Today, the theory of complex-valued functions finds widespread applications in various areas of mathematical research, as well as in electrical and mechanical engineering, aeronautics, and other disciplines. Complex analysis has become a basic course in mathematics, physics, and select engineering departments.
This concise textbook provides a thorough introduction to the function theory of one complex variable. It presents the fundamental concepts with clarity and rigor, offering concise proofs that avoid lengthy and tedious arguments commonly found in mathematics textbooks. It goes beyond traditional texts by exploring less common topics, including the different approaches to constructing analytic functions, the conformal mapping criterion, integration of analytic functions along arbitrary curves, global analytic functions and their Riemann surfaces, the general inverse function theorem, the Lagrange–Bürmann formula, and Puiseux series.
Drawing from several decades of teaching experience, this book is ideally suited for one or two semester courses in complex analysis. It also serves as a valuable companion for courses in topology, approximation theory, asymptotic analysis, and functional analysis. Abundant examples and exercises make it suitable for self-study as well.
Author(s): Taras Mel'nyk
Edition: 1
Publisher: Springer
Year: 2023
Language: English
Commentary: Publisher PDF | Publishing Due: 05 November 2023
Pages: 262
City: Cham
Tags: Complex Variable Functions; Conformal Mappings; Analytic Functions; Power Series; Meromorphic Functions; Entire Functions Representations; Riemann Surfaces; Analytic Continuation; Lagrange-Bürmann Series; Puiseux Series; Analytic Functions Integration
Preface
What Advantages Does This Book Offer over Competitive Titles and What Is Unique About It?
Description of the Contents
For Which Courses Would the Textbook Be Suitable?
Acknowledgements
Instructions for Readers
Contents
1 Complex Numbers and Complex Plane
1.1 Complex Numbers
1.2 Sequences in the Complex Plane: Extended Complex Plane
1.3 Complex-Valued Functions of a Real Variable
1.4 Curves in the Complex Plane
1.5 Basic Topological Concepts of the Complex Plane
2 Analytic Functions
2.1 Structure of Complex-Valued Functions of a Complex Variable
2.2 Differentiability of Complex-Valued Functions of a Complex Variable
2.3 Conjugate Harmonic Functions
2.4 Hydrodynamic Interpretation of Analytical Functions
2.5 Conformal Mappings: Geometric Meaning of the Modulus and Argument of the Derivative
3 Elementary Analytic Functions
3.1 Linear and Fractional-Linear Functions and Their Simplest Properties
3.2 Group and Circular Properties of Fractional-Linear Functions
3.3 Preservation of Symmetric Points by Fractional-Linear Mappings
3.4 Fractional-Linear Isomorphisms and Automorphisms
3.5 Power Functions with Natural Exponents
3.6 The Inverse to a Power Function and Its Riemann Surface
3.7 Exponential Function, Logarithmic Function and Its Riemann Surface
3.8 Joukowsky Function
3.9 Trigonometric and Hyperbolic Functions and Their Inverses
4 Integration of Functions of a Complex Variable
4.1 Line Integrals and Their Simplest Properties
4.2 An Antiderivative: Cauchy-Goursat Theorem
4.3 Local Existence of an Antiderivative: Antiderivative Along a Curve
4.4 The Cauchy Integral Theorem and Corollaries
4.5 The Cauchy Integral Formula
5 Complex Power Series
5.1 Basic Definitions and Properties of Function Series and Power Series
5.2 Expansion of a Differentiated Function Into a Power Series
5.3 Analyticity of the Sum of a Power Series
5.4 Uniqueness of Power Series Expansions: Morera's Theorem
5.5 Uniqueness Theorem for Analytic Functions: Zeros of Analytic Functions
6 Laurent Series: Isolated Singularities of Analytic Functions
6.1 Expansion of an Analytic Function Into a Laurent Series
6.2 Relationship Between Laurent Series and Fourier Series
6.3 Isolated Singularities of Analytic Functions
Isolated Singularity at Infinity
6.4 Behavior of an Analytic Function Near Its Essential Singularity
6.5 Classification of Analytic Functions with Respect to Their Isolated Singularities: Theorem on a Meromorphic Function
7 Residue Calculus
7.1 Cauchy's Residue Theorem
7.2 Formulas for Calculating Residues
7.3 Methods for Calculating Integrals
Integrals Over Closed Curves
Trigonometric Integrals
Integrals Along the Real Line
Fourier Transform Type Integrals -∞+∞ f(x) eiλx dx (λ> 0)
7.4 Argument Principle: Rouché's Theorem and Its Applications
7.5 Partial Fraction Decomposition of a Meromorphic Function
7.6 Factorization of an Entire Function Into an Infinite Product
8 Analytic Continuations
8.1 Analytic Function Elements
8.2 Methods of Analytic Continuation: Schwarz's Reflection Principle
8.3 Analytic Continuation Along a Curve: The Monodromy Theorem
8.4 Global Analytic Functions
8.5 Riemann Surfaces of Global Analytic Functions
8.6 Singularities of Global Analytic Functions
9 Qualitative Properties of Analytic Functions
9.1 Open Mapping Theorem, Maximum Modulus Principle, Schwarz Lemma
9.2 Inverse Function Theorem: Puiseux Series
9.3 Conformal Isomorphisms and Automorphisms
9.4 Montel's Theorem
9.5 Riemann Mapping Theorem
References
Index
