Math 503: Complex Analysis, in the Fall, 1997, semester
Preface Background Syllabus Homework Links to 503 technical pages The text

Preface

This is an introductory graduate course in complex analysis (also known as complex variables or function theory). The subject is one of the most beautiful and useful in mathematics. Most of the topics are "standard", and numerous adequate (and even good!) texts exist. There are contrasting and valid approaches to the subject. The first lecture illustrates some of the central themes of complex analysis and the last lecture gives a surprising application of complex analysis to twentieth century mathematics. The utility of this subject results from the fact that many methods for defining and guaranteeing differentiability of complex-valued functions of a complex variable agree. This surprising coincidence can be exploited to produce wonderful techniques and interesting results both in the field itself and in other areas of mathematics and science.

Much work has been done to make complex analysis accessible to a wide population since the techniques are so useful in applications to science and engineering. Undergraduate and graduate courses in complex analysis tend to have a large overlap in subject matter. The distinguishing mark of the graduate course is attention to the details of proofs, especially the sometimes irritating niceties of convergence arguments (varieties of uniform convergence and their consequences) and the necessary intricacies of plane topology (including winding number and some aspects of homology and homotopy). This careful underpinning should help a student to apply and if necessary extend the results of complex analysis.

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