Complex Analysis

1. Advanced Topics
   1. Analytic Continuation
      1. Definition and Purpose
         1. Extending the domain of analytic functions
         2. Connection with power series
      2. Methods of Analytic Continuation
         1. Direct approach and power series extension
         2. Sheaf theory and patches
         3. Monodromy and covering spaces
      3. Applications
         1. Piercing through branch cuts in complex analysis
         2. Continuation of special functions (e.g., Riemann Zeta function)
         3. Influence in number theory and other fields
   2. Monodromy Theorem
      1. Basics of Monodromy
         1. Concept of analytic continuation along paths
         2. Monodromy groups and their characteristics
      2. Monodromy Theorem Definition
         1. Formal statement of the theorem
         2. Consequences for multi-valued functions
      3. Applications
         1. Role in the theory of Riemann surfaces
         2. Usage in Reshetikhin-Turaev invariants
   3. Mittag-Leffler's Theorem
      1. Introduction and Background
         1. Origin and historical significance
         2. Connections to partial fraction decomposition
      2. Statement of the Theorem
         1. The formulation for meromorphic functions
         2. Conditions and implications
      3. Applications and Examples
         1. Constructing functions with specified poles
         2. Solutions to differential equations
   4. Weierstrass Factorization
      1. Basics and Background
         1. Factorization of entire functions
         2. Weierstrass Product Theorem
      2. Process and Methods
         1. Construction of elementary factors
         2. Application in the fundamental theorem of algebra
      3. Examples and Importance
         1. Illustrations with canonical products
         2. Influence on the theory of entire functions
   5. Picard's Theorems
      1. Introduction to Picard's Theorems
         1. Small and Great Picard Theorems
         2. Historical context and development
      2. Picard's Little Theorem
         1. Statement and proof outline
         2. Implications for entire functions
      3. Picard's Great Theorem
         1. Statement and significance
         2. Application in determining function behavior
      4. Consequences and Applications
         1. Application in transcendental functions
         2. Influence on modern complex analysis theory
         3. Role in studying the distribution of values of meromorphic functions