Complex Analysis

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