Special Functions in Physics with MATLAB

✍ Scribed by Wolfgang Schweizer

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 287
• Category: Library

✦ Synopsis

This handbook focuses on special functions in physics in the real and complex domain. It covers more than 170 different functions with additional numerical hints for efficient computation, which are useful to anyone who needs to program with other programming languages as well. The book comes with MATLAB-based programs for each of these functions and a detailed html-based documentation. Some of the explained functions are: Gamma and Beta functions; Legendre functions, which are linked to quantum mechanics and electrodynamics; Bessel functions; hypergeometric functions, which play an important role in mathematical physics; orthogonal polynomials, which are largely used in computational physics; and Riemann zeta functions, which play an important role, e.g., in quantum chaos or string theory. The book’s primary audience are scientists, professionals working in research areas of industries, and advanced students in physics, applied mathematics, and engineering.

✦ Table of Contents

Preface
Content Overview
Function List
The SpecFunPhys Toolbox
Contents
1 Gamma Functions, Beta Functions, and Related
1.1 Function Overview
1.2 Gamma Function
1.2.1 Fundamental Equations
1.2.2 Computation of the Gamma Function
1.2.3 The Class gammalnC and gammaC
1.3 The Pochhammer Symbol
1.4 The psi Function
1.5 The Incomplete Gamma Functions
1.5.1 Fundamental Equations
1.5.2 Computational Aspects
1.5.3 Related Programs
1.6 Applications: Examples
1.6.1 Examples
1.7 The Beta Function
1.7.1 Basics
1.7.2 The Class betalnC and betaC
1.8 The Incomplete Beta Function
1.8.1 Fundamental Equations
1.8.2 Computational Aspects
1.8.3 The Class incbetaC
References
2 Error Functions and Fresnel Integrals
2.1 Function Overview
2.2 Error Functions
2.2.1 Fundamental Equations and Computation
2.2.2 erfComp and Applications
2.3 Voigt Profile
2.4 Fresnel Integrals
References
3 Legendre Polynomials and Legendre Functions
3.1 Function Overview
3.2 Legendre Polynomials
3.3 Legendre Functions of Integer Type
3.3.1 Fundamental Equations and Computation
3.3.2 Application: Spherical Harmonics
3.4 Associate Legendre Functions with Complex Indices
3.4.1 Fundamental Equations
3.4.2 Conical or Mehler Functions
3.4.3 Complex Legendre Functions
References
4 Bessel and Airy Functions
4.1 Function Overview
4.2 Airy Functions and Related Functions
4.2.1 The Scorer Functions
4.3 Bessel Functions and Related Functions
4.3.1 Bessel and Hankel Functions
4.3.2 Modified Bessel Functions
4.3.3 Spherical Bessel Functions
4.3.4 Kelvin Functions
4.4 The SpecFunPhys-Class Bessel
References
5 Struve Functions and Related Functions
5.1 Function Overview
5.2 Struve Functions
5.2.1 Struve Functions H and K
5.2.2 Modified Struve Functions L and M
5.2.3 The Class Struve
5.3 The Anger Function and the Weber Function
5.3.1 Anger Function
5.3.2 Weber Function
5.3.3 The Class Angweb
Reference
6 Confluent Hypergeometric Function
6.1 Confluent Hypergeometric Function of 1st and 2nd Kind
6.1.1 The Function 1F1
6.1.2 The Function U and 2F0
6.2 The Confluent Hypergeometric Limit Function
6.3 The Whittaker Functions
6.4 The SpecFunPhys Class conhyp
References
7 Coulomb Wave Functions
7.1 Function Overview
7.2 The Coulomb Wave Functions
7.2.1 Partial Wave Coulomb Function
7.2.2 Coulomb Wave Function: Parabolic Coordinates
7.3 Program Details and Applications
7.3.1 The SpecFunPhys Class coulombwave
7.3.2 The SpecFunPhys Class coulombscatt
References
8 Hypergeometric Functions
8.1 The Hypergeometric Series
8.2 Method of Computation
8.3 The SpecFunPhys-Class Gausshyp
References
9 Functions
9.1 Function Overview
9.2 Jacobi Functions
9.2.1 General Equations and Computational Aspects
9.2.2 TheSpecFunPhys Class jacobiTheta
9.3 Dedekind's η Function
9.4 The Jacobi Symbol
References
10 Jacobi Elliptic Functions
10.1 Function Overview
10.2 The Elliptic Functions
10.2.1 Equations and Computational Aspects
10.3 The Class ellipFun
References
11 Elliptic Integrals
11.1 Function Overview
11.2 Equations and Computation
11.3 Programs
References
12 Weierstraß Functions
12.1 Function Overview
12.2 Fundamental Equations and Computation
12.3 The Class ellipWeier
Reference
13 Parabolic Cylinder Functions
13.1 Equations and the Class paracylFun
References
14 Mathieu Functions
14.1 Function Overview
14.2 Fundamental Equations and Computation
14.2.1 Mathieu Differential Equation
14.2.2 Modified Mathieu Differential Equation
14.3 Programs
14.3.1 The SpecFunPhys Class Mathieufun
14.3.2 Additional SpecFunPhys Functions
References
15 Orthogonal Polynomials: General Aspects
15.1 Function Overview
15.2 General Definitions
15.3 The Classes orthpoly and polymeth
15.3.1 The Class orthpoly
15.3.2 The Class polymeth
15.4 Gegenbauer Polynomials
15.4.1 The SpecFunPhys Class gegenbauerpoly and gegenbauerx
15.5 Jacobi Polynomials
15.5.1 The SpecFunPhys Class Jacobipoly and Jacobix
References
16 Hermite Polynomials
16.1 Function Overview
16.2 Hermite Polynomials
16.2.1 The SpecFunPhys Class hermitepoly
16.2.2 Evaluating Hermite Polynomials
16.3 Oscillator Systems
16.3.1 Applications
References
17 Laguerre Polynomials
17.1 Laguerre Polynomials
17.1.1 General Aspects
17.1.2 Related Programs
References
18 Chebyshev Polynomials
18.1 Basic Properties and Formulae
18.1.1 Definition and Recurrence Formula
18.1.2 The Shifted Chebyshev Polynomials
18.1.3 Miscellaneous
18.2 Computational Aspects and Programs
18.2.1 Computational Aspects
18.2.2 Programs
References
19 Bernoulli and Euler Polynomials
19.1 Bernoulli Numbers and Bernoulli Polynomials
19.1.1 Equations
19.1.2 Programs
19.2 Euler Numbers and Euler Polynomials
19.2.1 Equations
19.2.2 Programs
References
20 Riemann Zeta Function
20.1 Equations and Evaluation
20.1.1 Equations
20.1.2 Evaluation
References
21 Piecewise Interpolation Polynomials
21.1 Interpolation Polynomials
21.1.1 Lagrange Interpolation Polynomials
21.1.2 Hermite Interpolation Polynomials
21.1.3 Extended Hermite Interpolation Polynomials
21.2 Computational Aspects
21.3 The Finite Element Method
21.3.1 Example: The Hydrogen Atom
References
22 Wigner- and Clebsch-Gordan Coefficients
22.1 Fundamental Equations and Computation
22.1.1 Basic Aspects
22.1.2 Clebsch-Gordan Coefficients and Wigner 3j-Symbols
22.1.3 Wigner 6j-Symbols
22.1.4 Wigner 9j-Symbols
22.2 Evaluation
22.2.1 Clebsch-Gordan Coefficients and Wigner 3j-symbols
22.2.2 Wigner 6j-Symbols
22.2.3 Wigner 9j-Symbols
References
23 Coordinate Systems
23.1 Separability in Three Dimensions
23.2 Programs and Computational Aspects
23.2.1 The SpecFunPhys Class CoordTrafo
23.2.2 Ellipsoidal Coordinates and Cone Coordinates
23.2.3 Hyperspherical Coordinates
References
Index