Complex Analysis

1. Analytic Functions
   1. Definitions and Properties
      1. Local Differentiability
         1. Concept of differentiability for complex functions
         2. Comparison with real differentiability
         3. Examples of functions differentiable at a point
      2. Cauchy-Riemann Equations
         1. Derivation of the equations
         2. Necessary and sufficient conditions for analyticity
         3. Applications and examples
         4. Interpretation in terms of coordinate systems
      3. Harmonic Functions
         1. Definition and properties of harmonic functions
         2. Relationship with analytic functions
         3. Solutions to Laplace's equation
         4. Examples of harmonic functions in complex variables
   2. Power Series Representation
      1. Basic Concepts
         1. Definition of power series
         2. Convergence in the complex plane
         3. Comparison with power series in real analysis
      2. Radius and Interval of Convergence
         1. Methods to determine radius of convergence
            1. Ratio test
            2. Root test
         2. Examples illustrating computation of radius and interval
         3. Understanding boundary behavior
      3. Taylor and Laurent Series
         1. Taylor Series
            1. Definition and derivation
            2. Conditions for existence
            3. Examples and applications
         2. Laurent Series
            1. Expansion for functions with singularities
            2. Region of convergence for Laurent series
            3. Connection to residue theorem
            4. Applications in complex analysis
   3. Mapping and Transformation Properties
      1. Understanding the concept of complex mapping
      2. Effects of analytic functions on domains
      3. Conformal Mapping as a special case
      4. The role of analytic functions in geometry
   4. Analytic Continuation
      1. The process of extending analytic functions beyond their domain
      2. Examples of analytic continuation
      3. Introduction to branch cuts and Riemann surfaces through analytic continuation
   5. Entire Functions
      1. Definition and characterization
      2. Properties and examples
      3. Liouville’s theorem for bounded entire functions
      4. Applications and relevance in complex analysis
   6. Weierstrass Theorem
      1. Weierstrass product theorem’s application in representation
      2. Construction of functions with prescribed zeros
      3. Conceptual understanding and proof outline