Complex Analysis: A Functional Analytic Approach (De Gruyter Textbook)

✍ Scribed by Friedrich Haslinger

• Publisher: De Gruyter
• Year: 2017
• Tongue: English
• Leaves: 349
• Edition: 1
• Category: Library

✦ Synopsis

In this textbook, a concise approach to complex analysis of one and several variables is presented. After an introduction of Cauchy‘s integral theorem general versions of Runge‘s approximation theorem and Mittag-Leffler‘s theorem are discussed. The fi rst part ends with an analytic characterization of simply connected domains. The second part is concerned with functional analytic methods: Fréchet and Hilbert spaces of holomorphic functions, the Bergman kernel, and unbounded operators on Hilbert spaces to tackle the theory of several variables, in particular the inhomogeneous Cauchy-Riemann equations and the d-bar Neumann operator.

Contents
Complex numbers and functions
Cauchy’s Theorem and Cauchy’s formula
Analytic continuation
Construction and approximation of holomorphic functions
Harmonic functions
Several complex variables
Bergman spaces
The canonical solution operator to
Nuclear Fréchet spaces of holomorphic functions
The -complex
The twisted -complex and Schrödinger operators

✦ Table of Contents

Preface
Contents
1. Complex numbers and functions
1.1 Complex numbers
1.2 Some topological concepts in ℝ^n
1.3 Holomorphic functions
1.4 The Cauchy–Riemann equations
1.5 A geometric interpretation of the complex derivative
1.6 Uniform convergence
1.7 Power series
1.8 Line integrals
1.9 Primitive functions
1.10 Elementary functions
1.11 Exercises
1.12 Notes
2. Cauchy’s Theorem and Cauchy’s formula
2.1 Winding numbers
2.2 The theorem of Cauchy–Goursat and Cauchy’s formula
2.3 Important consequences of Cauchy’s Theorem
2.4 Isolated singularities
2.5 The maximum principle and Cauchy’s estimates
2.6 Open mappings
2.7 Holomorphic parameter integrals
2.8 Complex differential forms
2.9 The inhomogeneous Cauchy formula
2.10 General versions of Cauchy’s Theorem and Cauchy’s formula
2.11 Laurent series and meromorphic functions
2.12 The residue theorem
2.13 Exercises
2.14 Notes
3. Analytic continuation
3.1 Regular and singular points
3.2 Analytic continuation along a curve
3.3 The Monodromy Theorem
3.4 Exercises
3.5 Notes
4. Construction and approximation of holomorphic functions
4.1 A partition of unity
4.2 The inhomogeneous Cauchy–Riemann differential equations
4.3 The Hahn–Banach Theorem
4.4 Runge’s approximation theorems
4.5 Mittag-Leffler’s Theorem
4.6 The Weierstraß Factorization Theorem
4.7 Some applications of the Mittag-Leffler and Weierstraß Theorems
4.8 Normal families
4.9 The Riemann Mapping Theorem
4.10 Characterization of simply connected domains
4.11 Exercises
4.12 Notes
5. Harmonic functions
5.1 Definition and important properties
5.2 The Dirichlet problem
5.3 Jensen’s formula
5.4 Subharmonic functions
5.5 Exercises
5.6 Notes
6. Several complex variables
6.1 Complex differential forms and holomorphic functions
6.2 The inhomogeneous CR equations
6.3 Domains of holomorphy
6.4 Exercises
6.5 Notes
7. Bergman spaces
7.1 Elementary properties
7.2 Examples
7.3 Biholomorphic mappings
7.4 Exercises
7.5 Notes
8. The canonical solution operator to ∂̄
8.1 Compact operators on Hilbert spaces
8.2 The canonical solution operator to 𝜕 restricted to A^2(𝔻)
8.3 (0, 1)-forms with holomorphic coefficients
8.4 Exercises
9. Nuclear Fréchet spaces of holomorphic functions
9.1 General properties of Fréchet spaces
9.2 The space ℋ(DR (0)) and its dual space
9.3 Exercises
9.4 Notes
10. The ∂̄-complex
10.1 Unbounded operators on Hilbert spaces
10.2 Distributions and Sobolev spaces
10.3 Friedrichs’ lemma
10.4 A finite dimensional analog
10.5 The 𝜕-Neumann operator
10.6 Density in the graph norm
10.7 Properties of the 𝜕-Neumann operator
10.8 Exercises
10.9 Notes
11. The twisted ∂̄-complex and Schrödinger operators
11.1 An exact sequence of unbounded operators
11.2 The twisted basic estimates
11.3 Hörmander’s L^2-estimates
11.4 The 𝜕-Neumann operator on weighted (0, q)-forms
11.5 Weighted spaces of entire functions
11.6 Spectral analysis of self-adjoint operators
11.7 Real differential operators
11.8 Dirac and Pauli operators
11.9 Compact resolvents
11.10 Exercises
11.11 Notes
Bibliography
Index