, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer's addition formula.

2019

To meet strict requirements of the information society technologies, antennas and circuit elements are becoming increasingly complex. Frequently, their electromagnetic (EM) properties cannot be anymore expressed in closed-form analytical expressions mainly because of the multitude of irregular geometries found in actual devices. Therefore, accurate and efficient (in terms of computational time and memory) electromagnetic models coupled with the robust optimization techniques, are needed in order to be able to predict and optimize the behavior of the innovative antennas in complex environments. The contribution of this thesis consists in the development and improvement of accurate electromagnetic modeling and optimization algorithms for an ubiquitous class of antennas, the planar printed antennas. The approach most commonly used to model and analyze this type of structures is the Integral Equation (IE) technique numerically solved using the Method of Moments (MoM). From the computational point of view, the main challenge is to develop techniques for efficient numerical evaluation of spatial-domain Green's functions, which are commonly expressed in terms of the well-known Sommerfeld integrals (SIs), i.e., semi-infinite range integrals with Bessel function kernels. Generally, the analytical solution of the SIs is not available, and their numerical evaluation is notoriously difficult and time-consuming because the integrands are both oscillatory and slowly decaying, and might possess singularities on and/or near the integration path. Due to the key role that SIs play in many EM problems, the development of fast and accurate techniques for their evaluation is of paramount relevance. This problem is studied in detail and several efficient methods are developed. Finally, the applicability of one of these methods, namely the Weighted Averages (WA) technique, is extended to the challenging case appearing in many practical EM problems: the evaluation of semi-infinite integrals involving products of Bessel functions. However, the development of effective analysis codes is only one aspect. At least equally important is the availability of reliable optimization techniques for an adequate design of antennas. For that purpose, the Particle Swarm Optimization (PSO) algorithm is introduced in the context of our analysis codes. Moreover, the innovative hybrid version of the PSO algorithm, called the Tournament Selection PSO, has been proposed with the aim of even further improving convergence performances of the classical PSO algorithm. Detailed theoretical description of this socially inspired evolutionary algorithm is given in the thesis. Finally, the characteristics of both algorithms are compared throughout several EM optimization problems.

EPFL2011

3D Face Recognition with Sparse Spherical Representations

Pascal Frossard, Effrosyni Kokiopoulou, Roser Sala Llonch, Ivana Tosic

This paper addresses the problem of 3D face recognition using simultaneous sparse approximations on the sphere. The 3D face point clouds are first aligned with a novel and fully automated registration process. They are then represented as signals on the 2D sphere in order to preserve depth and geometry information. Next, we implement a dimensionality reduction process with simultaneous sparse approximations and subspace projection. It permits to represent each 3D face by only a few spherical functions that are able to capture the salient facial characteristics, and hence to preserve the discriminant facial information. We eventually perform recognition by effective matching in the reduced space, where Linear Discriminant Analysis can be further activated for improved recognition performance. The 3D face recognition algorithm is evaluated on the FRGC v.1.0 data set, where it is shown to outperform classical state-of-the-art solutions that work with depth images.

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EPFL2011