Complex Analysis

Complex Analysis is the branch of mathematics that investigates functions of complex variables, extending the familiar concepts of calculus—such as differentiation and integration—to the complex plane. Central to the field is the study of "analytic" or "holomorphic" functions, which are complex-differentiable and exhibit remarkably elegant properties not found in their real counterparts, such as being infinitely differentiable if they are differentiable just once. This unique structure leads to powerful tools like Cauchy's Integral Theorem and the Residue Theorem, which not only provide deep insights into the nature of functions but also have profound applications in solving problems in physics, engineering, and number theory.

1. Foundations of Complex Numbers
   1. The Field of Complex Numbers
      1. Definition and Construction
         1. Definition of Complex Numbers
         2. The Imaginary Unit i
         3. Standard Form a + bi
         4. Real and Imaginary Parts
         5. Equality of Complex Numbers
      2. Algebraic Operations
         1. Addition and Subtraction
            1. Component-wise Operations
            2. Geometric Interpretation as Vector Addition
         2. Multiplication
            1. Algebraic Rules
            2. Geometric Interpretation
            3. Properties of Multiplication
         3. Division
            1. Division Algorithm
            2. Rationalization of Denominators
            3. Division by Zero
      3. Complex Conjugate
         1. Definition and Notation
         2. Properties of Conjugation
         3. Conjugation and Arithmetic Operations
         4. Geometric Interpretation
      4. Modulus (Absolute Value)
         1. Definition and Properties
         2. Triangle Inequality
         3. Multiplicative Property
   2. The Complex Plane
      1. Cartesian Representation
         1. Plotting Complex Numbers
         2. Distance Formula
         3. Geometric Operations
      2. Polar Representation
         1. Modulus and Argument
         2. Principal Value of Argument
         3. Conversion Between Forms
         4. Geometric Meaning
      3. Exponential Form
         1. Euler's Formula
         2. Exponential Representation
         3. Properties of Exponential Form
   3. Powers and Roots
      1. De Moivre's Theorem
         1. Statement and Proof
         2. Applications to Trigonometry
      2. Roots of Complex Numbers
         1. nth Roots Formula
         2. Roots of Unity
         3. Geometric Distribution of Roots
      3. Complex Exponentiation
         1. Integer Powers
         2. Rational Powers
   4. Topology of the Complex Plane
      1. Basic Topological Concepts
         1. Open and Closed Sets
         2. Neighborhoods
         3. Interior, Exterior, and Boundary Points
         4. Accumulation Points
      2. Connectedness
         1. Connected Sets
         2. Path-Connected Sets
         3. Simply Connected Domains
      3. Compactness
         1. Bounded and Closed Sets
         2. Heine-Borel Theorem
      4. The Extended Complex Plane
         1. Point at Infinity
         2. Riemann Sphere
         3. Stereographic Projection