Geometric Function Theory: Explorations in Complex Analysis
Geometric Function Theory: Explorations in Complex Analysis
Pages: 314
Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geometry, the Bergman metric, the automorphism groups of domains, harmonic measure, boundary regularity of conformal maps, the Poisson kernel, the Hilbert transform, the boundary behavior of harmonic and holomorphic functions, the inhomogeneous Cauchy–Riemann equations, and the corona problem.
The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.
This methodically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis—and also to spark the interest of seasoned workers in the field—the book imparts a solid education both in complex analysis and in how modern mathematics works.
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.............................................. Harcourt Mathematics 12 - Geometry and Discrete Mathematics
Harcourt Mathematics 12 - Geometry and Discrete Mathematics
This new, student friendly text has been developed through a partnership between Ontario Classroom teachers and the University of Waterloo's Centre for Education in Mathematics and Computing to help prepare your students for post secondary studies at the College or University level. Specifically developed for Ontario Students and Teachers Reflects philosophy, spirit, and expectations of new Ontario curriculum Rich Learning Links, and Extending and Investigating opportunities. Review of Prerequisite Skills help introduce new concepts in context Key concepts reviewed to support a wide variety of learners Technology integrated in a flexible, optional manner.
Table of Contents
CHAPTER 1
Introduction to Proof
Rich Learning Link
1.1 What Is Proof?
1.2 An Introduction to Deductive Proof
1.3 Proof in Geometry
1.4 Proof With Analytic Geometry
Rich Learning Link Wrap-Up
Chapter 1 Test
CHAPTER 2
Plane Figures and Proof
Review of Prerequisite Skills
Rich Learning Link
2.1 Proofs Using Congruent Triangles
2.2 Conditional Statements
2.3 The Converse of a Conditional Statement
2.4 Indirect Proof and Parallel Lines
2.5 Ratio and Proportion
2.6 Similar Figures
Key Concepts Review
Review Exercise
Rich Learning Link Wrap-Up
Chapter 2 Test
Extending and Investigating
CHAPTER 3
Properties of Circles
Review of Prerequisite Skills
Rich Learning Link
3.1 Properties of Chords
3.2 Angles in a Circle
3.3 Cyclic Quadrilaterals
3.4 Tangents to a Circle
3.5 More About Tangents
Key Concepts Review
Rich Learning Link Wrap-Up
Review Exercise
Chapter 3 Test
Extending and Investigating
Cumulative Review Chapters 1-3
CHAPTER 4
Vectors
Review of Prerequisite Skills
Rich Learning Link
4.1 Vector Concepts
4.2 Vector Laws
4.3 Force as a Vector
4.4 Velocity as a Vector
Key Concepts Review
Rich Learning Link Wrap-Up
Review Exercise
Chapter 4 Test
Extending and Investigating
CHAPTER 5
Algebraic Vectors and Applications
Rich Learning Link
5.1 Algebraic Vectors
5.2 Operations With Algebraic Vectors
5.3 The Dot Product of Two Vectors
5.4 The Cross Product of Two Vectors
5.5 Applications of Dot and Cross Products
Review Exercise
Rich Learning Link Wrap-Up
Chapter 5 Test
Extending and Investigating
CHAPTER 6
Linear Combinations
Review of Prerequisite Skills
Rich Learning Link
6.1 Linear Combinations of Vectors
6.2 Linear Dependence and Independence
6.3 Division of a Line Segment
6.4 Vector Proofs in Geometry
Key Concepts Review
Review Exercise
Rich Learning Link Wrap-Up
Chapter 6 Test
Extending and Investigating
CHAPTER 7
Lines in a Plane
Review of Prerequisite Skills
Rich Learning Link
7.1 Parametric and Vector Equations of a Line in a Plane
7.2 The Scalar Equation of a Line in a Plane
7.3 Equations of a Line in 3-Space
7.4 The Intersection of Two Lines
Key Concepts Review
Review Exercise
Rich Learning Link Wrap-Up
Chapter 7 Test
Extending and Investigating
CHAPTER 8
Equations of Planes
Review of Prerequisite Skills
Rich Learning Link
8.1 The Vector Equation of a Plane in Space
8.2 The Scalar Equation of a Plane in Space
8.3 The Intersection of a Line and a Plane
8.4 The Intersection of Two Planes
8.5 The Intersection of Three Planes
Key Concepts Review
Rich Learning Link Wrap-Up
Review Exercise
Chapter 8 Test
Extending and Investigating
Cumulative Review Chapters 4-8
CHAPTER 9
Proof Using Different Approaches
Review of Prerequisite Skills
Rich Learning Link
9.1 Using Analytic Methods
9.2 Proof Using Analytic Geometry
9.3 Different Techniques of Proof
9.4 Locus
Key Concepts Review
Rich Learning Link Wrap-Up
Review Exercise
Chapter 9 Test
Extending and Investigating
CHAPTER 10
Introduction to Counting
Review of Prerequisite Skills
Rich Learning Link
10.1 Sets
10.2 Combining Subsets
10.3 The Sum Rule
10.4 The Product Rule
10.5 The Use of Cases
Key Concepts Review
Rich Learning Link Wrap-Up
Review Exercise
Chapter 10 Test
Extending and Investigating
CHAPTER 11
Counting Methods
Rich Learning Link
11.1 Counting of Sequences With Distinct Elements
11.2 Sequences With Unlimited Repeated Values
11.3 Counting Subsets
11.4 Counting Sequences With Repeated Elements
11.5 A Strategy for Counting Problems
Key Concepts Review
Rich Learning Link Wrap-Up
Review Exercise
Chapter 11 Test
Extending and Investigating
CHAPTER 12
Sequences
Review of Prerequisite Skills
Rich Learning Link
12.1 Sequences
12.2 Partial Sums and Sigma Notation
12.3 Mathematical Induction
12.4 The Binomial Theorem
Key Concepts Review
Rich Learning Link Wrap-Up
Review Exercise
Chapter 12 Test
Extending and Investigating
Cumulative Review Chapters 10-12
Technology Appendix
Glossary
Answers
Index
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