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Fuchs–Frobenius theory

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1: 31.18 Methods of Computation
Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their FuchsFrobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …
2: 31.3 Basic Solutions
§31.3(i) FuchsFrobenius Solutions at z = 0
§31.3(ii) FuchsFrobenius Solutions at Other Singularities
3: 16.23 Mathematical Applications
A variety of problems in classical mechanics and mathematical physics lead to Picard--Fuchs equations. … …
§16.23(iv) Combinatorics and Number Theory
4: 31.11 Expansions in Series of Hypergeometric Functions
Let w ( z ) be any FuchsFrobenius solution of Heun’s equation. … Every FuchsFrobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. …Then the FuchsFrobenius solution at belonging to the exponent α has the expansion (31.11.1) with … Such series diverge for FuchsFrobenius solutions. …
5: 2.7 Differential Equations
§2.7(i) Regular Singularities: FuchsFrobenius Theory
In theory either pair may be used to construct any other solution …
6: Bibliography F
  • P. Falloon (2001) Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.
  • P. J. Forrester and N. S. Witte (2001) Application of the τ -function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Comm. Math. Phys. 219 (2), pp. 357–398.
  • P. J. Forrester and N. S. Witte (2002) Application of the τ -function theory of Painlevé equations to random matrices: P V , P III , the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55 (6), pp. 679–727.
  • R. Fuchs (1907) Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63 (3), pp. 301–321.
  • Y. V. Fyodorov (2005) Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.
  • 7: 26.19 Mathematical Applications
    §26.19 Mathematical Applications
    Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). Other areas of combinatorial analysis include graph theory, coding theory, and combinatorial designs. These have applications in operations research, probability theory, and statistics. …
    8: 31.10 Integral Equations and Representations
    FuchsFrobenius solutions W m ( z ) = κ ~ m z - α H ( 1 / a , q m ; α , α - γ + 1 , α - β + 1 , δ ; 1 / z ) are represented in terms of Heun functions w m ( z ) = ( 0 , 1 ) Hf m ( a , q m ; α , β , γ , δ ; z ) by (31.10.1) with W ( z ) = W m ( z ) , w ( z ) = w m ( z ) , and with kernel chosen from …
    9: 6.17 Physical Applications
    §6.17 Physical Applications
    Geller and Ng (1969) cites work with applications from diffusion theory, transport problems, the study of the radiative equilibrium of stellar atmospheres, and the evaluation of exchange integrals occurring in quantum mechanics. …Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.
    10: 32.16 Physical Applications
    Statistical physics, especially classical and quantum spin models, has proved to be a major area for research problems in the modern theory of Painlevé transcendents. … For the Ising model see Barouch et al. (1973), Wu et al. (1976), and McCoy et al. (1977). … For applications in string theory see Seiberg and Shih (2005).