An Introduction to Complex Analysis and Geometry provides the reader with a deep appreciation of complex analysis and how this subject fits into mathematics. The book developed from courses given in the Campus Honors Program at the University of Illinois Urbana-Champaign. These courses aimed to share with students the way many mathematics and physics problems magically simplify when viewed from the perspective of complex analysis. The book begins at an elementary level but also contains advanced material.
The first four chapters provide an introduction to complex analysis with many elementary and unusual applications. Chapters 5 through 7 develop the Cauchy theory and include some striking applications to calculus. Chapter 8 glimpses several appealing topics, simultaneously unifying the book and opening the door to further study.
The 280 exercises range from simple computations to difficult problems. Their variety makes the book especially attractive.
A reader of the first four chapters will be able to apply complex numbers in many elementary contexts. A reader of the full book will know basic one complex variable theory and will have seen it integrated into mathematics as a whole. Research mathematicians will discover several novel perspectives.
From the Real Numbers to the Complex Numbers
Inequalities and ordered fields
The complex numbers
Alternative definitions of C
A glimpse at metric spaces
Existence of square roots
Convergent infinite series
Uniform convergence and consequences
The unit circle and trigonometry
The geometry of addition and multiplication
Complex Numbers and Geometry
Lines, circles and balls
Linear fractional transformations
The Riemann sphere
Power Series Expansions
The radius of convergence
An application of power series
Definitions of complex analytic function
The Cauchy-Riemann equations
Orthogonal trajectories and harmonic functions
A glimpse at harmonic functions
What is a differential form?
The Cauchy integral formula
A return to the definition of complex analytic function
Applications of Complex Integration
Singularities and residues
Evaluating real integrals using complex variables methods
The Gamma function
The minimum-maximum theorem
The fundamental theorem of algebra
Winding numbers, zeroes, and poles
Higher-dimensional complex analysis