A Complete Solution Guide to Real and Complex Analysis II

✍ Scribed by Kit-Wing Yu

• Publisher: 978-988-74156-5-7
• Year: 2021
• Tongue: English
• Leaves: 290
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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