Complex Analysis is the theory of functions of a single complex variable. (That is: Functions that map from complex numbers to complex numbers.) This is advanced undergraduate mathematics with applications to even more advanced mathematics, physics, and engineering.
Topics include:
- Arithmetic in the complex plane
- Conjugates and absolute value
- Polar form
- Complex functions
- Graphing
- Limits and derivatives
- Cauchy-Riemann equations
- Holomorphic functions - Regions, open sets, connected sets
- Paths
- Analytic functions
- Integration along a path
- Fundamental Theorem of Complex Integration
- Piecewise smooth paths
- Paths with corners or cusps
- Cauchy’s theorem
- Closed paths
- Simple paths
- Cauchy–Goursat theorem
- Trig functions, exponents, and logs in the complex plane
- Argument of a polar complex function
- Morera’s theorem
- Three kinds of isolated singular points
- Order of a pole
- Proving L’Hospital’s rule in the complex plane
- Meromorphic functions
- Casorati-Weierstrass theorem
- Picard’s great theorem
- Cauchy’s formula
- Jordan curve theorem
- Liouville theorem
- Proof of the Fundamental Theorem of Algebra
- Taylor series and Laurent series
- Residue theorem
- Analytic continuation
- Riemann Zeta function
- Conformal Mappings
- Homotopic paths
- Harmonic functions
Unlike most other
Complex Analysis programs, this one also has:
• Fred's scheduling issues
• Red Robin Hood
Life of Fred Complex Analysis is a hardcover textbook containing 176 pages. This book is not consumable. All answers are written on separate paper or in a notebook.
Number of Lessons: 24 Chapters. It is not broken up into daily lessons.