Special Functions in Physics (SpecFunPhys) Toolbox

The toolbox comes with more than 170 special functions in the complex domain, covering in addition non-integer indices where appropriate.

The toolbox covers the following topics:

Bessel and Airy Functions:
Bessel and Airy functions are in many areas of physics used. The toolbox
functions evaluate in addition modulus and phase of the Airy functions,
the Scorer functions and, e.g., the Kelvin functions for complex indices.

Struve Functions and Related Functions
The Struve, Weber, and Anger functions are solutions of the generalized
Bessel differential equation.

Confluent Hypergeometric Function
Hypergeometric functions are of importance for many numerical
applications in physics. The class conhyp supports the evaluation of the
confluent hypergeometric function of first and second kind, the
evaluation of the Whittaker function, the confluent hypergeometric limit
function and the function 2F_0(z).

Coulomb Wave Functions
Evaluation of the regular and irregular Coulomb functions and scattering
states.

Gauss Hypergeometric Functions
play an important in role in many area of mathematical physics and numerical
evaluations.

Theta Functions
Jacobi or classical theta function play an important role in evaluating
modular and elliptic functions. Additional topics are Dedekind's Eta
function and the Jacobi index.

Jacobi Elliptic Functions
Jacobi Elliptic Functions appear in a variety of applications in
engineering and physics, e.g., in hydrodynamics, general relativity, or
classical dynamics.

Elliptic Integrals
Evaluation of the complete and incomplete elliptic integrals of first,
second and third kind in the complex domain for various representations
(Jacobi, Carlson, Bulirsch), and some related functions.

Weierstrass Functions
Evaluation of the Weierstrass elliptic functions and some related
functions.

Parabolic Cylinder Functions
are related to the solution of the Laplace Beltrami
operator or Helmholtz equation in parabolic cylinder coordinates.

Mathieu Functions
are related to the solutions of the Helmholtz
equation in elliptic cylindrical coordinates. Applications are, e.g.,
related to the dynamical trapping of particles in a Paul trap.
Additionally the solutions of the modified or radial Mathieu
equations are evaluated.

Orthogonal Polynomials
Orthogonal polynomials play an important role in physics. E.g., Jacobi and
Gegenbauer polynomials are evaluated as well as
some general methods developed to support various computations with orthogonal
polynomials.

Hermite Polynomials
are related to the eigenfunctions of the quantum
harmonic oscillator. Thus play an important role, especially in quantum
dynamics.

Laguerre Polynomials
play an important role in solving the radial
Schrödinger equation for Coulomb systems. Here we evaluate generalized
or associate Laguerre polynomials.

Chebychev Polynomials
are orthogonal polynomials. Applications are, e.g.,
in approximation or in constructing wavelets. Supported are Chebychev
polynomials and arbitrarily shifted Chebychev polynomials of 1st, 2nd,
3rd, and 4th
kind in the complex domain and for non-integer indices (Chebychev
functions).

Bernoulli and Euler Polynomials
and the corresponding numbers
are frequently used in statistical physics.

Riemann Zeta Function
Target are the Riemann Zeta function and its non-trivial zeros.
Applications can be found, e.g., in quantum theory or string theory.

Piecewise Interpolation Polynomials
are useful for interpolation
techniques, e.g., for finite elements (example for the Hydrogen atom
included). The programs allow to compute the
interpolation polynomial coefficients for Lagrange interpolation
polynomials, Hermite interpolation polynomials and extended Hermite
interpolation polynomials of arbitrary degree.

Wigner- and Clebsch-Gordan Coefficients
Angular momentum and their coupling are an important concept for quantum
systems. Topics are Clebsch-Gordan coefficients, Wigner 3j-symbols,
Wigner 6j-symbols, and Wigner 9j-symbols.

Coordinate Systems
play an important role in treating physical systems.
E.g., the three dimensional Laplace-Beltrami operator is separable in
11 curvilinear coordinates. Under this title transformation equations
between various coordinate systems are programmed. E.g., between all 11
curvilinear coordinates mentioned above and other frequently used
coordinate systems in physics.

An alphabetic list can be found on the website listed above.