Complex Analysis

1. Techniques and Methods
   1. Contour Integration Techniques
      1. Parametrization Methods
         1. Unit Circle Parametrization
         2. Line Segment Parametrization
      2. Deformation of Contours
         1. Cauchy’s Deformation Principle
         2. Jordan's Lemma
      3. Common Contour Paths
         1. Semi-Circular Contours
         2. Keyhole Contours
         3. Indented Contours
   2. Schwarz Reflection Principle
      1. Statement of the Principle
      2. Symmetric Domain Mapping
         1. Real Axis Reflection
         2. Circular Arc Reflection
      3. Applications to Half-Plane Problems
         1. Solving Boundary Value Problems
         2. Extension Across Real and Imaginary Axes
   3. Asymptotic Expansions and Approximations
      1. Basic Concepts
         1. Definition and Importance
         2. Asymptotic Expansion vs. Convergence
      2. Methods of Derivation
         1. Laplace’s Method
         2. Method of Stationary Phase
         3. Saddle Point Method
      3. Application Examples
         1. Asymptotic Series for Special Functions
         2. Asymptotic Methods in Quantum Mechanics
         3. Expansion of Integrals
   4. Additional Techniques
      1. Argument Principle
         1. Application in Finding Roots
         2. Relation with the Counting of Zeroes and Poles
      2. Maximum Modulus Principle
         1. Application in Function Behavior Analysis
         2. Consequences for Bounded Entire Functions
      3. Method of Steepest Descent
         1. Evaluating Integrals Using Descent Paths
         2. Applications in Wave Mechanics