Twenty-One Lectures on Complex Analysis: A First Course

✍ Scribed by Alexander Isaev (auth.)

• Publisher: Springer International Publishing
• Year: 2017
• Tongue: English
• Leaves: 193
• Series: Springer Undergraduate Mathematics Series
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter ....Pages i-xii
Complex Numbers. The Fundamental Theorem of Algebra (Alexander Isaev)....Pages 1-8
ℝ- and ℂ-Differentiability (Alexander Isaev)....Pages 9-16
The Stereographic Projection. Conformal Maps. The Open Mapping Theorem (Alexander Isaev)....Pages 17-23
Conformal Maps (Continued). Möbius Transformations (Alexander Isaev)....Pages 25-32
Möbius Transformations (Continued). Generalised Circles. Symmetry (Alexander Isaev)....Pages 33-39
Domains Bounded by Pairs of Generalised Circles. Integration (Alexander Isaev)....Pages 41-48
Primitives Along Paths. Holomorphic Primitives. The Existence of a Holomorphic Primitive of a Function Holomorphic on a Disk. Goursat’s Lemma (Alexander Isaev)....Pages 49-55
Proof of Lemma 7.2. Constructible Primitives of Holomorphic Functions along Paths. Integration of Holomorphic Functions over Arbitrary Paths. Homotopy. Simply-Connected Domains. The Riemann Mapping Theorem (Alexander Isaev)....Pages 57-66
Cauchy’s Independence of Homotopy Theorem. Integration over Piecewise C1-paths. Jordan Domains and Integration over their Boundaries (Alexander Isaev)....Pages 67-76
Cauchy’s Integral Theorem. Proof of Theorem 3.1. Cauchy’s Integral Formula (Alexander Isaev)....Pages 77-85
Morera’s Theorem. Sequences and Series of Functions. Uniform Convergence Inside a Domain. Power Series. Abel’s Theorem. Disk of Convergence. Radius of Convergence (Alexander Isaev)....Pages 87-95
Proof of Theorem 11.9. Power Series (Continued). Taylor Series. Local Power Series Expansion of a Holomorphic Function. Cauchy’s Inequalities. The Uniqueness Theorem (Alexander Isaev)....Pages 97-105
Liouville’s Theorem. Laurent Series. Annulus of Convergence. Laurent Series Expansion of a Function Holomorphic on an Annulus. Cauchy’s Inequalities. Isolated Singularities of Holomorphic Functions (Alexander Isaev)....Pages 107-116
Isolated Singularities of Holomorphic Functions (Continued). Characterisation of an Isolated Singularity via the Laurent Series Expansion. Orders of Poles and Zeroes. Casorati-Weierstrass’ Theorem. Isolated Singularities of Holomorphic Functions at ∞ and their Characterisation via Laurent Series Expansions (Alexander Isaev)....Pages 117-125
Isolated Singularities of Holomorphic Functions at ∞ (Continued). Orders of Poles at ∞. Casorati-Weierstrass’ Theorem for an Isolated Singularity at ∞. Residues. Cauchy’s Residue Theorem. Computing Residues (Alexander Isaev)....Pages 127-135
Computing Residues (Continued). Computing Integrals over the Real Line Using Contour Integration. The Argument Principle (Alexander Isaev)....Pages 137-145
Index of a Path. The Argument Principle (Continued). Rouché’s Theorem. Theorem 1.1 Revisited. Proof of Theorem 3.2. The Maximum Modulus Principle. Proof of Theorem 3.3 (Alexander Isaev)....Pages 147-155
Schwarz’s Lemma. Conformal Maps of the Unit Disk and the Upper Half-Plane. (Pre)-Compact Subsets of a Metric Space. Continuous Linear Functionals on H(D). Arzelà-Ascoli’s Theorem. Montel’s Theorem. Hurwitz’s Theorem (Alexander Isaev)....Pages 157-165
Analytic Continuation (Alexander Isaev)....Pages 167-173
Analytic Continuation (Continued). The Monodromy Theorem (Alexander Isaev)....Pages 175-182
Proof of Theorem 8.3. Conformal Transformations of the Canonical Simply-Connected Domains (Alexander Isaev)....Pages 183-190
Back Matter ....Pages 191-194

✦ Subjects

Analysis