Complex Analysis

1. Complex Integration
   1. Line Integrals
      1. Definition and Properties
         1. Integration of complex functions along a contour
         2. Properties of line integrals in the complex plane
         3. Comparisons with real-variable integrals
      2. Parameterization of Contours
         1. Techniques for parameterizing paths and curves
         2. Examples of common contours: line segments, circles, and ellipses
         3. Smooth vs. piecewise smooth contours
      3. Path Independence
         1. Concept of independence from the path taken
         2. Conditions for path independence
         3. Connection with analyticity and exact differentials
   2. Cauchy's Integral Theorem
      1. Statement of the theorem
      2. Assumptions and prerequisites: simply connected regions, analytic functions
      3. Proofs and intuitive understanding
         1. Using Green's theorem and other techniques
         2. Insights from vector calculus
      4. Applications and implications
         1. Zero integral over closed contours
         2. Fundamental concept in complex analysis
   3. Cauchy's Integral Formula
      1. Statement of the formula
      2. Expressions for function values within a contour
      3. Derivatives of analytic functions
         1. Relationship to higher-order derivatives
         2. Generalized versions of the formula
      4. Practical applications
         1. Evaluating integrals through Cauchy's formulas
         2. Importance in the development of further theorems
   4. Morera's Theorem
      1. Statement and proof
         1. Relationship to determining analyticity
         2. Use of prior established theorems such as Cauchy's
      2. Implications and applications
         1. Knowing when a function is analytic based on integral conditions
         2. Use in diverse complex functions and topologies
   5. Contour Integration Techniques
      1. Definition and benefits
         1. Use of contours in evaluating definite integrals
         2. Advantages over real-variable techniques for certain integrals
      2. Types of contours
         1. Closed and open contours
         2. Common shapes and paths used in practice
      3. Jordan's Lemma and Similar Results
         1. Estimation techniques for contour integrals
         2. Asymptotic behavior and boundedness in the complex plane
      4. Use in evaluating real integrals
         1. Applications to real integrals and improper integrals
         2. Techniques and transformations simplifying real analysis problems
         3. Principal value integrals and their connections with complex analysis
   6. Advanced Concepts and Applications
      1. Relationship to harmonic and analytic functions
         1. Linkages between harmonicity and integration
         2. Applications in solving Laplace's equation
      2. Complex line integrals in physics and engineering
         1. Modeling electromagnetic fields and fluid flows
         2. Use in other applied sciences, like aerodynamics and control theory
      3. Software and numerical methods
         1. Algorithmic approaches to complex integration
         2. Error analysis and convergence in numerical integrals
         3. Applications and limitations in computational tools