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Non-Sinusoidal Harmonics and Special Functions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
In the context of the boundary value problems that typically appear in physics, one is led to the study of boundary value problems in the form of Sturm–Liouville eigenvalue problems. These lead to an appropriate set of basis vectors for the function space under consideration. We will touch a little on these ideas, leaving some of the deeper results for more advanced courses in mathematics. For now, we will turn to function spaces and explore some typical basis functions, many which originated from the study of physical problems. The common basis functions are often referred to as special functions in physics. Examples are the classical orthogonal polynomials (Legendre, Hermite, Laguerre, Tchebychef) and Bessel functions. But first we will introduce function spaces.
Optical Design and Aberrations
Published in Daniel Malacara-Hernández, Brian J. Thompson, Fundamentals and Basic Optical Instruments, 2017
Armando Gómez-Vieyra, Daniel Malacara-Hernández
There are several practical reasons why orthogonal polynomials are useful and desirable, for example, an easier numerical manipulation and an easier interpretation in interferometrical applications. If the wavefront aberration polynomials are designed to form an orthogonal (in a circular pupil with unit semi-diameter) set of polynomials, they are called the orthogonal Zernike polynomials [46], and recently, the orthonormal Zernike wavefront representation has been adopted as standard [47].
Programming and functions
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
We can use recursion relations to generate higher order orthogonal polynomials from the zeroth and first terms. (Note that the nth item in the list below is the (n-1)st order polynomial.) We use the Hermite polynomial example from the PolynomF.pdf reference manual, denoting them as He(n) to differentiate them from the differently scaled Hermite polynomials H(n) used by physicists (e.g., Abramowitz and Stegun, 1965):
Designing optimal models of nonlinear MIMO systems based on orthogonal polynomial neural networks
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Marko Milojković, Miroslav Milovanović, Saša S. Nikolić, Miodrag Spasić, Andjela Antić
Legendre polynomials were discovered in 1782 during an effort to find the force of attraction between the celestial bodies during their revolution [25]. After that, orthogonal polynomials and their properties were a subject of extensive study of many mathematicians and scientists from other fields where orthogonal functions found numerous applications, control systems engineers among them.
Investigating Univariate Dimension Reduction Model for Probabilistic Power Flow Computation
Published in Electric Power Components and Systems, 2019
Qing Xiao, Shaowu Zhou, Lianghong Wu, Yanming Zhao, You Zhou
With being the weighting function, a set of orthogonal polynomials can be derived where is the nth-order orthogonal polynomial, and