Introduction to Video Lectures

Course Topics

• Complex Analysis
• Differential Geometry

Reference Books

• Complex Analysis: Complex Variables and Applications by Brown and Churchill
• Differential Geometry: Elementary Differential Geometry by Barrett O'Neill

Complex Analysis

Basic Topics

• Complex Numbers and Functions
• Calculus of Complex Functions
  • Differential Calculus: Derivatives of Complex Functions
  • Integral Calculus: Cauchy's Theorem and its applications

Differential Geometry

Basic Topics

• Investigation of Geometry (Curves, Surfaces)
• Properties of Geometry that Differentiate

Introduction to Complex Numbers

Introduction to Numbers

• History: Necessity and Evolution of Numbers
• Primitive Number Systems: Use of Stone
• Introduction to Rational and Irrational Numbers

Limitations of Rational Numbers

• Problem: Measuring Hypotenuse of a Triangle
• Pythagoras Theorem: A^2 + B^2 = C^2
• Irrational Numbers and Real Numbers

Need for Complex Numbers

Equations and Complex Numbers

• Example: Linear and Quadratic Equations
• Problem: X^2 + 1 = 0
• Introduction to Iota: (i^2 = -1)
• Form of Complex Numbers: (a + ib)

History of Complex Numbers

• Cardano: Introduction to Complex Numbers
• Bombelli: Development of Complex Numbers
• Euler: Initiation of iota notation