TY  - BOOK
AU  - Isaev, Alexander
TI  - Twenty-one lectures on complex analysis: a first course
T2  - Springer undergraduate mathematics series
SN  - 9783319681696
U1  - 517 
PY  - 2017///
CY  - Cham
PB  - Springer
KW  - Analysis (Mathematics)
KW  - Mathematical analysis
KW  - Complex Analysis
KW  - Functions of Complex Variable
KW  -  Functional Analysis
KW  - Homotopy
KW  - Conformation transformations
KW  - Cauchy's Independence of Homotopy Theorem
KW  - Cauchy’s Integral Theorem
KW  - Cauchy’s Integral Formula
KW  - Fundamental Theorem of Algebra
KW  - Mobius Transformations
KW  - Riemann Mapping Theorem
N2  - 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
UR  - https://link.springer.com/content/pdf/bfm:978-3-319-68170-2/1
UR  - https://www.goodreads.com/book/show/36084516-twenty-one-lectures-on-complex-analysis#CommunityReviews
ER  -