Książki

At its core, this concise textbook presents standard material for a first course in complex analysis at the advanced undergraduate level. This distinctive text will prove most rewarding for students who have a genuine passion for mathematics as well as certain mathematical maturity. Primarily aimed at undergraduates with working knowledge of real analysis and metric spaces, this book can also be used to instruct a graduate course. The text uses a conversational style with topics purposefully apportioned into 21 lectures, providing a suitable format for either independent study or lecture-based teaching. Instructors are invited to rearrange the order of topics according to their own vision. A clear and rigorous exposition is supported by engaging examples and exercises unique to each lecture; a large number of exercises contain useful calculation problems. Hints are given for a selection of the more difficult exercises. This text furnishes the reader with a means of learning complex analysis as well as a subtle introduction to careful mathematical reasoning. To guarantee a student’s progression, more advanced topics are spread out over several lectures.   This text is based on a one-semester (12 week) undergraduate course in complex analysis that the author has taught at the Australian National University for over twenty years. Most of the principal facts are deduced from Cauchy’s Independence of Homotopy Theorem allowing us to obtain a clean derivation of Cauchy’s Integral Theorem and Cauchy’s Integral Formula.  Setting the tone for the entire book, the material begins with a proof of the Fundamental Theorem of Algebra to demonstrate the power of complex numbers and concludes with a proof of another major milestone, the Riemann Mapping Theorem, which is rarely part of a one-semester undergraduate course.

Twenty-One Lectures on Complex Analysis

• Preface


• Contents


• Lecture 1 Complex Numbers. The Fundamental Theorem of Algebra


• Exercises


• Lecture 2 R- and C-Differentiability


• Exercises


• Lecture 3 The Stereographic Projection. Conformal Maps. The Open Mapping Theorem


• Exercises


• Lecture 4 Conformal Maps (Continued). Möbius Transformations


• Exercises


• Lecture 5 Möbius Transformations (Continued). Generalised Circles. Symmetry


• Exercises


• Lecture 6 Domains Bounded by Pairs of Generalised Circles. Integration


• Exercises


• Lecture 7 Primitives Along Paths. Holomorphic Primitives The Existence of a Holomorphic Primitive of


• Exercises


• Lecture 8 Proof of Lemma 7.2. Constructible Primitives of Holomorphic Functions along Paths. Integra


• Exercises


• Lecture 9 Cauchy's Independence of Homotopy Theorem. Integration over Piecewise C1-paths. Jordan Dom


• Exercises


• Lecture 10 Cauchy's Integral Theorem. Proof of Theorem 3.1. Cauchy's Integral Formula


• Exercises


• Lecture 11 Morera?s Theorem. Sequences and Series of Functions. Uniform Convergence Inside a Domai


• Exercises


• Lecture 12 Proof of Theorem 11.9. Power Series (Continued). Taylor Series. Local Power Series Expans


• Exercises


• Lecture 13 Liouville?s Theorem. Laurent Series. Annulus of Convergence. Laurent Series Expansion


• Exercises


• Lecture 14 Isolated Singularities of Holomorphic Functions (Continued). Characterisation of an Isola


• Exercises


• Lecture 15 Isolated Singularities of Holomorphic Functionsat ? (Continued). Orders of Poles at ?


• Exercises


• Lecture 16 Computing Residues (Continued). Computing Integrals over the Real Line Using Contour Inte


• Exercises


• Lecture 17 Index of a Path. The Argument Principle (Continued). Rouch´e?s Theorem. Theorem 1.1 Re


• Exercises


• Lecture 18