vi
Preface
Chapter 4 covers Bessel functions, Airy functions, and related functions. In Chap. 5,
Struve functions and related functions are presented, which are solutions of nonhomogenous generalizations of Bessel’s differential equation.
Hypergeometric functions play an important role in many areas of mathematical
physics and numerical approximations. Chapters 6–8 are treating confluent hypergeometric functions, Coulomb wave functions, and Gauss hypergeometric functions
and related functions, for example, the Whittaker functions.
θ functions have been, for example, applied to soliton theory and quantum field
theory or in evaluating elliptic functions. Applications of Jacobi elliptic functions,
or elliptic integrals, appear in a variety of engineering and science problems.
Weierstraß functions can be found, for example, in astrophysical applications, in
solving supra-conductivity problems, and so forth. This family of functions and
related functions will be discussed in Chaps. 9–12.
Solving the Helmholtz differential equation in parabolic cylinder coordinates will
lead to parabolic cylinder functions, topic of Chap. 13. Physical questions related to
the solution of the Helmholtz equation in elliptical cylindrical coordinates lead to the
Mathieu functions. Mathieu functions and related functions are treated in Chap. 14.
In the realm of orthogonal polynomials are numerous applications in computational physics. A general discussion of orthogonal polynomials can be found in
Chap. 15, as well as Jacobi polynomials and Gegenbauer polynomials. Hermite
polynomials, Laguerre polynomials, Chebyshev polynomials, and related functions
are topics of Chaps. 16–18. The polynomials and, respectively, functions are as well
extended to the complex domain and to non-integer indices.
Bernoulli numbers and Euler numbers and the related polynomials are treated in
Chap. 19. Applications can be found, for example, in statistical physics. Computational applications in quantum physics, chaotic systems, and string theory, to list
only a few, are applying Riemann zeta functions, which is the topic of Chap. 20.
Piecewise interpolation polynomials are discussed in Chap. 21. One of the many
numerical applications of piecewise interpolation polynomials are finite element
approximations of the Schrödinger equation. As an example, this application of
piecewise interpolation polynomials will be discussed for the hydrogen atom.
Angular momenta and their coupling play a crucial role in quantum dynamics.
Clebsch-Gordan coefficients and Wigner symbols are presented in Chap. 22.
The last chapter is devoted to various coordinate systems, with the focus on those
coordinate systems for which the three dimensional Laplace–Beltrami operator
becomes separable. Many special functions are related to one of these coordinate
systems. In Chap. 23, we derive MATLAB code for the corresponding coordinate
transformations.
Despite all of my effort, the book is likely to contain some typos, hopefully no
errors, and you—the reader—might miss some special functions. Any comment or
suggestion will be appreciated, and I will maintain a list of errata on my home page:
https://wolfgang-schweizer.de.
Preface
Chapter 4 covers Bessel functions, Airy functions, and related functions. In Chap. 5,
Struve functions and related functions are presented, which are solutions of nonhomogenous generalizations of Bessel’s differential equation.
Hypergeometric functions play an important role in many areas of mathematical
physics and numerical approximations. Chapters 6–8 are treating confluent hypergeometric functions, Coulomb wave functions, and Gauss hypergeometric functions
and related functions, for example, the Whittaker functions.
θ functions have been, for example, applied to soliton theory and quantum field
theory or in evaluating elliptic functions. Applications of Jacobi elliptic functions,
or elliptic integrals, appear in a variety of engineering and science problems.
Weierstraß functions can be found, for example, in astrophysical applications, in
solving supra-conductivity problems, and so forth. This family of functions and
related functions will be discussed in Chaps. 9–12.
Solving the Helmholtz differential equation in parabolic cylinder coordinates will
lead to parabolic cylinder functions, topic of Chap. 13. Physical questions related to
the solution of the Helmholtz equation in elliptical cylindrical coordinates lead to the
Mathieu functions. Mathieu functions and related functions are treated in Chap. 14.
In the realm of orthogonal polynomials are numerous applications in computational physics. A general discussion of orthogonal polynomials can be found in
Chap. 15, as well as Jacobi polynomials and Gegenbauer polynomials. Hermite
polynomials, Laguerre polynomials, Chebyshev polynomials, and related functions
are topics of Chaps. 16–18. The polynomials and, respectively, functions are as well
extended to the complex domain and to non-integer indices.
Bernoulli numbers and Euler numbers and the related polynomials are treated in
Chap. 19. Applications can be found, for example, in statistical physics. Computational applications in quantum physics, chaotic systems, and string theory, to list
only a few, are applying Riemann zeta functions, which is the topic of Chap. 20.
Piecewise interpolation polynomials are discussed in Chap. 21. One of the many
numerical applications of piecewise interpolation polynomials are finite element
approximations of the Schrödinger equation. As an example, this application of
piecewise interpolation polynomials will be discussed for the hydrogen atom.
Angular momenta and their coupling play a crucial role in quantum dynamics.
Clebsch-Gordan coefficients and Wigner symbols are presented in Chap. 22.
The last chapter is devoted to various coordinate systems, with the focus on those
coordinate systems for which the three dimensional Laplace–Beltrami operator
becomes separable. Many special functions are related to one of these coordinate
systems. In Chap. 23, we derive MATLAB code for the corresponding coordinate
transformations.
Despite all of my effort, the book is likely to contain some typos, hopefully no
errors, and you—the reader—might miss some special functions. Any comment or
suggestion will be appreciated, and I will maintain a list of errata on my home page:
https://wolfgang-schweizer.de.