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This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs.  The choice of material and the flexible organization, including three different entryways into the study of the real numbers, making it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus.  The Real Numbers and Real Analysis is accessible to students who have prior experience with mathematical proofs and who have not previously studied real analysis. The text includes over 350 exercises.   Key features of this textbook:   - provides an unusually thorough treatment of the real numbers, emphasizing their importance as the basis of real analysis   - presents material in an order resembling that of standard calculus courses, for the sake of student familiarity, and for helping future teachers use real analysis to better understand calculus   - emphasizes the direct role of the Least Upper Bound Property in the study of limits, derivatives and integrals, rather than relying upon sequences for proofs; presents the equivalence of various important theorems of real analysis with the Least Upper Bound Property   - includes a thorough discussion of some topics, such as decimal expansion of real numbers, transcendental functions, area and the number p, that relate to calculus but that are not always treated in detail in real analysis texts   - offers substantial historical material in each chapter   This book will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.

Mathematical analysis --- Mathematics --- analyse (wiskunde) --- reeksen (wiskunde)

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Mathematical analysis --- Mathematics --- analyse (wiskunde) --- reeksen (wiskunde)

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This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17. Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook.

Geometry --- Mathematics --- landmeetkunde --- reeksen (wiskunde) --- functies (wiskunde) --- wiskunde

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This volume is the second of approximately four volumes that the authors plan to write on Ramanujan's lost notebook, which is broadly interpreted to include all material published in The Lost Notebook and Other Unpublished Papers in 1988. The primary topics addressed in the authors' second volume on the lost notebook are q-series, Eisenstein series, and theta functions. Most of the entries on q-series are located in the heart of the original lost notebook, while the entries on Eisenstein series are either scattered in the lost notebook or are found in letters that Ramanujan wrote to G.H. Hardy from nursing homes. About Ramanujan's Lost Notebook, Volume I: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society "...the results are organized topically with cross-references to the identities as they appear in the original Ramanujan manuscript. Particularly helpful are the extensive references, indicating where in the literature these results have been proven or independently discovered as well as where and how they have been used." - Bulletin of the American Mathematical Society "The mathematics community owes a huge debt of gratitude to Andrews and Berndt for undertaking the monumental task of producing a coherent presentation along with complete proofs of the chaotically written mathematical thoughts of Ramanujan during the last year of his life. Some 85 years after his death, beautiful "new" and useful results of Ramanujan continue to be brought to light." - Mathematical Reviews

Geometry --- Mathematics --- landmeetkunde --- reeksen (wiskunde) --- functies (wiskunde) --- wiskunde

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Geometry --- Mathematics --- landmeetkunde --- reeksen (wiskunde) --- functies (wiskunde) --- wiskunde