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Complex Analysis

Book Description

Complex Analysis presents a comprehensive and student-friendly introduction to the important concepts of the subject. Its clear, concise writing style and numerous applications make the basics easily accessible to students, and serves as an excellent resource for self-study. Its comprehensive coverage includes Cauchy-Goursat theorem, along with the description of connected domains and its extensions and a separate chapter on analytic functions explaining the concepts of limits, continuity and differentiability.

Table of Contents

  1. Cover
  2. Title page
  3. Contents
  4. Preface
  5. Chapter 1. Complex Numbers
    1. 1.1 Introduction
    2. 1.2 Complex Numbers
    3. 1.3 Graphical Representation of a Complex Number
    4. 1.4 Vector Form of Complex Numbers
    5. 1.5 Absolute Value and Conjugate
    6. 1.6 Triangle Inequality
    7. 1.7 Polar Form of a Complex Number
    8. 1.8 Exponential Form of a Complex Number
    9. 1.9 De Moivre’s Theorem
    10. 1.10 Roots of Complex Numbers
    11. 1.11 Stereographic Projection
    12. 1.12 Regions in the Complex Plane
    13. Summary
  6. Chapter 2. Analytic Functions
    1. 2.1 Introduction
    2. 2.2 Functions of a Complex Variable
    3. 2.3 Limit
    4. 2.4 Continuity
    5. 2.5 Differentiability
    6. 2.6 Analytic Functions
    7. 2.7 Cauchy-Riemann Equations
    8. 2.8 Harmonic Functions
    9. 2.9 Construction of Analytic Function
    10. 2.10 Orthogonal System
    11. Summary
  7. Chapter 3. Elementary Functions
    1. 3.1 Introduction
    2. 3.2 Elementary Functions
    3. 3.3 Periodic Functions
    4. 3.4 Exponential Function
    5. 3.5 Trigonometric Functions
    6. 3.6 Hyperbolic Functions
    7. 3.7 Branches, Branch Point and Branch Line
    8. 3.8 Logarithmic Function
    9. 3.9 Complex Exponents
    10. 3.10 Inverse Trigonometric Functions
    11. Inverse Hyperbolic Functions
    12. Summary
  8. Chapter 4. Complex Integration
    1. 4.1 Introduction
    2. 4.2 Derivative of Function w(t)
    3. 4.3 Definite Integrals of Functions
    4. 4.4 Contours
    5. 4.5 Contour Integrals
    6. 4.6 Moduli of Contour Integrals
    7. 4.7 Indefinite Integral
    8. 4.8 Cauchy’s Theorem
    9. 4.9 Cauchy-Goursat Theorem
    10. 4.10 Winding Number
    11. 4.11 Cauchy’s Integral Formula
    12. 4.12 Consequences of Cauchy’s Integral Formula
    13. 4.13 Maximum Moduli of Functions
    14. Summary
  9. Chapter 5. Sequence and Series
    1. 5.1 Introduction
    2. 5.2 Convergence of Sequence
    3. 5.3 Convergence of Series
    4. 5.4 Sequence of Functions
    5. 5.5 Series of Function
    6. 5.6 Power Series
    7. 5.7 Taylor Series
    8. 5.8 Laurent Series
    9. 5.9 Uniqueness of Series Representation
    10. 5.10 Multiplication and Division of Power Series
    11. Summary
  10. Chapter 6. Singularities and Residues
    1. 6.1 Introduction
    2. 6.2 Classification of Singularities
    3. 6.3 Zeros of an Analytic Function
    4. 6.4 Poles and Zeros
    5. 6.5 Behaviour at Infinity
    6. 6.6 Casorati-Weierstrass Theorem
    7. 6.7 Residues
    8. 6.8 Residue at Infinity
    9. 6.9 Meromorphic Functions
    10. 6.10 Mittag-Leffler Theorem
    11. Summary
  11. Chapter 7. Applications of Residues
    1. 7.1 Introduction
    2. 7.2 Definite Integrals Involving Sines and Cosines
    3. 7.3 Improper Integrals
    4. 7.4 Indented Contours
    5. 7.5 Other Types of Contours
    6. 7.6 Summation of Series
    7. 7.7 Inverse Laplace Transforms
    8. Summary
  12. Chapter 8. Bilinear and Conformal Transformations
    1. 8.1 Introduction
    2. 8.2 Linear Transformations
    3. 8.3 Transformation w = 1/z
    4. 8.4 Bilinear Transformation
    5. 8.5 Cross Ratio
    6. 8.6 Special Bilinear Transformations
    7. 8.7 Transformation W = z2
    8. 8.8 Transformation W = eZ
    9. 8.9 Trigonometric Transformations
    10. 8.10 Angle of Rotation
    11. 8.11 Conformal Transformation
    12. 8.12 Transformation
    13. 8.13 Transformation of Multivalued Functions
    14. 8.14 Riemann Surfaces
    15. 8.15 Mapping of Real Axis onto a Polygon
    16. 8.16 Schwarz–Christoffel Transformation
    17. Summary
  13. Chapter 9. Special Topics
    1. 9.1 Introduction
    2. 9.2 Analytic Continuation
    3. 9.3 Reflection Principle
    4. 9.4 Infinite Products
    5. 9.5 Infinite Product of Functions
    6. 9.6 Some Special Infinite Products
    7. 9.7 Boundary Value Problems
    8. Summary
  14. Appendix
  15. Glossary
  16. Acknowledgement
  17. Copyright