A Pathway to Complex Analysis

Description

The book is an honest attempt to establish a long cherished belief that Complex Analysis is a fine meeting ground analysis, geometry and topology. In the last five decades the curriculum has improved a lot and students in India have an exposure to real analysis, linear algebra, metric spaces and topology. Keeping the Improved background of the students in mind, we develop the subject.

The salient features of the book are:

• A careful treatment of argument and logarithms.
• Use of triangles and piecewise smooth paths in place of rectifiable paths.
• Geometric treatments of C-R equations, winding numbers and fractional linear transformations.
• A geometric treatment of analytic continuation explaining the subtleties to the beginner.
• The treatment of conformal property of holomorphic functions using the motion of oriented angles.
• Many insightful remarks which give a wider perspective to the readers and show them the ‘ big picture’.

We emphasize the central point of the Cauchy theory, viz, the existence of local primitives and that of the local power series expansion. We point out how the properties of the holomorphic functions are the local/global manifestations of the corresponding results for the power series.

Many students approach the subject ” Complex Analysis ” with a sense of wonder, mystery and awe. They are mystified by the infinite differentiability of “analytic” functions and so-called multivalued functions. This book will demystify them.

Table of Contents 

Preface

Suggestions for the Students

Suggestions for a Semester Course

1. Complex Numbers
2. Sequences and Series
3. Continuity
4. Exponential Function
5. Differentiation
6. Complex integration
7. Cauchy theory
8. Cauchy integral formula
9. Isolated Singularities and Laurent Series
10. Winding Numbers of Closed Curves
11. Residue Theorem and its Applications
12. Mapping Properties
13. Extended Complex Plane
14. Conformal Maps
15. Real Integrals
16. Global Versions
17. Harmonic Functions
18. Analytic Continuation
19. Riemann Mapping Theorem

A   An Outline of Metric Spaces

Index