Twenty-One Lectures on Complex Analysis

Clear and rigorous exposition is supported by engaging examples and exercises

Provides a means to learn complex analysis as well as subtle introduction to careful mathematical reasoning
Topics purposefully apportioned into 21 lectures, providing a suitable format for either independent study or lecture-based teaching

Clear and rigorous exposition is supported by engaging examples and exercises Provides a means to learn complex analysis as well as subtle introduction to careful mathematical reasoning Topics purposefully apportioned into 21 lectures, providing a suitable format for either independent study or lecture-based teaching Includes supplementary material: sn.pub/extras

Autorentext

Alexander Isaev is a professor of mathematics at the Australian National University. Professor Isaev's research interests include several complex variables, CR-geometry, singularity theory, and invariant theory. His extensive list of publications includes three additional Springer books: Introduction to Mathematical Methods in Bioinformatics (ISBN: 978-3-540-21973-6), Lectures on the Automorphism Groups of Kobayashi-Hyberbolic Manifolds (ISBN: 978-3-540-69151-8), and Spherical Tube Hypersurfaces (ISBN: 978-3-642-19782-6).

Inhalt

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.