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### Problems and Solutions for Undergraduate Analysis (2E) by Rami Shakarchi, Serge Lang

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Undergraduate Analysis (2E Solution) written by Rami Shakarchi, Serge Lang
The present volume contains all the exercises and their solutions for Lang's second edition of Undergraduate Analysis. The wide variety of exe􀐑cises, which range from computational to more conceptual and which are of varying difficulty, cover the following subjects and more: real numbers, limits, continuous functions, differentiation and elementary integration, normed vector spaces, compactness, series, integration in one variable, improper integrals, convolutions, Fourier series and the Fourier integral, functions in n-space, derivatives in vector spaces, the inverse and implicit mapping theorem, ordinary differential equations, multiple integrals, and differential forms. My objective is to offer those learning and teaehing analysis at the undergraduate level a large number of completed exercises and I hope that this book, which contains over 600 exercises covering the topics mentioned above, will achieve my goal.
The exercises are an integral part of L􀐒ng's book and I encourage the reader to work through all of them. In some cases, the problems in the beginning chapters are used in later ones, for example, in Chapter IV when one constructs-bump functions, which are used to smooth out singularities,. and prove that the space of functions is dense in the space of regulated maps. The numbering of the problems is as follows .. Exercise IX.5. 7 indicates Exercise 7, §5, of Chapter IX.
(Rami Shakarchi)

Book Detail :-
Title: Problems and Solutions for Undergraduate Analysis
Edition: 2nd
Author(s): Rami Shakarchi, Serge Lang
Publisher: Springer
Year: 1998
Pages: 381
Type: PDF
Language: English
ISBN: 0387982353,9780387982359
Country: US
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Author Serge Lang is Professor of Mathematics from Department of Mathematics, Yale University, New Haven, CT 06520, US.
Serge Lang was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra.
Lang was born in Saint-Germain-en-Laye, close to Paris, in 1927. Lang moved with his family to California as a teenager, where he graduated in 1943 from Beverly Hills High School. He subsequently graduated from the California Institute of Technology in 1946, and received a doctorate from Princeton University in 1951. He held faculty positions at the University of Chicago, Columbia University, and Yale University.
Lang studied under Emil Artin at Princeton University, writing his thesis on quasi-algebraic closure, and then worked on the geometric analogues of class field theory and diophantine geometry. Later he moved into diophantine approximation and transcendental number theory, proving the Schneider–Lang theorem. A break in research while he was involved in trying to meet 1960s student activism halfway caused him difficulties in picking up the threads afterwards. He wrote on modular forms and modular units, the idea of a 'distribution' on a profinite group, and value distribution theory. He made a number of conjectures in diophantine geometry: Mordell–Lang conjecture, Bombieri–Lang conjecture, Lang–Trotter conjecture, and the Lang conjecture on analytically hyperbolic varieties. He introduced the Lang map, the Katz–Lang finiteness theorem, and the Lang–Steinberg theorem (cf. Lang's theorem) in algebraic groups.
Lang was a prolific writer of mathematical texts, often completing one on his summer vacation. Most are at the graduate level. He wrote calculus texts and also prepared a book on group cohomology for Bourbaki. Lang's Algebra, a graduate-level introduction to abstract algebra, was a highly influential text that ran through numerous updated editions. His Steele prize citation stated, "Lang's Algebra changed the way graduate algebra is taught...It has affected all subsequent graduate-level algebra books." It contained ideas of his teacher, Artin; some of the most interesting passages in Algebraic Number Theory also reflect Artin's influence and ideas that might otherwise not have been published in that or any form.

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Book Contents :-
Undergraduate Analysis (2E Solution) written by Rami Shakarchi, Serge Lang cover the following topics.
0. Sets and Mappings
1. Real Numbers
2. Limits and Continuous Functions
3. Differentiation
4. Elementary Functions
5. The Elementary Real Integral
6. Normed Vector Spaces
7. Limits
8. Compactness
9. Series
10. The Integral in One Variable
11. Approximation with Convolutions
12. Fourier Series
13. Improper Integrals
14. The Fourier Integral
15. Functions on n-Space
16. The Winding Number and Global Potential Functions
17. Derivatives in Vector Spaces
18. Inverse Mapping Theorem
19. Ordinary Differential Equations
20. Multiple Integrals
21. Differential Forms

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