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 Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …

2: 31.3 Basic Solutions

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§31.3(i) Fuchs–Frobenius Solutions at $z=0$

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§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

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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

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4: 31.11 Expansions in Series of Hypergeometric Functions

… ►Let $w(z)$ be any Fuchs–Frobenius solution of Heun’s equation. …The Fuchs-Frobenius solutions at $\mathrm{\infty }$ are … ►Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. …Then the Fuchs–Frobenius solution at $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … ►Such series diverge for Fuchs–Frobenius solutions. …

5: 2.7 Differential Equations

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§2.7(i) Regular Singularities: Fuchs–Frobenius Theory

… ►In theory either pair may be used to construct any other solution …

6: Bibliography F

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Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.

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Notes:

Contains Mathematica package for spheroidal wave functions of complex arguments with complex parameters.

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FullText

Referenced by:

§30.12, 1st item, 2nd item.

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BibTeX

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Application of the $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Comm. Math. Phys. 219 (2), pp. 357–398.

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External Links:

ISSN 0010-3616, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§32.10(iv), §32.14, §32.6(v).

Encodings:

BibTeX

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{V}}$, ${\mathrm{P}}_{\mathrm{III}}$, the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55 (6), pp. 679–727.

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External Links:

ISSN 0010-3640, Document, MathReview (Mark Adler), ZentralBlatt

Referenced by:

§32.10(iii), §32.10(v), §32.14, §32.6(iv), §32.6(v).

Encodings:

BibTeX

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Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63 (3), pp. 301–321.

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External Links:

ISSN 0025-5831, Document, ZentralBlatt

Referenced by:

§32.2(iv).

Encodings:

BibTeX

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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.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

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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

… ►Fuchs–Frobenius solutions $W_m(z)={\tilde{\kappa }}_m{z}^{-\alpha }H\mathrm{\ell }(1/a,q_m;\alpha ,\alpha -\gamma +1,\alpha -\beta +1,\delta ;1/z)$ are represented in terms of Heun functions $w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$ 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).