Complex Analysis

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These notes consist of the following
chapters in an easy and detailed manner:
Chapter 1: Basic Concepts and Complex Numbers
Chapter 2: Analytic or Regular or Holomorphic Functions
Chapter 3: Elementary Transcendental Functions
Chapter 4: Complex Integration
Chapter 5: Power Series and Related Theorems
Chapter 1: Basic Concepts and Complex Numbers

Introduction to Complex Numbers
Complex Plane (Argand Diagram)
Real and Imaginary Parts
Complex Conjugates
Modulus (Absolute Value) and Argument
Polar Form of Complex Numbers
Operations on Complex Numbers (Addition, Subtraction, Multiplication, Division)
Complex Exponentiation
Roots of Complex Numbers
Complex Plane Geometry
Complex Conjugate and Absolute Value Properties
Euler's Formula
Applications in Engineering and Physics
Chapter 2: Analytic or Regular or Holomorphic Functions

Definitions and Terminology
The Cauchy-Riemann Equations
Analytic Functions and Holomorphic Functions
Examples of Analytic Functions
Harmonic Functions
Conformal Mapping
Mapping Properties of Analytic Functions
Analyticity of Elementary Functions
Chapter 3: Elementary Transcendental Functions

Exponential Functions
Logarithmic Functions
Trigonometric Functions
Hyperbolic Functions
Inverse Trigonometric and Hyperbolic Functions
Branch Cuts and Branch Points
Analytic Continuation
The Gamma Function
The Zeta Function
Chapter 4: Complex Integration

Line Integrals in the Complex Plane
Path Independence and Potential Functions
Contour Integrals
Cauchy's Integral Theorem
Cauchy's Integral Formula
Applications of Cauchy's Theorem
Morera's Theorem
Estimations of Integrals
Chapter 5: Power Series and Related Theorems

Power Series Representation of Analytic Functions
Taylor Series and Taylor's Theorem
Laurent Series
Singularities and the Residue Theorem
Analyticity on the Boundary
Applications of Power Series
Chapter 6: Singularities and Calculus of Residues

Classification of Singularities (Isolated Singularities, Essential Singularities)
Residues and Residue Theorem
Evaluation of Residues
Residue at Infinity
Applications of the Residue Theorem
Principal Value Integrals
Chapter 7: Conformal Mapping

Conformal Mappings and their Properties
Möbius Transformations
Conformal Mapping of Simple Regions
Conformal Mapping Applications (e.g., solving physical problems)
Chapter 8: Contour Integration

Contour Integration Techniques
Integration Along Real Axis (Jordan's Lemma)
Residues at Poles
Cauchy's Residue Theorem Revisited
Evaluation of Real Integrals Using Contour Integration
Complex Integration in Physics and Engineering
Chapter 6: Singularities and Calculus of Residues
Chapter 7: Conformal Mapping
Chapter 8: Contour Integration

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