Advanced Complex Analysis - A Comprehensive Course in Analysis, Part 2B

Main subject categories: • Complex analysis • Functions of a complex variable • Special functions • Ordinary differential equations • Number theory • Univalent and multivalent functions of one complex variable • Value distribution of meromorphic functions of one complex variable • Nevanlinna theory • Classical hypergeometric functions, 2F1 • Stochastic (Schramm-)Loewner evolution (SLE)A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis.Part 2B provides a comprehensive look at a number of subjects of complex analysis not included in Part 2A. Presented in this volume are the theory of conformal metrics (including the Poincaré metric, the Ahlfors-Robinson proof of Picard's theorem, and Bell's proof of the Painlevé smoothness theorem), topics in analytic number theory (including Jacobi's two- and four-square theorems, the Dirichlet prime progression theorem, the prime number theorem, and the Hardy-Littlewood asymptotics for the number of partitions), the theory of Fuchsian differential equations, asymptotic methods (including Euler's method, stationary phase, the saddle-point method, and the WKB method), univalent functions (including an introduction to SLE), and Nevanlinna theory. The chapters on Fuchsian differential equations and on asymptotic methods can be viewed as a minicourse on the theory of special functions. A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Part 1 is devoted to real analysis. From one point of view, it presents the infinitesimal calculus of the twentieth century with the ultimate integral calculus (measure theory) and the ultimate differential calculus (distribution theory). From another, it shows the triumph of abstract spaces: topological spaces, Banach and Hilbert spaces, measure spaces, Riesz spaces, Polish spaces, locally convex spaces, Fréchet spaces, Schwartz space, and LpLp spaces. Finally it is the study of big techniques, including the Fourier series and transform, dual spaces, the Baire category, fixed point theorems, probability ideas, and Hausdorff dimension. Applications include the constructions of nowhere differentiable functions, Brownian motion, space-filling curves, solutions of the moment problem, Haar measure, and equilibrium measures in potential theory. Part 2A is devoted to basic complex analysis. It interweaves three analytic threads associated with Cauchy, Riemann, and Weierstrass, respectively. Cauchy's view focuses on the differential and integral calculus of functions of a complex variable, with the key topics being the Cauchy integral formula and contour integration. For Riemann, the geometry of the complex plane is central, with key topics being fractional linear transformations and conformal mapping. For Weierstrass, the power series is king, with key topics being spaces of analytic functions, the product formulas of Weierstrass and Hadamard, and the Weierstrass theory of elliptic functions. Subjects in this volume that are often missing in other texts include the Cauchy integral theorem when the contour is the boundary of a Jordan region, continued fractions, two proofs of the big Picard theorem, the uniformization theorem, Ahlfors's function, the sheaf of analytic germs, and Jacobi, as well as Weierstrass, elliptic functions. Part 2B provides a comprehensive look at a number of subjects of complex analysis not included in Part 2A. Presented in this volume are the theory of conformal metrics (including the Poincaré metric, the Ahlfors-Robinson proof of Picard's theorem, and Bell's proof of the Painlevé smoothness theorem), topics in analytic number theory (including Jacobi's two- and four-square theorems, the Dirichlet prime progression theorem, the prime number theorem, and the Hardy-Littlewood asymptotics for the number of partitions), the theory of Fuchsian differential equations, asymptotic methods (including Euler's method, stationary phase, the saddle-point method, and the WKB method), univalent functions (including an introduction to SLE), and Nevanlinna theory. The chapters on Fuchsian differential equations and on asymptotic methods can be viewed as a minicourse on the theory of special functions. Part 3 returns to the themes of Part 1 by discussing pointwise limits (going beyond the usual focus on the Hardy-Littlewood maximal function by including ergodic theorems and martingale convergence), harmonic functions and potential theory, frames and wavelets, HpHp spaces (including bounded mean oscillation (BMO)) and, in the final chapter, lots of inequalities, including Sobolev spaces, Calderon-Zygmund estimates, and hypercontractive semigroups. Part 4 focuses on operator theory, especially on a Hilbert space. Central topics are the spectral theorem, the theory of trace class and Fredholm determinants, and the study of unbounded self-adjoint operators. There is also an introduction to the theory of orthogonal polynomials and a long chapter on Banach algebras, including the commutative and non-commutative Gel'fand-Naimark theorems and Fourier analysis on general locally compact abelian groups In the second half of 2015, the American Math Society will publish a five volume (total about 3000 pages) set of books that is a graduate analysis text with lots of additional bonus material. Included are hundreds of problems and copious notes which extend the text and provide historical background. Efforts have been made to find simple and elegant proofs and to keeping the writing style clear. Conformal metric methods, topics in analytic number theory, Fuchsian ODEs and associated special functions, asymptotic methods, univalent functions, Nevanlinna theory. Selected topics include Poincar metric, Ahlfors-Robinson proof of Picards theorem, Bergmann kernel, Painlevs conformal mapping theorem, Jacobi 2- and 4-squares theorems, Dirichlet series, Dirichlets prime progression theorem, zeta function, prime number theorem, hypergeometric, Bessel and Airy functions, Hankel and Sommerfeld contours, Laplaces method, stationary phase, steepest descent, WKB, Koebe function, Loewner evolution and introduction to SLE, Nevanlinnas First and Second Main theorems. Cover......Page 1 Title page......Page 4 Contents......Page 8 Preface to the series......Page 10 Preface to Part 2......Page 16 Chapter 12. Riemannian metrics and complex analysis......Page 18 Chapter 13. Some topics in analytic number theory......Page 54 Chapter 14. Ordinary differential equations in the complex domain......Page 112 Chapter 15. Asymptotic methods......Page 178 Chapter 16. Univalent functions and Loewner evolution......Page 248 Chapter 17. Nevanlinna theory......Page 274 Bibliography......Page 302 Symbol index......Page 326 Subject index......Page 328 Author index......Page 332 Index of capsule biographies......Page 338 Back Cover......Page 339 Part 1. Real Analysis -- Part 2a. Basic Complex Analysis -- Part 2b. Advanced Complex Analysis -- Part 3. Harmonic Analysis -- Part 4. Operator Theory. Barry Simon. Includes Bibliographical References And Index.