Tasty Bits of Several Complex Variables

by Jiří Lebl

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Graduate level mathematics textbook on Several Complex Variables (complex analysis in several variables). A free online textbook based on class notes for the graduate course at Oklahoma State University.

Description from LibreTexts.

Publisher
Oklahoma State University
Subjects
Mathematics, Analysis
Licence
CC BY-SA 4.0, as LibreTexts states it
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

LibreTexts writes equations in TeX, and here they show as TeX; the PDF has them typeset.

  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Table of Contents
  5. Licensing
  6. 0: Introduction
  7. 0.1: Motivation, Single Variable, and Cauchy's Formula
  8. 1: Holomorphic Functions in Several Variables
  9. 1.1: Onto Several Variables
  10. 1.2: Power Series Representation
  11. 1.3: Derivatives
  12. 1.4: Inequivalence of Ball and Polydisc
  13. 1.5: Cartan's Uniqueness Theorem
  14. 1.6: Riemann Extension Theorem, Zero Sets, and Injective Maps
  15. 2: Convexity and Pseudoconvexity
  16. 2.1: Domains of Holomorphy and Holomorphic Extensions
  17. 2.2: Tangent Vectors, the Hessian, and Convexity
  18. 2.3: Holomorphic Vectors, the Levi Form, and Pseudoconvexity
  19. 2.4: Harmonic, Subharmonic, and Plurisubharmonic Functions
  20. 2.5: Hartogs Pseudoconvexity
  21. 2.6: Holomorphic Convexity
  22. 3: CR Functions
  23. 3.1: Real-Analytic Functions and Complexification
  24. 3.2: CR Functions
  25. 3.3: Approximation of CR Functions
  26. 3.4: Extension of CR Functions
  27. 4: The ∂-Problem
  28. 4.1: The Generalized Cauchy Integral Formula
  29. 4.2: Simple Case of The ∂-Problem
  30. 4.3: The General Hartogs Phenomenon
  31. 4.4: Solvability of ∂-Problem in the Polydisc
  32. 5: Integral Kernels
  33. 5.1: The Bochner-Martinelli Kernel
  34. 5.2: The Bergman Kernel
  35. 5.3: The Szegö Kernel
  36. 6: Complex Analytic Varieties
  37. 6.1: The Ring of Germs
  38. 6.2: Weierstrass Preparation and Division Theorems
  39. 6.3: The Dependence of Zeros on Parameters
  40. 6.4: Properties of the Ring of Germs
  41. 6.5: Varieties
  42. 6.6: Hypervarieties
  43. 6.7: Irreducibility, Local Parametrization, and Puiseux Theorem
  44. 6.8: Segre Varieties and CR Geometry
  45. 7: Appendix
  46. 7.1: A- Basic Notation and Terminology
  47. 7.2: B- Results from one Complex Variable
  48. 7.3: C- Differential Forms and Stokes' Theorem
  49. 7.4: D- Basic Terminology and Results from Algebra
  50. 7.5: Measure Theory Review
  51. Back Matter
  52. Further Reading
  53. Index
  54. Glossary
  55. Detailed Licensing