Lecture Notes
Antoine Cerfon
CIMS, NYU
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Notes for Complex Variables
The notes below follow fairly closely the textbook
Complex Analysis (3rd edition)
by Lars V. Ahlfors (McGraw-Hill). Some of the proofs and explanations were also inspired by great notes by Dan Romik (UC Davis).
Lecture 1: Complex numbers
Lecture 2: Analytic functions
Lecture 3: Usual complex functions
Lecture 4: Complex Integration
Lecture 5: Cauchy's Theorem
Lecture 6: Consequences of Cauchy's Theorem
Lecture 7: Local properties of analytic functions - Part 1
Lecture 8: Local properties of analytic functions - Part 2
Lecture 9: The general form of Cauchy's Theorem
Lecture 10: The calculus of residues
Lecture 11: Harmonic functions
Lecture 12: Series and product developments
Lecture 13: The Euler Gamma function and the Riemann Zeta function
Lecture 14: Conformal Mapping
Lecture 15: The Riemann Mapping Theorem
Lecture 16: Applications of Conformal Mapping
Lecture 17: Analytic Continuation