representation via plane algebraic curve

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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: 26.19 Mathematical Applications

§26.19 Mathematical Applications

►Combinatorics has applications to analysis, algebra, and geometry. …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)). …

3: 22.18 Mathematical Applications

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§22.18(i) Lengths and Parametrization of Plane Curves

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§22.18(iii) Uniformization and Other Parametrizations

… ► … ►The existence of this group structure is connected to the Jacobian elliptic functions via the differential equation (22.13.1). …The theory of elliptic functions brings together complex analysis, algebraic curves, number theory, and geometry: Lang (1987), Siegel (1988), and Serre (1973). …

4: 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. …

5: 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. …

6: 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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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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External Links:

Document

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§8.24(iii).

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A Treatise on Plane and Advanced Trigonometry. 7th edition, Cambridge University Press.

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Notes:

Reprinted by Dover Publications Inc., New York, 2004.

External Links:

MathReview, ZentralBlatt

Referenced by:

§4.16, §4.17, §4.19, §4.2(ii), §4.2(iii), §4.21(i), §4.21(ii), §4.21(iii), §4.21(iv), §4.22, §4.23(v), §4.23(vi), §4.24(i), §4.24(iii), §4.28, §4.30, §4.31, §4.35(i), §4.35(ii), §4.35(iii), §4.35(iv), §4.42(i), §4.42(ii), §4.43, §4.8(ii), Ch.4.

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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

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§24.14(ii).

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7: 15.19 Methods of Computation

… ►However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. ►

§15.19(iii) Integral Representations

►The representation (15.6.1) can be used to compute the hypergeometric function in the sector . … ►Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). ►For example, in the half-plane $\mathrm{\Re }z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …

8: Bibliography W

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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Uniform asymptotic expansion of $J_{\nu }(\nu a)$ via a difference equation. Numer. Math. 91 (1), pp. 147–193.

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ISSN 0029-599X, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.20(ii).

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

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§18.29.

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Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.

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External Links:

ISBN 3-540-33283-9, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(i).

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Four-figure tables of the Airy function in the complex plane. Philos. Mag. (7) 37, pp. 236–261.

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Document, MathReview (J. C. P. Miller), ZentralBlatt

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1st item.

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9: Bibliography

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SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.

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Notes:

SPHEREPACK 2.0 is a collection of Fortran programs that facilitates computer modeling of geophysical processes. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms. The package also contains utility programs for computing the associated Legendre functions. SPHEREPACK 3.1 was released in August 2003.

External Links:

FullText, SPHEREPACK 3.1

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1st item.

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Plane partitions. III. The weak Macdonald conjecture. Invent. Math. 53 (3), pp. 193–225.

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ISSN 0020-9910, Document, MathReview (Edward A. Bender), ZentralBlatt

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§26.12(i), §26.12(ii).

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Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.

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ISSN 0044-2275, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

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§10.61(v), §10.64.

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Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (2), pp. 411–437.

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External Links:

Document, MathReview (D. L. Colton), ZentralBlatt

Referenced by:

§18.17(iv).

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Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.

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ISSN 1095-7154, Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.18(iv).

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10: 23.20 Mathematical Applications

… ►For conformal mappings via modular functions see Apostol (1990, §2.7). … ►An algebraic curve that can be put either into the form … ►If $a,b\in ℝ$, then $C$ intersects the plane ${ℝ}^2$ in a curve that is connected if $\mathrm{\Delta }\equiv 4a^3+27b^2>0$; if , then the intersection has two components, one of which is a closed loop. … ► ►

§23.20(iii) Factorization

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