classical theta functions

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11: 18.17 Integrals

§18.17 Integrals

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§18.17(ii) Integral Representations for Products

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§18.17(v) Fourier Transforms

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§18.17(vi) Laplace Transforms

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§18.17(vii) Mellin Transforms

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12: Bibliography M

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Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 215–237.

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ISSN 0377-0427, Document, MathReview (Leonid B. Golinskiĭ), ZentralBlatt

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§32.15.

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Introduction to the Classical Theory of Abelian Functions. American Mathematical Society, Providence, RI.

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Notes:

Translated from the 1979 Russian original by G. Bluher

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ISBN 0-8218-4542-X, MathReview, ZentralBlatt

Referenced by:

§21.8.

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Classical transcendental solutions of the Painlevé equations and their degeneration. Tohoku Math. J. (2) 56 (4), pp. 467–490.

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ISSN 0040-8735, Document, MathReview (Oleg Gleizer), ZentralBlatt

Referenced by:

§32.10(v), §32.10(vi).

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Tata Lectures on Theta. II. Birkhäuser Boston Inc., Boston, MA.

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Notes:

Jacobian theta functions and differential equations

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ISBN 0-8176-3110-0, MathReview (M. Kh. Gizatullin), ZentralBlatt

Referenced by:

§21.7(i), §21.7(ii), §21.7(iii), Ch.21.

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Classical solutions of the third Painlevé equation. Nagoya Math. J. 139, pp. 37–65.

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ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iii).

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13: 18.5 Explicit Representations

§18.5 Explicit Representations

… ►With $x=\cos \theta =\frac{1}{2}(z+z^{-1})$, … ►

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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14: 18.18 Sums

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§18.18(ii) Addition Theorems

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§18.18(iii) Multiplication Theorems

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§18.18(v) Linearization Formulas

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§18.18(vii) Poisson Kernels

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15: Bibliography D

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Computing Riemann theta functions. Math. Comp. 73 (247), pp. 1417–1442.

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Notes:

Two of the authors (Deconinck and van Hoeij) have developed Maple code for the algorithms described in this paper.

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ISSN 0025-5718, Document, Code, MathReview, ZentralBlatt

Referenced by:

§21.10(i), 1st item, §21.2(i), §21.4, M. Heil (1995).

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Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162 (10), pp. 1793–1804.

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ISSN 0021-9045, Document, FullText, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

(18.16.12), (18.16.13), §18.16(ii), §18.16(ii), §18.16(iv).

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Inequalities for extreme zeros of some classical orthogonal and $q$-orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.

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ISSN 0973-5348, Document, FullText, MathReview (Ana Foulquié Moreno), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(iv), §18.16(ii), §18.16(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(v).

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Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

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ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

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§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

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Linear operators. Part II. Wiley Classics Library, John Wiley & Sons, Inc., New York.

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Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication

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ISBN 0-471-60847-5, MathReview Entry

Referenced by:

(1.18.68), §1.18(ix), §1.18(ix), §1.18(ix), §1.18(x).

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16: 18.10 Integral Representations

§18.10 Integral Representations

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Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

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$p_n(x)$	$g_0(x)$	$g_1(z,x)$	$g_2(z,x)$	$c$	Conditions
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Laguerre

… ►For the Bessel function $J_{\nu }\left(z\right)$ see §10.2(ii). … ►See also §18.17.

17: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

… ►as $n\to \mathrm{\infty }$, uniformly with respect to $\theta \in [\delta ,\pi -\delta ]$. … ►Also, when , the right-hand side of (18.15.12) with $M=\mathrm{\infty }$ converges; paradoxically, however, the sum is $2P_n\left(\cos \theta \right)$ and not $P_n\left(\cos \theta \right)$ as stated erroneously in Szegő (1975, §8.4(3)). … ►The asymptotic behavior of the classical OP’s as $x\to \pm \mathrm{\infty }$ with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. … ►See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).

18: Bibliography B

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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

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Reprint of the 1897 original, with a foreword by Igor Krichever.

External Links:

ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt

Referenced by:

§20.9(ii).

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Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A 58 (3), pp. 1799–1809.

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Document

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§24.18.

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A monotonicity property involving ${}_3{}^{}F_2^{}$ and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.

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ISSN 1095-7154, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§19.9(i), §19.9(i).

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Electromagnetic Fields and Interactions. Vol. I, Blaisdell, New York.

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Replaces Abraham & Becker, Classical Theory of Electricity and Magnetism.

Referenced by:

§19.33(iii), §19.34.

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A Brief Introduction to Theta Functions. Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York.

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MathReview (W. J. LeVeque), ZentralBlatt

Referenced by:

§20.10(i), §20.10(ii), §20.11(i), §20.13, §20.5(ii), §20.7(viii), §20.8.

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19: Bibliography

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Chapter $16$ of Ramanujan’s second notebook: Theta-functions and $q$-series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.

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ISSN 0065-9266, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§20.11(ii).

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The classical moment problem and some related questions in analysis. Classics in Applied Mathematics, Vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Reprint of the 1965 edition [ 0184042], Translated by N. Kemmer, With a foreword by H. J. Landau

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ISBN 978-1-611976-38-0, MathReview Entry

Referenced by:

§18.40(ii).

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On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.

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Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§17.6(ii).

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Classical Orthogonal Polynomials. In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.), Lecture Notes in Math., Vol. 1171, pp. 36–62.

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MathReview, ZentralBlatt

Referenced by:

(18.27.12_5), (18.27.6).

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Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York.

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Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition

External Links:

ISBN 0-387-96890-3, MathReview

Referenced by:

§21.5(i).

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20: 23.22 Methods of Computation

§23.22 Methods of Computation

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§23.22(i) Function Values

… ►For $\mathrm{\wp }\left(z\right)$ we apply (23.6.2) and (23.6.5), generating all needed values of the theta functions by the methods described in §20.14. … ► … ►The determination of suitable generators $2{\omega }_1$ and $2{\omega }_3$ is the classical inversion problem (Whittaker and Watson (1927, §21.73), McKean and Moll (1999, §2.12); see also §20.9(i) and McKean and Moll (1999, §2.16)). …