Complex Analysis

Complex Analysis is a branch of mathematics that studies functions of complex numbers and their properties. It explores concepts such as analytic functions, contour integrals, and conformal mappings, revealing the deep interplay between geometry and analysis. Central to this field are theorems like Cauchy's integral theorem and the residue theorem, which facilitate the evaluation of integrals and provide insights into the behavior of complex functions. Applications of complex analysis extend to various disciplines, including engineering, physics, and number theory, making it a vital area of mathematical study.

1. Complex Numbers
   1. Definitions
      1. Real and Imaginary Parts
         1. Definition of Real Part
            1. Notation and representation
            2. Interpretation on the complex plane
         2. Definition of Imaginary Part
            1. Notation and representation
            2. Relationship with the imaginary unit 'i'
      2. Modulus and Argument
         1. Definition of Modulus
            1. Calculation of modulus
            2. Representation as distance from the origin
         2. Definition of Argument
            1. Principal argument
            2. Computing argument using inverse tangent function
            3. Multiple values of argument
         3. Relationship with Polar Coordinates
            1. Conversion between Cartesian and polar forms
            2. Applications in physical problems
   2. Operations
      1. Addition and Subtraction
         1. Component-wise addition and subtraction
         2. Geometric interpretation on the complex plane
         3. Properties such as commutativity and associativity
      2. Multiplication and Division
         1. Multiplication
            1. Distributive property
            2. Geometric interpretation as rotation and scaling
         2. Division
            1. Division through multiplication by conjugate
            2. Importance of non-zero divisor
      3. Conjugation
         1. Definition and notation
         2. Properties of conjugates
            1. Relationship with modulus
            2. Multiplication and division of conjugates
         3. Use in simplifying division operations
   3. Polar and Exponential Forms
      1. Polar Form
         1. Conversion from rectangular to polar form
         2. Application in simplifying multiplications and powers
      2. Exponential Form
         1. Definition using Euler's Formula
         2. Expressing complex numbers as exponential expressions
         3. Applications in simplifying complex arithmetic
      3. Euler's Formula
         1. Derivation and interpretation
         2. Connection between trigonometric functions and exponential form
         3. Applications in physics and engineering
      4. De Moivre's Theorem
         1. Statement and proof
         2. Applications in finding roots of complex numbers
         3. Use in trigonometric identities and equations
   4. Complex Equations
      1. Solving quadratic equations with complex solutions
         1. Discriminant analysis
         2. Cases with real coefficients
      2. Higher order polynomial equations
         1. Use of complex conjugates
         2. Role of the Fundamental Theorem of Algebra
      3. Systems of equations involving complex numbers
         1. Gaussian elimination method adaptation
         2. Applications in electrical engineering
   5. Geometric Representation
      1. Argand Diagram
         1. Representation and interpretation of complex numbers
         2. Geometric transformations like translation, rotation, scaling
      2. Loci of Complex Numbers
         1. Circle and line representations
         2. Application in modeling oscillations and waves
      3. Visualizing operations
         1. Impact on magnitude and angle
         2. Vector representation of sums and differences