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11: 22.4 Periods, Poles, and Zeros

… ►The other poles are at congruent points, which is the set of points obtained by making translations by $2mK+2n\mathrm{i}{K}^{\prime }$, where $m,n\in \mathbb{Z}$. … ►Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form $2mK+2n\mathrm{i}{K}^{\prime }$, where $m,n\in \mathbb{Z}$. … ►The set of points $z=mK+n\mathrm{i}{K}^{\prime }$, $m,n\in \mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+n\mathrm{i}{K}^{\prime }$, where again $m,n\in \mathbb{Z}$. … ►

§22.4(iii) Translation by Half or Quarter Periods

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12: Bibliography V

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Newton polyhedra and estimates of oscillatory integrals. Funkcional. Anal. i Priložen. 10 (3), pp. 13–38 (Russian).

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Notes:

English translation in Functional Anal. Appl. 18 (1976), pp. 175-196.

External Links:

ISSN 0374-1990, Document, MathReview (Helmut Hamm), ZentralBlatt

Referenced by:

§36.6.

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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

Referenced by:

§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

Referenced by:

§18.38(iii).

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On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

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Notes:

In Russian. English translation: Differential Equations 1(1), pp. 58–59.

External Links:

MathReview, ZentralBlatt

Referenced by:

§32.8(ii).

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13: Bibliography G

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Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.

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Notes:

Translated from the Russian by Eugene Saletan. A paperbound reprinting by the same publisher appeared in 1977

External Links:

MathReview, ZentralBlatt

Referenced by:

§2.6(i).

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Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 19(1964), no. 4, pp. 1–89

External Links:

Document, MathReview (A. Korányi), ZentralBlatt

Referenced by:

§35.3(i).

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Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point. Translated from the Russian by J. Shorr-Kon, Office of Technical Services, U. S. Department of Commerce, Washington, D.C..

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Notes:

Published for the National Science Foundation by the Israel Program for Scientific Translations, 1960.

External Links:

MathReview, ZentralBlatt

Referenced by:

§22.19(iv).

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Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

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Notes:

In Russian. English translation: Differential Equations 18(3), pp. 317–326.

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).

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Asymptotische Entwicklungen für unvollständige Gammafunktionen. Forum Math. 3 (2), pp. 105–141 (German).

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Notes:

Translated title: Asymptotic expansions for incomplete gamma functions.

External Links:

ISSN 0933-7741, MathReview (T. Erber), ZentralBlatt

Referenced by:

§8.16.

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14: Bibliography P

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Tablitsy nepolnoi gamma-funktsii. Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).

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Notes:

In Russian. Translated title: Tables of the Incomplete Gamma Function.

External Links:

MathReview (John Todd)

Referenced by:

2nd item.

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An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166

External Links:

Document, MathReview (E. Lukacs), ZentralBlatt

Referenced by:

§8.11(iv).

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On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).

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Notes:

English translation in Math. Notes 65 (1999) pp. 571-581.

External Links:

ISSN 0025-567X, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.37.

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Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. Springer-Verlag, New York.

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Notes:

Pólya’s contribution translated from the German by Dorothee Aeppli

External Links:

ISBN 0-387-96413-4, MathReview (Daniel Turzík)

Referenced by:

§26.20.

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Integrals and Series: Elementary Functions, Vol. 1. Gordon & Breach Science Publishers, New York.

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Notes:

Translated from the Russian and with a preface by N. M. Queen. Table errata: Math. Comp. v. 66 (1997), no. 220, pp. 1765–1766, Math. Comp. v. 65 (1996), no. 215, pp. 1380–1381 .

External Links:

ISBN 2-88124-097-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §1.8(v), §24.7(iii), §4.10(iii), §4.11, §4.26(v), §4.27, §4.40(v), §4.41.

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15: 5.19 Mathematical Applications

… ►By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …

16: 20.2 Definitions and Periodic Properties

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§20.2(iii) Translation of the Argument by Half-Periods

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17: Bibliography R

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Tablitsy modifitsirovannykh funktsii Besselya $K_{\frac{1}{2}+i\beta }(x)$ . “Nauka”, Moscow (Russian).

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Notes:

In Russian. Translated title: Tables of Modified Bessel Functions $K_{\frac{1}{2}+i\beta }(x)$

External Links:

MathReview, ZentralBlatt

Referenced by:

2nd item.

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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67

External Links:

ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt

Referenced by:

§10.74(iii).

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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

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Notes:

English translation, AEC-TR-4491, United States Atomic Energy Commission

External Links:

MathReview (A. S. Householder)

Referenced by:

§3.11(iii).

Encodings:

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Functional analysis. Dover Books on Advanced Mathematics, Dover Publications, Inc., New York.

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Notes:

Translated from the second French edition by Leo F. Boron, Reprint of the 1955 original

External Links:

ISBN 0-486-66289-6, MathReview Entry

Referenced by:

§1.18(i), §1.4(v), §1.4(v).

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18: Bibliography S

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A Course in Arithmetic. Graduate Texts in Mathematics, Vol. 7, Springer-Verlag, New York.

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Notes:

Translated from the French.

External Links:

MathReview, ZentralBlatt

Referenced by:

§20.12(i), §20.7(viii), §20.9(i), §20.9(ii), §22.18(iv), §23.15(i), §23.18, §23.19, §23.20(v).

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Asymptotic Solutions of the One-dimensional Schrödinger Equation. American Mathematical Society, Providence, RI.

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Notes:

Translated from the 1990 Russian original by Vadim Khidekel

External Links:

ISBN 0-8218-0563-3, MathReview (Denise Huet), ZentralBlatt

Referenced by:

§31.13.

Encodings:

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Tables of Airy Functions and Special Confluent Hypergeometric Functions. Pergamon Press, New York.

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Notes:

Translated from the Russian by D. G. Fry

External Links:

MathReview (W. F. Freiberger), ZentralBlatt

Referenced by:

(9.13.19), (9.13.20), (9.13.21), §9.13(i), 1st item.

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Izbrannye Trudy. Analiticheskaya teoriya obyknovennykh differentsialnykh uravnenii. Izdatel’ stvo Sankt-Peterburgskogo Universiteta, St. Petersburg (Russian).

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Notes:

In Russian. Translated title: Selected Works. Analytic Theory of Ordinary Differential Equations

External Links:

ISBN 5-288-01337-3, MathReview (Alois Klíč)

Referenced by:

§31.16(i).

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Atombau und Spektrallinien. Vieweg, Braunschweig.

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Notes:

English translation: Wave Mechanics published by Methuen, London, 1930.

External Links:

ZentralBlatt

Referenced by:

§33.1.

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19: Preface

… ►Miller was responsible for information architecture, specializing LaTeX for the needs of the project, translation from LaTeX to MathML, and the search interface. …

20: Bibliography F

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Tablicy značeniĭ funkcii $w(z)=e^{-z^2}(1+\frac{2i}{\sqrt{\pi }}{\int }_0^z{e}^{t^2}𝑑t)$ ot kompleksnogo argumenta. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).

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Notes:

In Russian. English translation, see Faddeyeva and Terent’ev (1961)

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§7.21.

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Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

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The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

External Links:

ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§29.11.

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Asymptotics of the spectrum of the Heun equation and of Heun functions. Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv., 38(1992), no. 3, pp. 621–635 (1992)

External Links:

ISSN 0373-2436, Document, MathReview (Andreĭ Bolibrukh), ZentralBlatt

Referenced by:

§31.13.

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