Rami Shakarchi – författare

This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors then use the Baire category theorem to illustrate several points, including the existence of Besicovitch sets. The second half of the book introduces readers to other central topics in analysis, such as probability theory and Brownian motion, which culminates in the solution of Dirichlet's problem. The concluding chapters explore several complex variables and oscillatory integrals in Fourier analysis, and illustrate applications to such diverse areas as nonlinear dispersion equations and the problem of counting lattice points. Throughout the book, the authors focus on key results in each area and stress the organic unity of the subject.* A comprehensive and authoritative text that treats some of the main topics of modern analysis * A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables * Key results in each area discussed in relation to other areas of mathematics * Highlights the organic unity of large areas of analysis traditionally split into subfields * Interesting exercises and problems illustrate ideas * Clear proofs provided

The present volume contains all the exercises and their solutions of Lang's' Linear Algebra. Solving problems being an essential part of the learning process, my goal is to provide those learning and teaching linear algebra with a large number of worked out exercises. Lang's textbook covers all the topics in linear algebra that are usually taught at the undergraduate level: vector spaces, matrices and linear maps including eigenvectors and eigenvalues, determinants, diagonalization of symmetric and hermitian maps, unitary maps and matrices, triangulation, Jordan canonical form, and convex sets. Therefore this solutions manual can be helpful to anyone learning or teaching linear algebra at the college level. As the understanding of the first chapters is essential to the comprehension of the later, more involved chapters, I encourage the reader to work through all of the problems of Chapters I, II, III and IV. Often earlier exercises are useful in solving later problems. (For example, Exercise 35, §3 of Chapter II shows that a strictly upper triangular matrix is nilpotent and this result is then used in Exercise 7, §1 of Chapter X.) To make the solutions concise, I have included only the necessary arguments; the reader may have to fill in the details to get complete proofs. Finally, I thank Serge Lang for giving me the opportunity to work on this solutions manual, and I also thank my brother Karim and Steve Miller for their helpful comments and their support.

The present volume contains all the exercises and their solutions for Lang's second edition of Undergraduate Analysis. The wide variety of exercises, which range from computational to more conceptual and which are of vary­ ing 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 teaching 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 Lang'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 singulari­ ties, and prove that the space of functions is dense in the space of regu­ lated maps. The numbering of the problems is as follows. Exercise IX. 5. 7 indicates Exercise 7, §5, of Chapter IX. Acknowledgments I am grateful to Serge Lang for his help and enthusiasm in this project, as well as for teaching me mathematics (and much more) with so much generosity and patience.

This book contains all the exercises and solutions of Serge Lang's Complex Analy­ sis. Chapters I through VITI of Lang's book contain the material of an introductory course at the undergraduate level and the reader will find exercises in all of the fol­ lowing topics: power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings and har­ monic functions. Chapters IX through XVI, which are suitable for a more advanced course at the graduate level, offer exercises in the following subjects: Schwarz re­ flection, analytic continuation, Jensen's formula, the Phragmen-LindelOf theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and the Zeta function. This solutions manual offers a large number of worked out exercises of varying difficulty. I thank Serge Lang for teaching me complex analysis with so much enthusiasm and passion, and for giving me the opportunity to work on this answer book. Without his patience and help, this project would be far from complete. I thank my brother Karim for always being an infinite source of inspiration and wisdom. Finally, I want to thank Mark McKee for his help on some problems and Jennifer Baltzell for the many years of support, friendship and complicity. Rami Shakarchi Princeton, New Jersey 1999 Contents Preface vii I Complex Numbers and Functions 1 1. 1 Definition . . . . . . . . . . 1 1. 2 Polar Form . . . . . . . . . 3 1. 3 Complex Valued Functions . 8 1. 4 Limits and Compact Sets . . 9 1. 6 The Cauchy-Riemann Equations .

This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions. The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression. In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis.Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.

With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle. With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory. Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences.The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets.Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels. Also available, the first two volumes in the Princeton Lectures in Analysis:

The present volume contains all the exercises and their solutions of Lang''s'' Linear Algebra. Solving problems being an essential part of the learning process, my goal is to provide those learning and teaching linear algebra with a large number of worked out exercises. Lang''s textbook covers all the topics in linear algebra that are usually taught at the undergraduate level: vector spaces, matrices and linear maps including eigenvectors and eigenvalues, determinants, diagonalization of symmetric and hermitian maps, unitary maps and matrices, triangulation, Jordan canonical form, and convex sets. Therefore this solutions manual can be helpful to anyone learning or teaching linear algebra at the college level. As the understanding of the first chapters is essential to the comprehension of the later, more involved chapters, I encourage the reader to work through all of the problems of Chapters I, II, III and IV. Often earlier exercises are useful in solving later problems. (For example, Exercise 35, §3 of Chapter II shows that a strictly upper triangular matrix is nilpotent and this result is then used in Exercise 7, §1 of Chapter X.) To make the solutions concise, I have included only the necessary arguments; the reader may have to fill in the details to get complete proofs. Finally, I thank Serge Lang for giving me the opportunity to work on this solutions manual, and I also thank my brother Karim and Steve Miller for their helpful comments and their support.

This book contains all the exercises and solutions of Serge Lang''s Complex Analy­ sis. Chapters I through VITI of Lang''s book contain the material of an introductory course at the undergraduate level and the reader will find exercises in all of the fol­ lowing topics: power series, Cauchy''s theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings and har­ monic functions. Chapters IX through XVI, which are suitable for a more advanced course at the graduate level, offer exercises in the following subjects: Schwarz re­ flection, analytic continuation, Jensen''s formula, the Phragmen-LindelOf theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and the Zeta function. This solutions manual offers a large number of worked out exercises of varying difficulty. I thank Serge Lang for teaching me complex analysis with so much enthusiasm and passion, and for giving me the opportunity to work on this answer book. Without his patience and help, this project would be far from complete. I thank my brother Karim for always being an infinite source of inspiration and wisdom. Finally, I want to thank Mark McKee for his help on some problems and Jennifer Baltzell for the many years of support, friendship and complicity. Rami Shakarchi Princeton, New Jersey 1999 Contents Preface vii I Complex Numbers and Functions 1 1. 1 Definition . . . . . . . . . . 1 1. 2 Polar Form . . . . . . . . . 3 1. 3 Complex Valued Functions . 8 1. 4 Limits and Compact Sets . . 9 1. 6 The Cauchy-Riemann Equations .