Complex Analysis and Applications

Preface
Contents
About the Author
Acronyms
Glossary of Symbols
1 Complex Numbers and Metric Topology of mathbbC
	1.1 Introduction
	1.2 Complex Numbers
		1.2.1 Equality of Complex Numbers
		1.2.2 Fundamental Laws of Addition and Multiplication
		1.2.3 Difference and Division of Two Complex Numbers
	1.3 Modulus and Argument of Complex Numbers
	1.4 Geometrical Representations of Complex Numbers
	1.5 Modulus and Argument of Complex Numbers
		1.5.1 Polar Forms of Complex Numbers
		1.5.2 Conjugates
		1.5.3 Vector Representation of Complex Numbers
		1.5.4 Multiplication of a Complex Number by i
	1.6 Properties of Moduli
	1.7 Properties of Arguments
	1.8 Equations of Straight Lines
	1.9 Equations of Circles
		1.9.1 General Equation of a Circle
		1.9.2 Equations of Circles Through Three Points
	1.10 Inverse Points
		1.10.1 Inverse Points with Respect to Lines
		1.10.2 Inverse Points with Respect to Circles
	1.11 Relations Between Inverse Points with Respect To Circles
	1.12 Riemann Spheres and Point at Infinity
		1.12.1 Point at Infinity
		1.12.2 Riemann Spheres
	1.13 Cauchy–Schwarz's Inequality and Lagrange's Identity
	1.14 Metric Spaces and Topology of mathbbC
		1.14.1 Metric Spaces
		1.14.2 Dense Set
		1.14.3 Connectedness
		1.14.4 Convergence and Completeness
		1.14.5 Component
		1.14.6 Compactness
		1.14.7 Continuity
		1.14.8 Topological Spaces
		1.14.9 Metrizable Spaces
		1.14.10 Homeomorphism
2 Analytic Functions, Power Series,  and Uniform Convergence
	2.1 Introduction
	2.2 Functions of Complex Variables
		2.2.1 Limits of Functions
		2.2.2 Continuity
	2.3 Uniform Continuity
	2.4 Differentiability
	2.5 Analytic and Regular Functions
	2.6 Cauchy–Riemann Equations
		2.6.1 Conjugate Functions
		2.6.2 Harmonic Functions
		2.6.3 Polar Form of the Cauchy–Riemann Equations
	2.7 Methods of Constructing Analytic Functions
		2.7.1 Simple Methods of Constructing Analytic Functions (Without Using Integrals)
	2.8 Power Series
		2.8.1 Absolute Convergence of a Power Series
		2.8.2 Some Special Test for Convergence of Series
	2.9 Certain Theorems on Power Series
		2.9.1 Abel's Theorem
		2.9.2 Cauchy–Hadamard's Theorem
		2.9.3 Circle and Radius of Convergence of a Power Series
		2.9.4 Analyticity of the Sum Function of a Power Series
		2.9.5 Abel's Limit Theorem
	2.10 Elementary Functions of a Complex Variable
	2.11 Many-Valued Functions: Branches
	2.12 The Logarithm and Power Functions
	2.13 The Riemann Surface for Log z
	2.14 Uniform Convergence of a Sequence
		2.14.1 General Principle of Uniform Convergence  of a Sequence
	2.15 Uniform Convergence of a Series
		2.15.1 Principle of Uniform Convergence of a Series
		2.15.2 Sufficient Tests for Uniform Convergence of a Series
		2.15.3 Weierstrass M-Test
	2.16 Hardy's Tests for Uniform Convergence
	2.17 Continuity of the Sum Function of a Series
3 Complex Integrations
	3.1 Introduction
	3.2 Complex Integrations
		3.2.1 Some Definitions
		3.2.2 Rectifiable Curves
	3.3 Complex Integrals
		3.3.1 Evaluation of Some Integrals by the Direct Definition
		3.3.2 Some Elementary Properties of Complex Integrals
		3.3.3 Integrations Along Regular Arcs
		3.3.4 Complex Integrals as Sum of Two Real Line Integrals
		3.3.5 The Absolute Value of Complex Integrals
		3.3.6 Line Integrals as Functions of Arcs
	3.4 Cauchy's Theorem
		3.4.1 The Elementary Form of Cauchy's Theorem
		3.4.2 The Index of Closed Curves with Respect to a Point
		3.4.3 The General Form of Cauchy's Theorem
		3.4.4 The Second Proof of Cauchy–Goursat's Theorem
	3.5 Indefinite Integrals of Primitives
	3.6 Cauchy's Integral Formula
	3.7 Derivatives of Analytic Functions
	3.8 Higher Order Derivatives
	3.9 Morera's Theorem
	3.10 Poisson's Integral Formula for a Circle
	3.11 Cauchy's Inequality
	3.12 Liouville's Theorem
	3.13 Cauchy's Theorem and Integral Formulas
	3.14 Cauchy's Theorem and Simple Connectivity
		3.14.1 Homotopic Closed Curves
		3.14.2 The Homotopic Version of Cauchy's Theorem
		3.14.3 Simply Connected Region
	3.15 Term-by-Term Integration
	3.16 Analyticity of the Sum Function of a Series (Term-by-Term Differentiation)
	3.17 Uniform Convergence of Power Series
	3.18 Expansion of Analytic Functions as Power Series
		3.18.1 Taylor's Theorem
		3.18.2 Laurent's Theorem
4 Singularities of Complex Functions  and Principle of Argument
	4.1 Introduction
	4.2 Zeros of Analytic Functions
	4.3 Singular Points
		4.3.1 Definitions
		4.3.2 Poles, Isolated Essential Singularities, and Removable Singularities
		4.3.3 Meromorphic Functions
		4.3.4 Some Theorems on Poles and Other Singularities
		4.3.5 Limiting Point of Zeros
		4.3.6 Limit Point of Poles
		4.3.7 The ``Point at Infinity''
	4.4 Characterization of Polynomials
		4.4.1 Characterization of Rational Functions
	4.5 Argument Principle
	4.6 Rouché's Theorem
		4.6.1 The Fundamental Theorem of Algebra
	4.7 Maximum Modulus Principle
		4.7.1 Schwarz's Lemma
	4.8 The Inverse Functions
5 Calculus of Residues and Applications  to Contour Integration
	5.1 Introduction
	5.2 The Residues at Singularities
	5.3 Calculation of Residues in Some Special Cases
	5.4 Residues at Infinity
	5.5 Some Residue Theorems
	5.6 Evaluation of Definite Integrals by Contour Integration
	5.7 Integration Round the Unit Circle
	5.8 Evaluation of the Integral of the Type int-inftyinfty f(x)dx
	5.9 Jordan's Inequality
	5.10 Jordan's Lemma
	5.11 Evaluation of the Integrals of the Form …
	5.12 Case of Poles on the Real Axis
	5.13 Case of Poles on the Real Axis (Indenting Method)
	5.14 Integrals of Many-Valued Functions
	5.15 A Quadrant or a Sector of a Circle as the Contour
	5.16 Rectangular Contour
6 Bilinear Transformations  and Applications
	6.1 Introduction
	6.2 Mapping or Transformation
	6.3 Jacobian of a Transformation
	6.4 Superficial Magnification
	6.5 Some Elementary Transformations
	6.6 Linear Transformation
	6.7 Bilinear or Möbius Transformation
	6.8 Product or Resultant of Two Bilinear Transformations
	6.9 Every Bilinear Transformation Is the Resultant of Elementary Transformations
	6.10 Bilinear Transformation as the Resultant of an Even Number of Inversions
	6.11 The Linear Group
	6.12 Cross Ratio
	6.13 Preservation of Cross Ratio Under Bilinear Transformations
	6.14 Preservation of the Family of Circles and Straight Lines Under Bilinear Transformations
	6.15 Two Important Families of Circles
	6.16 Fixed Point of a Bilinear Transformation
	6.17 Normal Form of a Bilinear Transformation
	6.18 Elliptic, Hyperbolic and Parabolic Transformations
	6.19 Special Bilinear Transformations
7 Conformal Mappings and Applications
	7.1 Introduction
	7.2 Conformal Mapping
	7.3 Sufficient Condition for w = f(z) to Represent a Conformal Mapping
	7.4 Necessary Condition for w = f(z) to Represent  a Conformal Mapping
	7.5 The Transformation w = za (a, Any Complex Number)
	7.6 The Inverse Transformation z = sqrtw
	7.7 The Exponential Transformation w = ez
	7.8 The Logarithmic Transformation w =logz
	7.9 The Trigonometrical Transformation z = c sinw
	7.10 The Transformation w = tanz
	7.11 The Transformation w = tan2(π4asqrtz)
	7.12 The Transformation w = 12(z+1z)
	7.13 The Transformation z = 12(w+1w)
8 Spaces of Analytic Functions
	8.1 Introduction
	8.2 The Space of Continuous Functions C(G,Ω)
	8.3 Normality
	8.4 Equicontinuity
	8.5 Spaces of Analytic Functions
	8.6 Analytic Functions and Their Inverses
	8.7 The Riemann Mapping Theorem
9 Entire and Meromorphic Functions
	9.1 Introduction
	9.2 Weierstrass Factorization Theorem
	9.3 Gamma Function
	9.4 The Riemann Zeta Function
		9.4.1 Extension of Zeta Function
		9.4.2 Riemann's Functional Equation
	9.5 Application of Riemann Hypothesis in Number Theory
		9.5.1 The Prime Number Theorem
	9.6 Runge's Theorem
	9.7 Mittag-Leffler's Theorem
10 Analytic Continuation
	10.1 Introduction
	10.2 Analytic Continuation
	10.3 Uniqueness of Analytic Continuation
	10.4 Power Series Method of Analytic Continuation
	10.5 Schwarz's Reflection Principle
	10.6 Analytic Continuation Along a Path
	10.7 Monodromy Theorem and Its Consequences
11 Harmonic Functions and Integral Functions
	11.1 Introduction
	11.2 Harmonic Functions
	11.3 Basic Properties of Harmonic Functions
	11.4 Harmonic Functions on a Disk
	11.5 Space of Harmonic Functions
	11.6 Subharmonic and Superharmonic Functions
	11.7 The Dirichlet Problem
	11.8 Green's Function
	11.9 Formulas of Poisson, Hilbert, and Bromwich
	11.10 Functions Defined by Integrals
12 Canonical Products and Convergence  of Entire Functions
	12.1 Introduction
	12.2 Canonical Product
	12.3 The Jensen and Poisson–Jensen Formulas
	12.4 Growth, Order, and Exponents of Convergence  of Entire Functions
		12.4.1 Growth of Entire Functions
		12.4.2 The Maximum Modulus of an Entire Function
	12.5 Hadamard's Three-Circle Theorem
	12.6 Convex Functions
	12.7 The Genus and Order of an Entire Function
	12.8 Exponents of Convergence
13 The Range of an Analytic Function
	13.1 Introduction
	13.2 Bloch's Theorem
	13.3 The Little Picard Theorem
	13.4 Schottky's Theorem
	13.5 Montel-Carathéodory Theorem and the Great Picard Theorem
14 Univalent Functions and Applications
	14.1 Introduction
	14.2 Univalent Function
		14.2.1 Open Mapping Theorem
		14.2.2 Inverse Function Theorem
		14.2.3 Global Mapping Theorem
		14.2.4 Reflection Principle
	14.3 The Class of mathscrS
	14.4 Bieberbach Conjecture
		14.4.1 Subordination
		14.4.2 Starlike Functions
		14.4.3 Convex and Close-to-Convex Functions
		14.4.4 Non-univalent Analytic Functions with Real Coefficients
	14.5 14-Theorem
		14.5.1 An Application of ``14-Theorem''
15 Function Theory of Several Complex Variables
	15.1 Introduction
	15.2 Analytic Functions of Several Complex Variables
		15.2.1 Elementary Properties of Analytic Functions
	15.3 Power Series in Several Variables
	15.4 Complex Analysis in Several Variables
		15.4.1 Cauchy Integral Formula
		15.4.2 Higher Order Partial Derivatives
		15.4.3 Montel Theorem
	15.5 Cartan Theorem
	15.6 Groups of Analytic Automorphism of the Unit Ball and the Bidisk
	15.7 Poincaré Theorem
	15.8 Hartogs Theorem
A Solution of Selected Problems in Exercises
References
Index