Twentyone lectures on complex analysis : a first course
 Responsibility
 Alexander Isaev.
 Digital
 text file; PDF
 Publication
 Cham, Switzerland : Springer, [2017]
 Physical description
 1 online resource
 Series
 Springer undergraduate mathematics series.
Online
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Description
Creators/Contributors
 Author/Creator
 Isaev, Alexander, author.
Contents/Summary
 Contents

 1. Complex Numbers. The Fundamental Theorem of Algebra.
 2. R and CDifferentiability. 3 The Stereographic Projection. Conformal Maps. The Open Mapping Theorem.
 4. Conformal Maps (Continued). Moebius Transformations.
 5. Moebius Transformations (Continued). Generalised Circles. Symmetry.
 6. Domains Bounded by Pairs of Generalised Circles. Integration.
 7. Primitives Along Paths. Holomorphic Primitives on a Disk. Goursat's Lemma.
 8. Proof of Lemma 7
 .2. Homotopy. The Riemann Mapping Theorem.
 9. Cauchy's Independence of Homotopy Theorem. Jordan Domains.
 10. Cauchy's Integral Theorem. Proof of Theorem 3
 .1. Cauchy's Integral Formula.
 11. Morera's Theorem. Power Series. Abel's Theorem. Disk and Radius of Convergence.
 12. Power Series (Cont'd). Expansion of a Holomorphic Function. The Uniqueness Theorem.
 13. Liouville's Theorem. Laurent Series. Isolated Singularities.
 14. Isolated Singularities (Continued). Poles and Zeroes. Isolated Singularities at infinity.
 15. Isolated Singularities at infinity (Continued). Residues. Cauchy's Residue Theorem.
 16. Residues (Continued). Contour Integration. The Argument Principle 137.
 17. The Argument Principle (Cont'd). Rouche's Theorem. The Maximum Modulus Principle.
 18. Schwarz's Lemma. (Pre) Compactness. Montel's Theorem. Hurwitz's Theorem.
 19. Analytic Continuation.
 20. Analytic Continuation (Continued). The Monodromy Theorem.
 21. Proof of Theorem 8
 .3. Conformal Transformations of Simply Connected Domains. Index.
 (source: Nielsen Book Data)
 Publisher's summary

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 lecturebased 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 onesemester (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 onesemester undergraduate course.
(source: Nielsen Book Data)
Subjects
 Subjects
 Mathematical analysis.
Bibliographic information
 Publication date
 2017
 Series
 Springer undergraduate mathematics series
 ISBN
 3319681702 (electronic book)
 9783319681702 (electronic bk.)
 9783319681696
 3319681699
 DOI
 10.1007/9783319681702