Twenty-One Lectures on Complex Analysis A First Course /

• Κύριος συγγραφέας: Isaev, Alexander (Συγγραφέας)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Cham : Springer International Publishing : Imprint: Springer, 2017.
• Σειρά: Springer Undergraduate Mathematics Series,
• Θέματα: Mathematics.; Mathematical analysis.; Analysis (Mathematics).; Analysis.

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...

Πίνακας περιεχομένων:

• 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.