This book is intended to serve as a textbook for the following courses.
- Chapters 1-6 are intended for a one-term introductory course on the theory and applications of complex functions.
- Chapters 7-12 are intended for a one-term introductory course op partial differential equations and boundary value problems, that incorporates basic tools from complex variables such as contour integrals, residues, and Laurent series.
- Topics from Chapters 1-12 can be chosen to form a one-term introductory course on complex variables with applications to partial differential equations and boundary value problems.
The Book's Content and Organization
This book differs in many ways from other traditional textbooks on complex variables and differs differential equations. In outlining its contents, I will present some of these differences in four components of the book: The examples and exercises, the applications, the graphics, and the proofs.
The Proofs In writing this book, my goal was to reach out to all students with varying abilities to do mathematical proofs. For that reason, I have included complete proofs of most results so as to give the instructor the flexibility to choose the proofs that he or she wants to present to the class, while skipping others and referring the students to the book. While recognizing the importance of training students in proof writing, I have tried not to let this learning process hinder their ability to see and appreciate the applications behind the theory.
In this book, all the proofs are written in a style that is very flexible for classroom presentation. They are carefully arranged and illustrated in such a way that they are accessible to students at the undergraduate level. Some proofs are very basic (for example, those found in the early sections of each chapter); others require a deeper understanding of calculus (for example, use of differentiability in Section 2.3); and yet others propel the students to the graduate level of mathematics. The latter advanced proofs are found in optional sections, such as Sections 2.6, 3.5, 3.9, and 7.6. Typically, in my courses, I would cover some of each level of proofs. Even the most advanced proofs found a place in my courses, as assigned group projects and classroom presentation by students.
The Graphics
Drawing from my teaching experience, I learned that even the most abstract notions can benefit from graphics; and so I did use graphics liberally in this book. I inserted as many pictures in the book as I felt is necessary to clarify an argument, or a statement of a problem, or the result of an example. As a result, this book contains over seven hundred figures.
In the exercises, I found that a statement such as "Solve the Dirichlet problem in Figure X," when accompanied by Figure X, is a much more inviting statement than, say, trying to describe by words the Dirichlet problem. Moreover, it requires from the student to think about the problem, from the applied perspective and to write down the mathematical equations that describe the problem.
The figures are extremely useful in visualizing the applications that are at the heart of the matter: heat flow, isotherms, vibrations of strings and membranes. Graphics are also extremely useful in visualizing more abstract concepts, such as the maximum modulus principle (Figure 6, Section 3.7), the invariance of Laplace's equation (Figure 9, Section 2.5), the analyticity of composed mappings (Figure 2, Section 2.3), the more complicated topics of conformal mappings (Section 6.2), changes of variables using analytic functions, Green's functions (Figure 2, Section 6.5), the convergence of Fourier series, Gibbs phenomenon, and Bessel and Legendre series.
The Applications
I started writing this book with what is now Section 2.5. My goal was to show students the applications of complex analysis as quickly as possible. More importantly, I wanted students to realize the significance of the abstract notions from complex analysis before taking up their detailed study. As a result, the book is written in the style of Section 2.5, where the applications go hand in hand with the theory. Whenever possible, I tried to describe the methods in a sequence of steps that a student can follow systematically. For example, in Section 2.5 (following Theorem 3), I describe step by step how to solve a Dirichlet problem using conformal mappings. Then immediately after that I solve an interesting Dirichlet problem by following these steps. I have used the same approach to other important applications throughout the book.
The Examples and Exercises
I have included far more examples than can be covered in one course. The examples are presented in full detail. As with the proofs, the objective is to give the instructor he option to choose the examples that are best suitable for classroom discussion, while at the same time giving students a variety of completely worked examples to assist them in doing the exercises.
The exercises vary in difficulty from the straightforward ones that call on the application of a formula to the more involved Project Problems. All of the problems have been tested in the classroom, and the harder ones come with detailed hints to make them accessible to all students. Some of the longer exercises can be used by individual students, or as group projects, or as further illustrations by the instructor. Many sections in the book contain far more exercises than one would typically assign in a course. This allows a greater flexibility in the instructor's choice of problems, depending on the needs and backgrounds of the students.
Several exercises present interesting formulas and noteworthy results that are not found in many comparable books and that are more tractable with the availability of computer systems. Hopefully, even the experienced instructor will enjoy them as new material.
Exercises that require the use of the computer are preceded by a computer icon, such as the one in the margin. Typically these exercises ask the student to investigate problems using computer-generated graphics, and to generate numerical data that cannot be computed by hand with a reasonable effort. Occasionally, the computer is used to compute transforms, verify difficult identities, and aid in algebraic manipulations. Based on my teaching experience, I am convinced that such exercises are a great aid in understanding even the most abstract notions that are covered in the course.
Possible Course Outlines
The following are possible outlines of courses based on this book.
Basic undergraduate course in complex analysis
Chapter 1.
Chapter 2 (Section 2.6 is optional).
Chapter 3 (Sections 3.5, 3.8 and 3.9 are optional).
Chapter 4 (Section 4.7 is optional). Omit the proofs in Section 4.6.
Chapter 5 (Sections 5.5, 5.6, and 5.7 are optional).
Chapter 6 (Sections 6.4, 6.5, 6.6 are optional).
Depending on the background and need of the students, these sections can be covered at different speed. In a course with less emphasis on proofs, more applications from the optional Sections 3.8, 4.7, 6.4-6.6 can be presented.
A course in partial differential equations (to follow the basic course on complex analysis, as outlined previously).
Chapter 7 (Section 7.6 is optional).
Chapter 8 (Section 8.8 is optional).
Chapter 9: Sections 9.1-9.5 (appealing to Sections 9.6 and 9.7 as needed).
Chapter 10 Section 10.1-10.4 (appealing to Sections 10.5-10.7 as needed).
Chapters 11 and 12.
Refer to Sections 6.5 and 6.6 as needed to cover the related material on Green's functions and Neumann functions in Chapters 7-11.
A one-term course in complex analysis and partial differential equations
Complex Analysis Part:
Chapter 1
Chapter 2: Section 2.3 (refer to Sections 2.1 and 2.2 as needed), Section 2.4, Section 2.5. Cover Section 2.5 in detail as a substitute for Chapter 6.
Chapter 3: Sections 3.1 and 3.2. Section 3.4 (do only the version of Cauchy's theorem for simple paths, Theorems 5 and 6. Sections 3.6, 3.7, and 3.8.
Chapter 4: Section 4.1, Section 4.2 (Theorem 4 and its corollary). Skip the proofs in Sections 4.3-4.6, and just present the results and their applications from the examples. Do Section 4.7 followed by Section 7.2.
Chapter 5: Sections 5.1-5.3. Partial Differential Equations Part: Chapter 8: Refer to Section 7.4 for the half-range expansions.
Chapter 9: Sections 9.1 and 9.2 (refer to Sections 9.6 and 9.7 as needed).
Chapter 11: Sections 11.1-11.3.
Chapter 12: Sections 12.1-12.3.
Depending on the background and need of the students, these sections can be covered of different speed. In a course with less emphasis on proofs, more applications from Chapters 9 and 10 can be presented.
Basic graduate course in complex analysis
Cover the same topics as the basic undergraduate course, with more emphasis on proofs. In particular, I would cover Sections 2.6, 3.5, 4.6, and 5.7.
