An Intuitive Introduction to Complex Analysis: Volume I, draft version 1

THIS IS A DRAFT MANUSCRIPT VERSION. This version of the book is being made available to benefit students who wish to have a copy of the book before the final print version is available. This version preserves the beautifully hand-drawn illustrations of the original. THIS IS A DRAFT MANUSCRIPT VERSION.

After having taught the traditional senior-level undergraduate complex variables course many times, and after writing some dozen research papers incorporating the elements of this subject, the first author became aware of a need for a “down to earth” presentation of the important applicable features. The development here is intuitive and inductive as opposed to the usual rigorous and deductive presentations. Mathematical maturity of the reader is not required, as no use is made of epsilon-delta arguments. The inductive exposition offered here requires that the reader first study in detail specific concrete examples; she is then called upon to “conjecture” general truths based on her experience with special cases. In this way the essential facts needed for a good working knowledge of complex analysis are made to stand out clearly, and the intricacies of the subject are mastered from first-hand experience. The only background required of the reader is the usual three semester intuitive level calculus course given at most colleges and universities in the freshman and sophomore years. Even then, it is assumed that the reader has only a very vague appreciation for the more subtle aspects of the calculus such as infinite series and improper integrals.

While a rigorous formulation of the subject is absent from these pages, there is no attempt to “water down” the information needed in practical applications. Indeed, use is made of material and intuitive insights which the authors have gleaned from their own research in complex variables which is not usually found in text books. As an example, unusual stress is placed upon actually visualizing specific functions through graphical representations. The exact definition of an analytic function is not presented until the fourth chapter, even though the concept is used in the second and third chapters. The student is made to see that she can deal with a concept even though it is not precisely formulated, and that definitions often evolve slowly as experience is gained with special cases. Thus the reader gradually develops a “feel” for this subject.

It is hoped that this intuitive presentation will be of value to a wide audience of readers. It can be used as a text book for the usual one semester undergraduate complex variable course given in the junior or senior year. Since this intuitive presentation proceeds at a considerably faster pace than most rigorous texts, advanced topics not usually given in a one semester course can be included. Mathematical maturity is not required of the student, and even advanced sophomores should be able to profit from this course. If the professor also wishes to introduce a rigorous development of complex analysis, this text can serve as a tool for “anchoring the students’ feet to the ground” so that they will better appreciate the need for a deductive development. Engineers and physicists usually welcome intuitive developments of advanced mathematics, and this presentation might be of value in a one semester course for them. This book is also intended for self-study. There are many example problems, and every problem posed for the reader is solved in detail in an appendix. In addition, each chapter is followed by review problems which are also solved in full in the appendix. Students who have taken the traditional course in complex analysis might find that reading this book helps to add concreteness to the general theoretical development they have witnessed.