Introductory Complex Analysis – Richard A. Silverman

Introductory Complex Analysis is a scaled-down version of A. I. Markushevich’s masterly three-volume “Theory of Functions of a Complex Variable.” Dr. Richard Silverman, the editor and translator of the original, has prepared this shorter version expressly to meet the needs of a one-year graduate or undergraduate course in complex analysis. In his selection and adaptation of the more elementary topics from the original larger work, he was guided by a brief course prepared by Markushevich himself.

The book begins with fundamentals, with a definition of complex numbers, their geometric representation, their algebra, powers and roots of complex numbers, set theory as applied to complex analysis, and complex functions and sequences. The notions of proper and improper complex numbers and of infinity are fully and clearly explained, as is stereographic projection. Individual chapters then cover limits and continuity, differentiation of analytic functions, polynomials and rational functions, Mobius transformations with their circle-preserving property, exponentials and logarithms, complex integrals and the Cauchy theorem , complex series and uniform convergence, power series, Laurent series and singular points, the residue theorem and its implications, harmonic functions (a subject too often slighted in first courses in complex analysis), partial fraction expansions, conformal mapping, and analytic continuation.

Elementary functions are given a more detailed treatment than is usual for a book at this level. Also, there is an extended discussion of the Schwarz-Christolfel transformation, which is particularly important for applications.

There is a great abundance of worked-out examples, and over three hundred problems (some with hints and answers), making this an excellent textbook for classroom use as well as for independent study. A noteworthy feature is the fact that the parentage of this volume makes it possible for the student to pursue various advanced topics in more detail in the three-volume original, without the problem of having to adjust to a new terminology and notation .

In this way, IntroductoryComplex Analysis serves as an introduction not only to the whole field of complex analysis, but also to the magnum opus of an important contemporary Russian mathematician.

Contents:

Chapter 1. Complex numbers, functions and sequences
1. Introductory Remarks
2. Complex Numbers and Their Geometric Representation
3. Complex Algebra
4. Powers and Roots of Complex Numbers
5. Set Theory. Complex Functions
6. Complex Sequences
7. Proper and Improper Complex Numbers
8. Infinity and Stereographic Projection

Chapter 2. Limits and continuity
9. More Set Theory. The Heine-Borel Theorem
10. The Limit of a Function at a Point
11. Continuous Functions
12. Curves and Domains

Chapter 3. Differentiation. Analytic functions
13. Derivatives. Rules for Differentiating Complex Functions
14. The Cauchy-Riemann Equations. Analytic Functions
15. Geometric Interpretation of Arg f′(z) and |f′(z)|. Conformal Mapping
16. The Mapping images
17. Conformal Mapping of the Extended Plane

Chapter 4. Polynomials and rational functions
18. Polynomials. The Mapping w = Pn(z)
19. The Mapping w = (z – a)n
20. The Mapping w = images
21. Rational Functions
22. The Mapping images

Chapter 5. Möbius transformations
23. The Group Property of Möbius Transformations
24. The Circle-Preserving Property of Möbius Transformations
25. Fixed Points of a Möbius Transformation. Invariance of the Cross Ratio
26. Mapping of a Circle onto a Circle
27. Symmetry Transformations
28. Examples

Chapter 6. Exponentials and logarithms
29. The Exponential
30. The Mapping w = ez
31. Some Functions Related to the Exponential
32. The Logarithm
33. The Function zα. Exponentials and Logarithms to an Arbitrary Base

Chapter 7. Complex integrals. Cauchy’s integral theorem
34. Rectifiable Curves. Complex Integrals
35. The Case of Smooth Curves
36. Cauchy’s Integral Theorem. The Key Lemma
37. Proof of Cauchy’s Integral Theorem
38. Application to the Evaluation of Definite Integrals
39. Cauchy’s Integral Theorem for a System of Contours

Chapter 8. Cauchy’s integral formula and its implications
40. Indefinite Integrals
41. Cauchy’s Integral Formula
42. Morera’s Theorem. Cauchy’s Inequalities

Chapter 9. Complex series. Uniform convergence
43. Complex Series
44. Uniformly Convergent Series and Sequences
45. Series and Sequences of Analytic Functions

Chapter 10. Power series
46. The Cauchy-Hadamard Theorem
47. Taylor Series. The Uniqueness Theorem for Power Series
48. Expansion of an Analytic Function in a Power Series
49. Liouville’s Theorem. The Uniqueness Theorem for Analytic Functions
50. A-Points and Zeros
51. Weierstrass’ Double Series Theorem
52. Substitution of One Power Series into Another
53. Division of Power Series

Chapter 11. Laurent series. Singular points
54. Laurent Series
55. Laurent’s Theorem
56. Poles and Essential Singular Points
57. Behavior at an Essential Singular Point. Picard’s Theorem
58. Behavior at Infinity

Chapter 12. The residue theorem and its implications
59. The Residue Theorem. Residues at Infinity
60. Jordan’s Lemma. Evaluation of Definite Integrals
61. The Argument Principle. The Theorems of Rouché and Hurwitz
62. Local Behavior of Analytic Mappings. The Maximum Modulus Principle and Schwarz’s Lemma

Chapter 13. Harmonic functions
63. Laplace’s Equation. Conjugate Harmonic Functions
64. Poisson’s Integral. Schwarz’s Formula
65. The Dirichlet Problem

Chapter 14. Infinite product and partial fraction expansions
66. Preliminary Results. Infinite Products
67. Weierstrass’ Theorem
68. Mittag-Leffier’s Theorem
69. The Gamma Function
70. Cauchy’s Theorem on Partial Fraction Expansions

Chapter 15. Conformal mapping
71. General Principles of Conformai Mapping
72. Mapping of the Upper Half-Plane onto a Rectangle
73. The Schwarz-Christoffel Transformation

Chapter 16. Analytic continuation
74. Elements and Chains
75. General and Complete Analytic Functions
76. Analytic Continuation Across an Arc
77. The Symmetry Principle
78. More on Singular Points
79. Riemann Surfaces
Bibliography
Index

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