Kooperativer Bibliotheksverbund

UID:

Format: XVI, 664 Seiten : , graph. Darst.

Edition: 1. publication

ISBN: 0-521-62321-9 , 0-521-78988-5 , 978-0-521-62321-6 , 978-0-521-78988-2

Series Statement: Encyclopedia of Mathematics and Its Applications 71

Note: Hier auch später erschienene, unveränderte Nachdrucke

Language: English

Subjects: Mathematics

Keywords: Spezielle Funktion ; Hypergeometrische Reihe ; Bessel-Funktionen ; Gammafunktion ; Orthogonale Polynome

URL: Table of contents

URL: Inhaltsverzeichnis

URL: Inhaltsverzeichnis

URL: Inhaltstext

URL: Table of contents

URL: Inhaltsverzeichnis

Author information: Andrews, George E., 1938-

UID:

Format: XVI, 664 Seiten , graph. Darst.

Edition: 1. publication

ISBN: 0521623219 , 0521789885 , 9780521623216 , 9780521789882

Series Statement: Encyclopedia of Mathematics and Its Applications 71

Note: Hier auch später erschienene, unveränderte Nachdrucke

Language: English

Subjects: Mathematics

Keywords: Spezielle Funktion ; Hypergeometrische Reihe ; Bessel-Funktionen ; Gammafunktion ; Orthogonale Polynome

URL: Table of contents

URL: Inhaltsverzeichnis

Author information: Andrews, George E. 1938-

UID:

Format: 1 online resource (xix, 974 pages) : , digital, PDF file(s).

Edition: 1st ed.

ISBN: 1-316-08658-5 , 1-139-63551-4 , 1-283-29584-9 , 1-139-12260-6 , 9786613295842 , 0-511-84419-0 , 1-139-11686-X , 1-139-12752-7 , 1-139-11250-3 , 1-139-11469-7

Content: The discovery of infinite products by Wallis and infinite series by Newton marked the beginning of the modern mathematical era. It allowed Newton to solve the problem of finding areas under curves defined by algebraic equations, an achievement beyond the scope of the earlier methods of Torricelli, Fermat and Pascal. While Newton and his contemporaries, including Leibniz and the Bernoullis, concentrated on mathematical analysis and physics, Euler's prodigious accomplishments demonstrated that series and products could also address problems in algebra, combinatorics and number theory. In this book, Ranjan Roy describes many facets of the discovery and use of infinite series and products as worked out by their originators, including mathematicians from Asia, Europe and America. The text provides context and motivation for these discoveries, with many detailed proofs, offering a valuable perspective on modern mathematics. Mathematicians, mathematics students, physicists and engineers will all read this book with benefit and enjoyment.

Note: Title from publisher's bibliographic system (viewed on 05 Oct 2015). , Machine generated contents note: 1. Power series in fifteenth-century Kerala; 2. Sums of powers of integers; 3. Infinite product of Wallis; 4. The binomial theorem; 5. The rectification of curves; 6. Inequalities; 7. Geometric calculus; 8. The calculus of Newton and Leibniz; 9. De Analysi per Aequationes Infinitas; 10. Finite differences: interpolation and quadrature; 11. Series transformation by finite differences; 12. The Taylor series; 13. Integration of rational functions; 14. Difference equations; 15. Differential equations; 16. Series and products for elementary functions; 17. Solution of equations by radicals; 18. Symmetric functions; 19. Calculus of several variables; 20. Algebraic analysis: the calculus of operations; 21. Fourier series; 22. Trigonometric series after 1830; 23. The gamma function; 24. The asymptotic series for ln [Gamma] (x); 25. The Euler-Maclaurin summation formula; 26. L-series; 27. The hypergeometric series; 28. Orthogonal polynomials; 29. q-Series; 30. Partitions; 31. q-Series and q-orthogonal polynomials; 32. Primes in arithmetic progressions; 33. Distribution of primes: early results; 34. Invariant theory: Cayley and Sylvester; 35. Summability; 36. Elliptic functions: eighteenth century; 37. Elliptic functions: nineteenth century; 38. Irrational and transcendental numbers; 39. Value distribution theory; 40. Univalent functions; 41. Finite fields. , English

Additional Edition: ISBN 0-521-11470-5

Language: English

URL: Volltext

URL: Inhaltstext

URL: https://doi.org/10.1017/CBO9780511844195

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Format: 1 Online-Ressource (XXV, 411 p. 71 illus., 65 illus. in color).

Edition: 1st ed. 2023

ISBN: 978-3-031-40128-2

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-40127-5

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-40129-9

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-40130-5

Language: English

Keywords: Public Health ; Bevölkerungsentwicklung ; Hygiene

DOI: 10.1007/978-3-031-40128-2

URL: Volltext  (URL des Erstveröffentlichers)

URL: https://doi.org/10.1007/978-3-031-40128-2

UID:

Format: 1 Online-Ressource (XXV, 411 p. 71 illus., 65 illus. in color).

Edition: 1st ed. 2023

ISBN: 978-3-031-40128-2

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-40127-5

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-40129-9

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-40130-5

Language: English

Keywords: Public Health ; Bevölkerungsentwicklung ; Hygiene

DOI: 10.1007/978-3-031-40128-2

URL: Volltext  (URL des Erstveröffentlichers)

UID:

Format: XIX, 974 Seiten : , Illustrationen.

ISBN: 978-0-521-11470-7

Later: Fortgesetzt durch Ranjan Roy Series and products in the development of mathematics

Language: English

Subjects: Mathematics

URL: Cover image

URL: Contributor biographical information

URL: Publisher description

URL: Table of contents only

URL: http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027430095&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA

URL: Cover

URL: Autorenbiografie

URL: Verlagsangaben

URL: Inhaltsverzeichnis

URL: Inhaltstext

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Format: 1 online resource (xviii, 459 pages) : , digital, PDF file(s).

Edition: Second edition.

ISBN: 9781108671620 (ebook)

Uniform Title: Sources in the development of mathematics

Content: This is the second volume of a two-volume work that traces the development of series and products from 1380 to 2000 by presenting and explaining the interconnected concepts and results of hundreds of unsung as well as celebrated mathematicians. Some chapters deal with the work of primarily one mathematician on a pivotal topic, and other chapters chronicle the progress over time of a given topic. This updated second edition of Sources in the Development of Mathematics adds extensive context, detail, and primary source material, with many sections rewritten to more clearly reveal the significance of key developments and arguments. Volume 1, accessible even to advanced undergraduate students, discusses the development of the methods in series and products that do not employ complex analytic methods or sophisticated machinery. Volume 2 examines more recent results, including deBranges' resolution of Bieberbach's conjecture and Nevanlinna's theory of meromorphic functions.

Note: Title from publisher's bibliographic system (viewed on 19 Mar 2021).

Additional Edition: Print version: ISBN 9781108709378

Language: English

URL: https://doi.org/10.1017/9781108671620

UID:

Format: 1 online resource (xiii, 475 pages) : , digital, PDF file(s).

ISBN: 1-108-13210-3 , 1-108-13378-9 , 1-316-67150-X

Content: This thorough work presents the fundamental results of modular function theory as developed during the nineteenth and early-twentieth centuries. It features beautiful formulas and derives them using skillful and ingenious manipulations, especially classical methods often overlooked today. Starting with the work of Gauss, Abel, and Jacobi, the book then discusses the attempt by Dedekind to construct a theory of modular functions independent of elliptic functions. The latter part of the book explains how Hurwitz completed this task and includes one of Hurwitz's landmark papers, translated by the author, and delves into the work of Ramanujan, Mordell, and Hecke. For graduate students and experts in modular forms, this book demonstrates the relevance of these original sources and thereby provides the reader with new insights into contemporary work in this area.

Note: Title from publisher's bibliographic system (viewed on 21 Apr 2017). , Cover -- Half title -- Title -- Copyright -- Contents -- Preface -- 1 The Basic Modular Forms of the Nineteenth Century -- 1.1 The Modular Group -- 1.2 Modular Forms -- 1.3 Exercises -- 2 Gauss's Contributions to Modular Forms -- 2.1 Early Work on Elliptic Integrals -- 2.2 Landen and Legendre's Quadratic Transformation -- 2.3 Lagrange's Arithmetic-Geometric Mean -- 2.4 Gauss on the Arithmetic-Geometric Mean -- 2.5 Gauss on Elliptic Functions -- 2.6 Gauss: Theta Functions and Modular Forms -- 2.7 Exercises -- 3 Abel and Jacobi on Elliptic Functions -- 3.1 Preliminary Remarks -- 3.2 Jacobi on Transformations of Orders 3 and 5 -- 3.3 The Jacobi Elliptic Functions -- 3.4 Transformations of Order n and Infinite Products -- 3.5 Jacobi's Transformation Formulas -- 3.6 Equivalent Forms of the Transformation Formulas -- 3.7 The First and Second Transformations -- 3.8 Complementary Transformations -- 3.9 Jacobi's First Supplementary Transformation -- 3.10 Jacobi's Infinite Products for Elliptic Functions -- 3.11 Jacobi's Theory of Theta Functions -- 3.12 Jacobi's Triple Product Identity -- 3.13 Modular Equations and Transformation Theory -- 3.14 Exercises -- 4 Eisenstein and Hurwitz -- 4.1 Preliminary Remarks -- 4.2 Eisenstein's Theory of Trigonometric Functions -- 4.3 Eisenstein's Derivation of the Addition Formula -- 4.4 Eisenstein's Theory of Elliptic Functions -- 4.5 Differential Equations for Elliptic Functions -- 4.6 The Addition Theorem for the Elliptic Function -- 4.7 Eisenstein's Double Product -- 4.8 Elliptic Functions in Terms of the Φ Function -- 4.9 Connection of Φ with Theta Functions -- 4.10 Hurwitz's Fourier Series for Modular Forms -- 4.11 Hurwitz's Proof That ∆(w) Is a Modular Form -- 4.12 Hurwitz's Proof of Eisenstein's Result -- 4.13 Kronecker's Proof of Eisenstein's Result -- 4.14 Exercises. , 5 Hermite's Transformation of Theta Functions -- 5.1 Preliminary Remarks -- 5.2 Hermite's Proof of the Transformation Formula -- 5.3 Smith on Jacobi's Formula for the Product of Four Theta Functions -- 5.4 Exercises -- 6 Complex Variables and Elliptic Functions -- 6.1 Historical Remarks on the Roots of Unity -- 6.2 Simpson and the Ladies Diary -- 6.3 Development of Complex Variables Theory -- 6.4 Hermite: Complex Analysis in Elliptic Functions -- 6.5 Riemann: Meaning of the Elliptic Integral -- 6.6 Weierstrass's Rigorization -- 6.7 The Phragmén-Lindelöf Theorem -- 7 Hypergeometric Functions -- 7.1 Preliminary Remarks -- 7.2 Stirling -- 7.3 Euler and the Hypergeometric Equation -- 7.4 Pfaff's Transformation -- 7.5 Gauss and Quadratic Transformations -- 7.6 Kummer on the Hypergeometric Equation -- 7.7 Riemann and the Schwarzian Derivative -- 7.8 Riemann and the Triangle Functions -- 7.9 The Ratio of the Periods K'/K as a Conformal Map -- 7.10 Schwarz: Hypergeometric Equation with Algebraic Solutions -- 7.11 Exercises -- 8 Dedekind's Paper on Modular Functions -- 8.1 Preliminary Remarks -- 8.2 Dedekind's Approach -- 8.3 The Fundamental Domain for SL[sub(2)] (ℤ) -- 8.4 Tesselation of the Upper Half-plane -- 8.5 Dedekind's Valency Function -- 8.6 Branch Points -- 8.7 Differential Equations -- 8.8 Dedekind's η Function -- 8.9 The Uniqueness of k[sup(2)] -- 8.10 The Connection of η with Theta Functions -- 8.11 Hurwitz's Infinite Product for η(w) -- 8.12 Algebraic Relations among Modular Forms -- 8.13 The Modular Equation -- 8.14 Singular Moduli and Quadratic Forms -- 8.15 Exercises -- 9 The η Function and Dedekind Sums -- 9.1 Preliminary Remarks -- 9.2 Riemann's Notes -- 9.3 Dedekind Sums in Terms of a Periodic Function -- 9.4 Rademacher -- 9.5 Exercises -- 10 Modular Forms and Invariant Theory -- 10.1 Preliminary Remarks. , 10.2 The Early Theory of Invariants -- 10.3 Cayley's Proof of a Result of Abel -- 10.4 Reduction of an Elliptic Integral to Riemann's Normal Form -- 10.5 The Weierstrass Normal Form -- 10.6 Proof of the Infinite Product for ∆ -- 10.7 The Multiplier in Terms of 12√∆ -- 11 The Modular and Multiplier Equations -- 11.1 Preliminary Remarks -- 11.2 Jacobi's Multiplier Equation -- 11.3 Sohnke's Paper on Modular Equations -- 11.4 Brioschi on Jacobi's Multiplier Equation -- 11.5 Joubert on the Multiplier Equation -- 11.6 Kiepert and Klein on the Multiplier Equation -- 11.7 Hurwitz: Roots of the Multiplier Equation -- 11.8 Exercises -- 12 The Theory of Modular Forms as Reworked by Hurwitz -- 12.1 Preliminary Remarks -- 12.2 The Fundamental Domain -- 12.3 An Infinite Product as a Modular Form -- 12.4 The J-Function -- 12.5 An Application to the Theory of Elliptic Functions -- 13 Ramanujan's Euler Products and Modular Forms -- 13.1 Preliminary Remarks -- 13.2 Ramanujan's τ Function -- 13.3 Ramanujan: Product Formula for ∆ -- 13.4 Proof of Identity (13.2) -- 13.5 The Arithmetic Function τ(n) -- 13.6 Mordell on Euler Products -- 13.7 Exercises -- 14 Dirichlet Series and Modular Forms -- 14.1 Preliminary Remarks -- 14.2 Functional Equations for Dirichlet Series -- 14.3 Theta Series in Two Variables -- 14.4 Exercises -- 15 Sums of Squares -- 15.1 Preliminary Remarks -- 15.2 Jacobi's Elliptic Functions Approach -- 15.3 Glaisher -- 15.4 Ramanjuan's Arithmetical Functions -- 15.5 Mordell: Spaces of Modular Forms -- 15.6 Hardy's Singular Series -- 15.7 Hecke's Solution to the Sums of Squares Problem -- 15.8 Exercises -- 16 The Hecke Operators -- 16.1 Preliminary Remarks -- 16.2 The Hecke Operators T(n) -- 16.3 The Operators T(n) in Terms of Matrices λ(n) -- 16.4 Euler Products -- 16.5 Eigenfunctions of the Hecke Operators -- 16.6 The Petersson Inner Product. , 16.7 Exercises -- Appendix: Translation of Hurwitz's Paper of 1904 -- 1. Equivalent Quantities -- 2. The Modular Forms G[sub(n)](ω[sub(1)], ω[sub(2)]) -- 3. The Representation of the Function G[sub(n)] by Power Series -- 4. The Modular Form ∆(ω[sub(1)], ω[sub(2)]) -- 5. The Modular Function J(ω) -- 6. Applications to the Theory of Elliptic Functions -- Bibliography -- Index.

Additional Edition: ISBN 1-107-15938-5

Language: English

URL: https://doi.org/10.1017/9781316671504

UID:

Format: 1 online resource (xviii, 758 pages) : , digital, PDF file(s).

Edition: Second edition.

ISBN: 1-108-57318-5 , 1-108-62770-6

Uniform Title: Sources in the development of mathematics

Content: This is the first volume of a two-volume work that traces the development of series and products from 1380 to 2000 by presenting and explaining the interconnected concepts and results of hundreds of unsung as well as celebrated mathematicians. Some chapters deal with the work of primarily one mathematician on a pivotal topic, and other chapters chronicle the progress over time of a given topic. This updated second edition of Sources in the Development of Mathematics adds extensive context, detail, and primary source material, with many sections rewritten to more clearly reveal the significance of key developments and arguments. Volume 1, accessible to even advanced undergraduate students, discusses the development of the methods in series and products that do not employ complex analytic methods or sophisticated machinery. Volume 2 treats more recent work, including deBranges' solution of Bieberbach's conjecture, and requires more advanced mathematical knowledge.

Note: Title from publisher's bibliographic system (viewed on 22 Feb 2021).

Additional Edition: ISBN 1-108-70945-1

Language: English

URL: https://doi.org/10.1017/9781108709453

UID:

Format: 1 online resource (xvi, 664 pages) : , digital, PDF file(s).

ISBN: 9781107325937 (ebook)

Series Statement: Encyclopedia of mathematics and its applications ; volume 71

Content: Special functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for this beautiful and useful area of mathematics. This treatise presents an overview of special functions, focusing primarily on hypergeometric functions and the associated hypergeometric series, including Bessel functions and classical orthogonal polynomials, using the basic building block of the gamma function. In addition to relatively new work on gamma and beta functions, such as Selberg's multidimensional integrals, many important but relatively unknown nineteenth century results are included. Other topics include q-extensions of beta integrals and of hypergeometric series, Bailey chains, spherical harmonics, and applications to combinatorial problems. The authors provide organizing ideas, motivation, and historical background for the study and application of some important special functions. This clearly expressed and readable work can serve as a learning tool and lasting reference for students and researchers in special functions, mathematical physics, differential equations, mathematical computing, number theory, and combinatorics.

Note: Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Additional Edition: Print version: ISBN 9780521623216

Language: English

URL: https://doi.org/10.1017/CBO9781107325937

URL: Volltext

URL: Inhaltstext