Introduction to Complex Analysis Course Syllabus
Full curriculum breakdown — modules, lessons, estimated time, and outcomes.
Overview: This course provides a comprehensive introduction to complex analysis, designed for learners seeking to build a strong foundation in mathematical theory with applications in engineering and physics. The curriculum spans eight modules, totaling approximately 32 hours of content, featuring interactive exercises and quizzes to reinforce understanding. Each module builds progressively from basic concepts to advanced applications, ensuring a structured and engaging learning experience. Lifetime access allows flexible pacing, ideal for both academic and professional development.
Module 1: Introduction to Complex Numbers
Estimated time: 4 hours
- History and algebra of complex numbers
- Geometric representation in the complex plane
- Polar coordinates and their properties
Module 2: Complex Functions and Iteration
Estimated time: 3 hours
- Complex functions and their mappings
- Sequences and limits in the complex plane
- Introduction to complex dynamics
- Julia sets and the Mandelbrot set
Module 3: Analytic Functions
Estimated time: 4 hours
- Complex differentiation
- Cauchy-Riemann equations
- Definition and properties of analytic functions
Module 4: Conformal Mappings
Estimated time: 4 hours
- Introduction to conformal mappings
- Complex logarithm and complex roots
- Möbius transformations
- Riemann mapping theorem
Module 5: Complex Integration
Estimated time: 4 hours
- Complex path integrals
- Cauchy’s integral theorem
- Fundamental Theorem of Algebra and applications
Module 6: Power Series
Estimated time: 4 hours
- Power series representations of analytic functions
- Convergence of power series
- Riemann zeta function
Module 7: Laurent Series and the Residue Theorem
Estimated time: 4 hours
- Laurent series expansions
- Isolated singularities
- Residue theorem and its applications
Module 8: Applications of Complex Analysis
Estimated time: 4 hours
- Solving problems in physics using complex analysis
- Engineering applications of complex integration
- Real-world problem solving with analytic functions
Prerequisites
- Basic knowledge of calculus
- Familiarity with real analysis concepts
- Understanding of high school algebra and geometry
What You'll Be Able to Do After
- Understand and manipulate complex numbers and functions
- Apply the Cauchy-Riemann equations to test analyticity
- Evaluate complex integrals using Cauchy’s theorem
- Expand functions into Taylor and Laurent series
- Solve applied problems in physics and engineering using residue theory