representation via Schottky group

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1: 21.10 Methods of Computation

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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).

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

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Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

2: 13.27 Mathematical Applications

§13.27 Mathematical Applications

►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. The elements of this group are of the form …Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. … …

3: 18.38 Mathematical Applications

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Quadrature

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Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►

Group Representations

… ►Algebraic structures were built of which special representations involve Dunkl type operators. …

4: 15.17 Mathematical Applications

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§15.17(iii) Group Representations

… ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►

§15.17(v) Monodromy Groups

… ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

5: Bibliography S

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The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.

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Notes:

First edition published by Wadsworth & Brooks/Cole Advanced Books & Software, 1991.

External Links:

ISBN 0-387-95067-2, MathReview, ZentralBlatt

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§26.19.

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Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

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§18.38(i).

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Über die Moduln der Thetafunctionen. Acta Math. 27 (1), pp. 235–288.

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ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

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§21.9.

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On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.

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ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt

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§29.17(i), §29.8.

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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MathReview (M. C. Gray), ZentralBlatt

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§12.13(ii).

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6: 9.17 Methods of Computation

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§9.17(iii) Integral Representations

►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. … ►

§9.17(iv) Via Bessel Functions

… ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …

7: Bibliography V

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Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1466-2, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§18.38(iii).

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Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1493-X, MathReview (P. D. F. Ion), ZentralBlatt

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§18.38(iii), Ch.35.

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Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt

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§18.38(iii).

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Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.

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External Links:

MathReview, ZentralBlatt

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§13.27.

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Fourier series representation of Ferrers function $𝖯$ .

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FullText

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(14.13.1).

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8: Morris Newman

… ►Newman wrote the book Matrix Representations of Groups, published by the National Bureau of Standards in 1968, and the book Integral Matrices, published by Academic Press in 1972, which became a classic. …

9: Bibliography H

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Lie groups, Lie algebras, and representations. Second edition, Graduate Texts in Mathematics, Vol. 222, Springer, Cham.

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Notes:

An elementary introduction

External Links:

ISBN 978-3-319-13466-6; 978-3-319-13467-3, Document, Link, MathReview Entry

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§1.2(vi).

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Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. 2nd edition, Noordhoff International Publishing, Leyden.

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Notes:

The first edition was published in 1965 by Prentice-Hall. The 2nd edition was reissued in 1983 by Springer, and in 1986 by Martinus Nijhoff Publishers (Kluwer Academic Publishers Group).

External Links:

MathReview (Chia-Shun Yih), ZentralBlatt

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§10.73(i).

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Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Internat. J. Quantum Chem. 88 (6), pp. 701–734.

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Document

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§8.24(iii).

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Sums of powers of integers via generating functions. Fibonacci Quart. 34 (3), pp. 244–256.

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ISSN 0015-0517, MathReview (Noriaki Kimura), ZentralBlatt

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§24.4(iii).

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Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.

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ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.14(ii).

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10: 16.24 Physical Applications

… ►The $\mathit{3}j$ symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …