A Basic Course in Complex Variables
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In this book, you will need to have a working knowledge of not only algebra, but of calculus. Multivariable calculus becomes necessary in order to appreciate the material in Chapter 3. The derivative of a complex function is introduced—the same one you learned in calculus. We find out, for example, that the derivative of sin z is cos z, and the derivative of ln z is 1/z, just like ordinary calculus. Fortunately, you will not have to learn new formulas. Even the methods for differentiation are the same, such as the derivative of a product and quotient, and the chain rule.
So if all the basic theory is the same, why study complex variables? The fun does not really begin until the concept of the integral is introduced in Chapters 4 and 5. One result of this development is that if the first derivative of a function exists, then its second derivative also exists, unlike ordinary calculus. This result is a consequence of a famous theorem in complex variables due to A. Cauchy (1789–1857). Cauchy’s theorem states that if you integrate a differentiable function about a closed curve, you always get zero. It is surprising how much can be gleaned from this one simple (and seemingly trivial) result.
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About the Book
The calculus of real numbers can be extended to complex numbers, where the definitions and techniques one learns in calculus carry over to complex variables.
David C. Kay, who has written several books geared for college students, explains this development in his new book. A short review of basic concepts from real variable calculus appears with each new topic. Differentiation and integration in complex variables is clearly explained, with numerical examples.
Other topics include infinite series of complex variables, uniform convergence, the Taylor and Laurent series, and methods for evaluating difficult integrals.
Charts, tables, and drawings throughout the book make even tough concepts easy to understand, and problems have been carefully crafted to cover the main concepts while maintaining your interest.
Whether you're an educator seeking to provide an additional resource for your students or a student seeking a self-help guide to understand complex variables, the developmental in this book is a refreshing treatment that can be a stand-alone tutorial or companion guide to another textbook.
About the Author
David C. Kay received his doctorate from Michigan State University in 1963 and has held teaching positions in the mathematics departments at the University of Wyoming, the University of Oklahoma, the University of North Carolina at Asheville (UNCA), and, after retirement, at North Georgia State University. He has written four books on geometry and its applications. He and his wife, Katie, live in Watkinsville, Georgia. They have thirteen grandchildren and ten great-grandchildren.