Introductory Complex Analysis by Silverman, Richard A. -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page 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

Chapter 4 - Polynomials and Rational Functions

18. Polynomials. The Mapping w = Pn(z) 19. The Mapping w = (z – a)n 20. The Mapping w = 21. Rational Functions 22. The Mapping

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

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