Classical Complex Analysis: a geometric approach (2 vols)

I-Hsiung Lin, Classical Complex Analysis: A Geometric Approach Vol.1 (2011)_9789813101104
	Preface
	Contents
	CHAPTER 1 Complex Numbers
		1.1. How to Visualize Geometrically the Existence of the So-Called Complex Numbers in Our Daily Life
		1.2. Complex Number and Its Geometric Representations
		1.3. Complex Number System (Field) C
		1.4. Algebraic Operations and Their Geometric Interpretations (Applications)
			1.4.1. Conjugate complex numbers
			1.4.2. Inequalities
			1.4.3. Applications in (planar) Euclidean geometry
			1.4.4. Steiner circles and symmetric points with respect to a circle (or line)
		1.5. de Moivre Formula and nth Roots of Complex Numbers
		1.6. Spherical Representations of Complex Numbers: Riemann Sphere and Extended Complex Plane
		1.7. Complex Sequences
		1.8. Elementary Point Sets
		1.9. Completeness of the Complex Field C
	CHAPTER 2 Complex-Valued Functions of a Complex Variable
		2.1. Limits of Functions
		2.2. Continuous Functions
		2.3. Uniform Convergence of a Sequence or Series of Functions
		2.4. Curves
		2.5. Elementary Rational Functions
			2.5.1. Polynomials
			2.5.2. The power function w = zn (n>=2)
			2.5.3. Rational functions
			2.5.4. Linear fractional (or bilinear or Mobius) transformations
			2.5.5. Joukowski function w=1/2 (z+1/z)
		2.6. Elementary Transcendental Functions
			2.6.1. The exponential function ez
			2.6.2. Trigonometric functions cos z, sin z, and tan z
		2.7. Elementary Multiple-Valued Functions
			2.7.1. The origin of multiple-valuedness: argz
			2.7.2. w = n√z (n>=2) and its Riemann surface (etc.)
			2.7.3. w = logz (the natural logarithm function with base e) and its Riemann surface
			2.7.4. w = cos^{-1} z and w = tan^{-1} z and their Riemann surfaces
		2.8. Differentiation in Complex Notation
		2.9. Integration in complex notation
	CHAPTER 3 Fundamental Theory: Differentiation, Integration, and Analytic Functions
		3.1. (Complex) Differentiation
		3.2. Differentiability: Cauchy-Riemann Equations, their Equivalents and Meanings
			3.2.1. (Linearly) Algebraic viewpoint
			3.2.2. Analytic viewpoint
			3.2.3. Geometric viewpoint
			3.2.4. Physical viewpoint
		3.3. Analytic Functions
			3.3.1. Basic examples
			3.3.2. The analyticity of functions defined by power series
			*3.3.3. Analyticity of multiple-valued functions and the Riemann surfaces (revisited)
		3.4. Analytic Properties of Analytic Functions
			3.4.1. Elementary properties derived, from definition
			3.4.2. Cauchy integral theorem and formula (simple forms) : The continuity (analyticity) of the derivative of an analytic function and its Taylor series representation
			3.4.3. The real and imaginary parts of an analytic function: Harmonic functions
			3.4.4. The maximum-minimum principle and the open mapping property
			3.4.5. Schwarz’s lemma
			3.4.6. The symmetry (or reflection) principle
			*3.4.7. The inverse and implicit function theorems
		3.5. Geometric Properties of Analytic Functions
			3.5.1. Local behavior of an analytic function at a point: Conformality, etc.
			3.5.2. The winding number: Its integral representation and geometric meaning
			3.5.3. The argument principle
			3.5.4. The Rouche’s theorem
			3.5.5. Some sufficient conditions for analytic functions to be univalent
			*3.5.6. The inverse and implicit function theorems (revisited)
			3.5.7. Examples of (univalently) conformal mappings
	CHAPTER 4 Fundamental Theory: Integration (Advanced)
		4.1. Complex Integration Independent of Paths: Primitive Functions
		4.2. The General Form of Cauchy Integral Theorem: Homotopy
			4.2.1. The line integral of an analytic function along a continuous curve
			4.2.2. Homotopy of curves
			4.2.3. Homotopic invariance of the winding numbers
			4.2.4. Homotopic form of Cauchy integral theorem
		4.3. The General Form of Cauchy Integral Theorem: Homology
			4.3.1. Cycles and homology of two cycles
			4.3.2. Simply and finitely connected domains: Homology basis
			4.3.3. Homologous form of Cauchy integral theorem
			4.3.4. Artin’s proof
		4.4. Characteristic Properties of Simply Connected Domains (a Review): The Single-Valuedness of a Primitive Function
		4.5. The Branches of a Multiple-Valued Primitive Function on a Multiple-Connected Domain
		4.6. The General Form of Cauchy Integral Formula
		4.7. Integrals of Cauchy Type and Cauchy Principal Value
		4.8. Taylor Series (Complicated Examples)
		4.9. Laurent Series
			4.9.1. The Laurent series expansion of an analytic function in a (circular) ring domain
			4.9.2. Examples
		4.10. Classification and Characteristic Properties of Isolated Singularities of an Analytic Function
			4.10.1. Removable singularity
			4.10.2. Pole
			4.10.3. Essential singularity
		4.11. Residues and Residue Theorem
			4.11.1. Definition of residues
			4.11.2. The computation of residues and examples
			4.11.3. The residue theorem
		4.12. The Applications of the Residue Theorem in Evaluating the Integrals
			4.12.1. /\\int^{2\\pi}_0 (cosθ, sinθ)dθ, where f(x,y) is a function in x and y, etc.
			4.12.2. /\\int^1_0 x^{α-1} (1-x)^{-α}dx (0 < α < 1), etc.
			4.12.3. Improper integrals over (-∞, ∞)
				4.12.3.1. \\int^{\\infty}_{-\\infty} f(x)dx
				4.12.3.2. \\int^{\\infty}_{-\\infty} f(x)e^{imx}dx ( m \\in R)
				4.12.3.3. \\int^{\\infty}_{-\\infty} f(x)dx (with periodic f(x))
			4.12.4. Improper integrals over (0, ∞)
				4.12.4.1. \\int^{\\infty}_0 f(x)dx
				4.12.4.2.  \\int^{\\infty}_0 f(x)(log x)^p dx (p, positive integers)
				4.12.4.3.  \\int^{\\infty}_0 x^α f(x) dx  (a \\in C)
		4.13. The Integral \\int^{x_0+i \\infty}_{x_0+i \\infty} f(z)dz along a Line Re z = X_0
			4.13.1. Fourier transforms
			4.13.2. Laplace transforms
		4.14. Asymptotic Function and Expansion of Functions Defined by Integrals with a Parameter
		4.15. The Summation of Series by Residues
			4.15.1. \\sum^{\\infty}_{\\infty} f(n)
			4.15.2. \\sum^{\\infty}_{\\infty} (-1)^n f(n)
	Appendix
		Appendix A. The Real Number System R
	References
	21
	52
	80
	Index of Notations
	Index
		A
		B
		C
		DE
		F
		G
		HIJ
		KL
		MNOP
		R
		S
		TUV
		WY
I-Hsiung Lin, Classical Complex Analysis: A Geometric Approach Vol.2 (2010)I.Lin_9789813101074
	Preface
	Contents
	CHAPTER 5 Fundamental Theory: Sequences, Series, and Infinite Products
		5.1 Power Series
			5.1.1 Algorithm of power series
			5.1.2 Basic properties of an analytic function defined by a power series in an open disk
			5.1.3 Boundary behavior of a power series on its circle of convergence
		5.2 Analytic Continuation
			5.2.1 Analytic continuation along a curve
			5.2.2 Homotopy and monodromy theorem
		5.3 Local Uniform Conver gence of a Sequence or a Series of Analytic Functions
			5.3.1 Analyticity of the limit function: Weierstrass\'s theorem
			5.3.2 Zeros of the limit function: Hurwitz\'s theorem
			*5.3.3 Some sufficient criteria for local uniform convergence
			*5.3.4 An application: The fixed points of an analytic function and its iterate functions
		5.4 Meromorphic Functions: Mittag-Leffier\'s Partial Fractions Theorem
			5.4.1 Mittlag-Leffler\'s partial fractions expansion for meromorphic functions
			5.4.2 Cauchy\'s residue method
		5.5 Entire Functions: Weierstrass\'s Factorization Theorem and Hadamard\'s Order Theorem
			5.5.1 In finite products (of complex numbers and f unctions)
			5.5.2 Weierstrass\'s factorization theorem
			5.5.3 Hadamard\'s order theorem
		5.6 The Gamma Function Γ(z)
			5.6.1 Definition and representations
			5.6.2 Basic and char acteristic properties
			5.6.3 The asymptotic function of r (z); Stirlin g\'s f ormula
		5.7 The Riemann zeta Function ζ(z)
		5.8 Normal Families of Analytic (Meromorphic) Functions
			5.8.1 Criteria for normality
			5.8.2 Examples
			5.8.3 The elliptic modu lar function: Mantel\'s normality criterion and Picard\'s theorems
			*5.8.4 Remarks on Schottky\'s theorems and Schwarz - Ahlfors\' Lemma: Other proofs of Mantel\'s criterion and Picard\'s theorems
			*5.8.5 An application: Some results in complex dynamical system
	CHAPTER 6. Conformal Mapping and Dirichlet\'s Problems
		6.1 The Riemann Mapping Theorem
			6.1.1 Proof
			6.1.2 The boundary correspondence
		6.2 Conformal Mapping of Polygons: The Schwarz-Christoffel Formulas
			6.2.1 The Schwarz-Christoffel formulas for polygons
			6.2.2 Examples
			6.2.3 The Schwarz-Christoffel formula for the generalized polygon
		6.3 Harmonic Function and the Dirichlet Problem for a Disk
			6.3.1 Some further properties of harmonic functions: Green\'s function
			6.3.2 Poisson\'s formula and integral: The Dirichlet problem for a disk Harnack\'s principle
		6.4 Subharmonic Functions
		6.5 Perron\'s Method: Dirichlet\'s Problem for a Class of General Domains
		6.6 Canonical Mappings and Canonical Doma ins of Finitely Connected Domains
			6.6.1 Harmonic measures
			6.6.2 Canonical domains: The annuli with concentric circular slits
			6.6.3 Canonical domains: The parallel slit domains
			*6.6.4 Other canonical domains
	CHAPTER 7. Riemann Surfaces (Abstract)
		7.1 Riemann Surface: Definition and Examples
		7.2 Analytic Mappings and Meromorphic Functions on Riemann Surfaces
		7.3 Harmonic Functions and the Maximum Principle on Riemann Surfaces
			7.3.1 Harmonic measure
			7.3.2 Green\'s function
			7.3.3 A classification of Riemann surfaces
		7.4 The Fundamental Group
		7.5 Covering Spaces (or Surfaces) and Covering Transformations
			7.5.1 Definitions and examples
			7.5.2 Basic properties of covering spaces (or surfaces) :
The lifting of mappings and the monodromy theorem
			7.5.3 Characteristics and classifications of covering surfaces: The existence theorem
			7.5.4 Covering transformations
			7.5.5 The universal covering surface of a surface
		7.6 The Uniformization Theorem of Riemann Surfaces
			7.6.1 Simply connected Riemann surfaces
			7.6.2 Arbitrary Riemann surfaces
	Appendix
	Appendix B : Parabolic, Elliptic, and Hy perbolic Geome tries
	Appendix C. Quasiconformal Mappings
	References
	[21]
	[51]
	[78]
	Index of Notations
	Index
		AB
		C
		DE
		FG
		HI
		JKLMN
		OP
		QR
		ST
		UVW