A Complete Solution Guide to Real and Complex Analysis II

This is a complete solution guide to all exercises from Chapters 10 to 20 in Rudin's Real and Complex Analysis . The features of this book are as follows: It covers all the 221 exercises from Chapters 10 to 20 with detailed and complete solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied. There are 29 illustrations for explaining the mathematical concepts or ideas used behind the questions or theorems. Sections in each chapter are added so as to increase the readability of the exercises. Different colors are used frequently in order to highlight or explain problems, lemmas, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) Necessary lemmas with proofs are provided because some questions require additional mathematical concepts which are not covered by Rudin. Many useful or relevant references are provided to some questions for your future research. Preface List of Figures Elementary Properties of Holomorphic Functions Basic Properties of Holomorphic Functions Evaluation of Integrals Composition of Holomorphic Functions and Morera's Theorem Problems related to Zeros of Holomorphic Functions Laurent Series and its Applications Miscellaneous Problems Harmonic Functions Basic Properties of Harmonic Functions Harnack's Inequalities and Positive Harmonic Functions The Weak Convergence and Radial Limits of Holomorphic Functions Miscellaneous Problems The Maximum Modulus Principle Applications of the Maximum Modulus Principle Asymptotic Values of Entire Functions Further Applications of the Maximum Modulus Principle Approximations by Rational Functions Meromorphic Functions on S2 and Applications of Runge's Theorem Holomorphic Functions in the Unit Disc without Radial Limits Simply Connectedness and Miscellaneous Problems Conformal Mapping Basic Properties of Conformal Mappings Problems on Normal Families and the Class S Proofs of Conformal Equivalence between Annuli Constructive Proof of the Riemann Mapping Theorem Zeros of Holomorphic Functions Infinite Products and the Order of Growth of an Entire Function Some Examples Problems on Blaschke Products Miscellaneous Problems and the Müntz-Szasz Theorem Analytic Continuation Singular Points and Continuation along Curves Problems on the Modular Group and Removable Sets Miscellaneous Problems Hp-Spaces Problems on Subharmonicity and Harmonic Majoriants Basic Properties of Hp Factorization of f Hp A Projection of Lp onto Hp Miscellaneous Problems Elementary Theory of Banach Algebras Examples of Banach Spaces and Spectrums Properties of Ideals and Homomorphisms The Commutative Banach algebra H Holomorphic Fourier Transforms Problems on Entire Functions of Exponential Type Quasi-analytic Classes and Borel's Theorem Uniform Approximation by Polynomials Index Bibliography This is a complete solution guide to all exercises from Chapters 10 to 20 in Rudin's Real and Complex Analysis. The features of this book are as follows:1. It covers all the 221 exercises from Chapters 10 to 20 with detailed and complete solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied. 2. There are 29 illustrations for explaining the mathematical concepts or ideas used behind the questions or theorems. 3. Sections in each chapter are added so as to increase the readability of the exercises. 4. Different colors are used frequently in order to highlight or explain problems, lemmas, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) 5. Necessary lemmas with proofs are provided because some questions require additional mathematical concepts which are not covered by Rudin. 6. Many useful or relevant references are provided to some questions for your future research.