At its core, this concise textbook presents standard material for a
first course in complex analysis at the advanced undergraduate
level. This distinctive text will prove most rewarding for students
who have a genuine passion for mathematics as well as certain
mathematical maturity. Primarily aimed at undergraduates with
working knowledge of real analysis and metric spaces, this book can
also be used to instruct a graduate course. The text uses a
conversational style with topics purposefully apportioned into 21
lectures, providing a suitable format for either independent study
or lecture-based teaching. Instructors are invited to rearrange the
order of topics according to their own vision. A clear and rigorous
exposition is supported by engaging examples and exercises unique
to each lecture; a large number of exercises contain useful
calculation problems. Hints are given for a selection of the more
difficult exercises. This text furnishes the reader with a means of
learning complex analysis as well as a subtle introduction to
careful mathematical reasoning. To guarantee a student's
progression, more advanced topics are spread out over several
lectures. This text is based on a one-semester (12 week)
undergraduate course in complex analysis that the author has taught
at the Australian National University for over twenty years. Most
of the principal facts are deduced from Cauchy's Independence of
Homotopy Theorem allowing us to obtain a clean derivation of
Cauchy's Integral Theorem and Cauchy's Integral Formula. Setting
the tone for the entire book, the material begins with a proof of
the Fundamental Theorem of Algebra to demonstrate the power of
complex numbers and concludes with a proof of another major
milestone, the Riemann Mapping Theorem, which is rarely part of a
one-semester undergraduate course.
General
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