## 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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