Life of Fred Mathematics

Life of Fred: Complex Analysis
(For Math Majors)

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.

• 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

Click here to view a sample lesson (Opens in a new window or tab)

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.