Complex Analysis

1. Singularities and Residues
   1. Classification of Singularities
      1. Isolated Points
         1. Definition and Characteristics
         2. Examples in Common Functions
      2. Poles
         1. Simple Poles
            1. Definition and Identification
            2. Examples in Rational Functions
         2. Double Poles
            1. Definition and Identification
            2. Computational Techniques
         3. Higher-order Poles
            1. General Definition
            2. Laurent Series Approach
         4. Removal of Poles
            1. Techniques for Removable Singularities
            2. Conditions for Removability
      3. Essential Singularities
         1. Definition and Characteristics
         2. Examples in Complex Functions
         3. Picard's Theorem on Essential Singularities
         4. Behavior around Essential Points
   2. Residue Theorem
      1. Statement and Proof
         1. Detailed Theoretical Framework
         2. Illustration through Classical Examples
         3. Assumptions and Preconditions
      2. Calculation of Residues
         1. At Simple Poles
            1. Formula and Derivation
            2. Applications in Normal Integrals
         2. At Higher-order Poles
            1. Techniques for Different Cases
            2. Laurent Series Methods
         3. At Essential Singularities
            1. Theoretical Implications
            2. Case Studies and Examples
      3. Applications to Real Integrals
         1. Solving Real Integrals via Residue Calculus
         2. Techniques for Contour Deformation
         3. Interchangeability of Limits and Integrals
         4. Practical Examples in Physics and Engineering
   3. Laurent Series Expansion
      1. Definition and Convergence
         1. Theoretical Background
         2. Definition in the Context of Complex Functions
      2. Convergence Regions
         1. Annular Regions and Examples
         2. Relationship to Power Series
      3. Construction of Laurent Series
         1. Methods for Finding Terms
         2. Application in Complex Problem Solving
      4. Applications in Identifying Singularities
         1. Use in Classifying Singularities
         2. Key Case Studies and Examples