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Quantum Mechanics and Schrödinger Equation
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
It is well known that there is a Rodrigues’ formula for the Legendre function (see Section 4.10.2 of Chau, 2018). In general, this type of function also exists for other special functions, such as Bessel functions, Hermit polynomials, Chebyshev polynomials, Gegenbauer polynomials, and Jacobi polynomials. They are called generating functions. The generating function for Laguerre polynomials is exp{-xt/(1-t)}1-t=∑n=0∞Ln(x)tn $$ \frac{{\exp \{ - xt/(1 - t)\} }}{{1 - t}} = \sum\limits_{{n = 0}}^{\infty } {L_{n} (x)t^{n} } $$
Series Solutions of Second Order ODEs
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
The investigation of special functions probably started with Bernoulli and Euler in 1700s. They include elliptic integrals and Bessel functions. We have given an introduction on elliptic integrals in Chapter 2 when we discussed the pendulum problem. Further information on Jacobi’s elliptic integral is also given in Appendix A. Euler introduced gamma functions as a non-integer continuation of factorial and studied elliptic integrals related to pendulums, and Bessel functions related to vibrations of circular drums. Nearly all ofhis investigations are driven by everyday applications. Related to celestial mechanics and potential theory, Legendre polynomials emerged. There are also other special functions in terms of polynomials, such as Hermite polynomials, Laguerre polynomials, Chebyshev polynomials, and Jacobi polynomials. Because of the orthogonal properties of these polynomials, they are important as the basis of eigenfunction expansion (see Chapter 10). Some of them relate to the theory ofprobabilities, quantum mechanics and wave scattering theory.
Series Methods
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
Some of these properties may seem surprising. The gamma function, as do the Bessel functions, all belong to a broad class of functions that mathematicians, physicists, and engineers term special functions. Special functions are functions, usually defined in terms of a convergent power series or integral, that play a special role in the solution of some problems of practical importance. The gamma and Bessel functions we are studying in this section are perhaps new to you and their properties may seem surprising and unfamiliar. However, the class of special functions contains very familiar functions as well. One only needs to remember that the functions ex, sin x, and cos x are all defined as convergent power series (although this is not how you probably first learned of them), to realize that the “strangeness” of the properties of the gamma and Bessel functions is just an artifact of their newness to your collection of mathematical facts.
Degenerate 2D bivariate Appell polynomials: properties and applications
Published in Applied Mathematics in Science and Engineering, 2023
Shahid Ahmad Wani, Arundhati Warke, Javid Gani Dar
The expression (4) is a solution of the heat equation: In recent years, a number of generalizations of mathematical physics especially, special functions have seen a considerable evolution. The new advancement in the special functions theory provides the analytical basis for the solution of numerous mathematical physics problems, which have several wide-ranging applications. The significant advancement in the theory of generalized special functions is based on the introduction of multi-variable and multi-index special functions. The significance of special functions has been acknowledged in both pure mathematics and practical contexts. The need for multi- variable and multi-index special functions are realized to tackle the issues emerging in the theory of abstract algebra and partial differential equations. Hermite himself [3] first devised the notion of multiple-index, multiple-variable Hermite polynomials. The Hermite polynomials are found in physics, where they generate the eigenstates of the quantum harmonic oscillator and also appear in the solution of the Schrodinger equation for the harmonic oscillator. They are also used in the numerical analysis as Gaussian quadrature.
Arbogast : Higher order automatic differentiation for special functions with Modular C
Published in Optimization Methods and Software, 2018
Isabelle Charpentier, Jens Gustedt
Special functions and their derivatives play a crucial role in research fields of physics and mathematical analysis. As reported in [2,35], many of these functions are solutions of the general second-order ODE where the input z is either a real or a complex variable, and functions , , determine the mathematical function . In other words, seeds and , and functions , , may be used to evaluate the second-order derivative , then higher order derivatives. For the sake of generality in the presentation, we also adopt the same general seed formula and notation for the orthogonal polynomials (Table 7), the Bessel functions (Table 8) and the hypergeometric functions (Table 9). Note that other formulas are available and can be used to deal with special cases, see for instance [17].
Estimating the expansion coefficients of a geomagnetic field model using first-order derivatives of associated Legendre functions
Published in Optimization Methods and Software, 2018
H. Martin Bücker, Johannes Willkomm
The scientific analysis for the solution of physical problems encounters a variety of special functions. Well-known examples of special functions include Hermite polynomials, Legendre polynomials, and Bessel functions [1]. Therefore, it is not uncommon that computer models in computational science and engineering employ these special functions in some way or the other. If computer models involving these function are transformed by automatic differentiation (AD), the derivatives of these functions will be necessary. Assuming that the source code for the evaluation of these functions is available, these derivatives can be obtained by transforming the source code in a black-box AD approach. That is, the code implementing the special function is treated in the same way as any other code without considering its semantics. However, it is well known that the semantics can and should be exploited to apply the chain rule more judiciously. The corresponding techniques are sometimes called hierarchical AD approaches [5,6,11,18]. They are successfully used not only to enhance robustness, but also to increase performance in terms of computing time and storage in various different situations including the solution of systems of linear and nonlinear equations, or Fourier transforms to name a few.