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About A first course in complex analysis with applications by Dennis G. Zill and Patrick Shanahan Pdf
This text grew out of chapters 17-20 in Advanced Engineering Mathematics, Second Edition (Jones and Bartlett Publishers), by Dennis G. Zill and the late Michael R. Cullen. This present work represents an expansion and revision of that original material and is intended for use in either a one-semester or a one-quarter course. Its aim is to introduce the basic principle and applications of complex analysis to undergraduates who have no prior knowledge of this subject. The motivation to adapt the material from Advanced Engineering Mathematics into a stand-alone text sprang from our dissatisfaction with the succession of textbooks that we have used over the years in our departmental undergraduate course offering in complex analysis. It has been our experience that books claiming to be accessible to undergraduates were often written at a level that was too advanced for our audience. The “audience” for our junior level course consists of some majors in mathematics, some majors in physics, but mostly majors from electrical engineering and computer science. At our institution, a typical student majoring in science or engineering does not take theory-oriented mathematics courses in methods of proof, linear algebra, abstract algebra, advanced calculus, or introductory real analysis. Moreover, the only prerequisite for our undergraduate course in complex variables is the completion of the third semester of the calculus sequence. For the most part, then, calculus is all that we assume by way of preparation for a student to use this text, although some working knowledge of differential equations would be helpful in the sections devoted to applications. We have kept the theory in this introductory text to what we hope is a manageable level, concentrating only on what we feel is necessary. Many concepts are conveyed in an informal and conceptual style and driven by examples, rather than the formal definition/theorem/proof. We think it would be fair to characterize this text as a continuation of the study of calculus, but also the study of the
calculus of functions of a complex variable. Do not misinterpret the preceding words; we have not abandoned theory in favor of “cookbook recipes”; proofs of major results are presented and much of the standard terminology is used. Indeed, there are many problems in the exercise sets in which a student is asked to prove something. We freely admit that any student—not just majors in mathematics—can gain some mathematical maturity and insight by attempting a proof. But we know, too, that most students have no idea how to start a proof. Thus, in some of our “proof” problems, either the reader is guided through the starting steps or a strong hint on how to proceed is provided. The writing herein is straightforward and reflects the no-nonsense style of Advanced Engineering Mathematics.
We have purposely limited the number of chapters in this text to seven. This was done for two “reasons”: to provide an appropriate quantity of material so that most of it can reasonably be covered in a one-term course, and at the same time to keep the cost of the text within reason. Here is a brief description of the topics covered in the seven chapters.
Chapter 1: The complex number system and the complex plane are examined in detail.
Chapter 2: Functions of a complex variable, limits, continuity, and mappings are introduced.
Chapter 3: The all-important concepts of the derivative of a complex function and analyticity of a function are presented.
Chapter 4: The trigonometric, exponential, hyperbolic, and logarithmic functions are covered. The subtle notions of multiple-valued functions and branches are also discussed.
Chapter 5: The chapter begins with a review of real integrals (including line integrals). The definitions of real line integrals are used to motivate the definition of the complex integral. The famous Cauchy Goursat theorem and the Cauchy integral formulas are introduced in this chapter. Although we use Green’s theorem to prove Cauchy’s theorem, a sketch of the proof of Goursat’s version of this same theorem is given in an appendix.
Chapter 6: This chapter introduces the concepts of complex sequences and infinite series. The focus of the chapter is on Laurent series, residues, and the residue theorem. Evaluation of complex as well as real integrals, summation of infinite series, and calculation of inverse Laplace and inverse Fourier transforms are some of the applications of residue theory that are covered.
Chapter 7: Complex mappings that are con formal are defined and used to solve certain problems involving Laplace’s partial differential equation.