Functions of One Complex Variable I

1. Front Matter

   Pages i-xiii

   PDF

2. The Complex Number System

   • John B. Conway

   Pages 1-10

3. Metric Spaces and the Topology of ℂ

   • John B. Conway

   Pages 11-29

4. Elementary Properties and Examples of Analytic Functions

   • John B. Conway

   Pages 30-57

5. Complex Integration

   • John B. Conway

   Pages 58-102

6. Singularities

   • John B. Conway

   Pages 103-127

7. The Maximum Modulus Theorem

   • John B. Conway

   Pages 128-141

8. Compactness and Convergence in the Space of Analytic Functions

   • John B. Conway

   Pages 142-194

9. Runge’s Theorem

   • John B. Conway

   Pages 195-209

10. Analytic Continuation and Riemann Surfaces

    • John B. Conway

    Pages 210-251

11. Harmonic Functions

    • John B. Conway

    Pages 252-278

12. Entire Functions

    • John B. Conway

    Pages 279-291

13. The Range of an Analytic Function

    • John B. Conway

    Pages 292-301

14. Back Matter

    Pages 303-321

    PDF

This book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - 8 arguments. The actual pre­ requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differ­ entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as "An Introduction to Mathe­ matics" has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc.

• Argument principle
• Complex analysis
• Meromorphic function
• Monodromy
• Riemann surfaces
• analytic function
• gamma function

"This book presents a basic introduction to complex analysis in both an interesting and a rigorous manner. It contains enough material for a full year's course, and the choice of material treated is reasonably standard and should be satisfactory for most first courses in complex analysis. The approach to each topic appears to be carefully thought out both as to mathematical treatment and pedagogical presentation, and the end result is a very satisfactory book for classroom use or self-study."   --MathSciNet

• Mathematics Department, University of Tennessee, Knoxville, USA

  John B. Conway

Policies and ethics