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Heun equation

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1: 31.2 Differential Equations
§31.2(i) Heun’s Equation
§31.2(ii) Normal Form of Heun’s Equation
§31.2(v) Heun’s Equation Automorphisms
Composite Transformations
2: 31.12 Confluent Forms of Heun’s Equation
§31.12 Confluent Forms of Heun’s Equation
Confluent Heun Equation
Doubly-Confluent Heun Equation
Biconfluent Heun Equation
Triconfluent Heun Equation
3: 31.13 Asymptotic Approximations
§31.13 Asymptotic Approximations
For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
4: 31.1 Special Notation
Sometimes the parameters are suppressed.
5: Gerhard Wolf
 Schmidt) of the Chapter Double Confluent Heun Equation in the book Heun’s Differential Equations (A. …
6: 31.17 Physical Applications
§31.17 Physical Applications
§31.17(i) Addition of Three Quantum Spins
Heun functions appear in the theory of black holes (Kerr (1963), Teukolsky (1972), Chandrasekhar (1984), Suzuki et al. (1998), Kalnins et al. (2000)), lattice systems in statistical mechanics (Joyce (1973, 1994)), dislocation theory (Lay and Slavyanov (1999)), and solution of the Schrödinger equation of quantum mechanics (Bay et al. (1997), Tolstikhin and Matsuzawa (2001), and Hall et al. (2010)). For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …
7: 31.14 General Fuchsian Equation
Heun’s equation (31.2.1) corresponds to N = 3 .
Normal Form
8: 31.18 Methods of Computation
§31.18 Methods of Computation
The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 2830.
9: 31.10 Integral Equations and Representations
§31.10 Integral Equations and Representations
If w ( z ) is a solution of Heun’s equation, then another solution W ( z ) (possibly a multiple of w ( z ) ) can be represented as …
Kernel Functions
Kernel Functions
10: 31.16 Mathematical Applications
§31.16 Mathematical Applications
§31.16(i) Uniformization Problem for Heun’s Equation
It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. …