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Twenty-one lectures on complex analysis : a first course

Author: Alexander Isaev
Publisher: Cham, Switzerland : Springer, [2017]
Series: Springer undergraduate mathematics series.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Isaev, Alexander.
Twenty-one lectures on complex analysis.
Cham, Switzerland : Springer, [2017]
(OCoLC)1002128300
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Alexander Isaev
ISBN: 9783319681702 3319681702
OCLC Number: 1013820783
Description: 1 online resource.
Contents: 1. Complex Numbers. The Fundamental Theorem of Algebra.- 2. R- and C-Differentiability --
3 The Stereographic Projection. Conformal Maps. The Open Mapping Theorem.- 4. Conformal Maps (Continued). Möbius Transformations.- 5. Möbius 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). Rouché'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.
Series Title: Springer undergraduate mathematics series.
Responsibility: Alexander Isaev.

Abstract:

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  Read more...

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"This text furnishes the reader with a means of learning complex analysis as well as a subtle introduction to careful mathematical reasoning. ...There is no doubt that graduate students and seasoned Read more...

 
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