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11: 30.12 Generalized and Coulomb Spheroidal Functions
which reduces to (30.2.1) if α = 0 . … which also reduces to (30.2.1) when α = 0 . …
12: 14.29 Generalizations
As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when μ 1 = μ 2 = μ . …
13: 29.11 Lamé Wave Equation
In the case ω = 0 , (29.11.1) reduces to Lamé’s equation (29.2.1). …
14: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
The Fourier cosine and sine transform pairs (1.14.9) & (1.14.11) and (1.14.10) & (1.14.12) can be easily obtained from (1.18.57) as for ν = ± 1 2 the Bessel functions reduce to the trigonometric functions, see (10.16.1). … This representation has poles with residues | f ^ ( λ n ) | 2 at the discrete eigenvalues and a branch cut along [ 0 , ) with discontinuity, from below to above the cut, 2 π i | f ^ ( λ ) | 2 , as in (1.18.53), see Newton (2002, §7.1.1). Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies λ res i Γ res / 2 corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to 1 / Γ res . … Similar results hold for two, but not higher, dimensional quantum systems. …
15: Bibliography M
  • I. G. Macdonald (1982) Some conjectures for root systems. SIAM J. Math. Anal. 13 (6), pp. 988–1007.
  • I. G. Macdonald (2000) Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).
  • R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
  • Maxima (free interactive system)
  • MuPAD (commercial interactive system and Matlab toolbox) SciFace Software, Paderborn, Germany.
  • 16: 32.16 Physical Applications
    §32.16 Physical Applications
    Integrable Continuous Dynamical Systems
    17: Software Index
    Open Source With Book Commercial
    4.48(iii) General Precision a REDUCE
  • Open Source Collections and Systems.

    These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

  • 18: 21.10 Methods of Computation
  • Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).

  • Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.

  • 19: 18.39 Applications in the Physical Sciences
    Introduction and One-Dimensional (1D) Systems
    where x is a spatial coordinate, m the mass of the particle with potential energy V ( x ) , = h / ( 2 π ) is the reduced Planck’s constant, and ( a , b ) a finite or infinite interval. … As in classical dynamics this sum is the total energy of the one particle system. …
    1D Quantum Systems with Analytically Known Stationary States
    §18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom
    20: Stephen M. Watt
    His areas of research include algorithms and systems for computer algebra, programming languages and compilers, mathematical handwriting recognition and mathematical document analysis. He was one of the original authors of the Maple and Axiom computer algebra systems, principal architect of the Aldor programming language and its compiler at IBM Research, and co-author of the MathML and InkML W3C standards. …