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 complexanalysis 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.
