Basic properties of holomorphic functions
Cauchy integral formula/s and applications
Conformal mappings
Preliminaries: algebra, geometry, and topology of the complex numbers
Topology of open subsets of C
Holomorphic functions. Cauchy--Riemann equations
Power series and operations with them
Examples: exp, sin, cos, tan, ...
Integration along curves and basic properties of the curve integral
Primitives and curve integrals
Goursat's theorem
Local primitives.
Cauchy integral formulas in a disc. Estimates
Liouville's theorem. The Fundamental theorem of Algebra
Morera's theorem and Weierstrass theorem (conv of derivatives)
The topology of uniform convergence on compacta and its properties
Harmonic functions, harmonic conjugate (Mean value property, ...). Abel's theorem
Remarks about simply connected regions
Index and homotopy of closed loops
Generalized Cauchy formulae
Runge's theorem, I
Runge's theorem II. Schwarz reflection principle
Zeros of holomorphic functions
Laurent series. Classification of singularities.
Riemann theorem (removable singularities) and Weierstrass theorem (essential singularities)
Calculus of residues
Riemann sphere, rational functions
PSL(2,C) as the automorphisms of the Riemann sphere
The Argument principle and Rouche's theorem
Local properties of holomorphic maps. Open mapping theorem
Conformal mappings, some examples
Schwarz lemma. PSL(2,R) as automorphisms of the upper half plane / unit disc
Riemann's Mapping Theorem - intro
Normal families and Montel's theorem. Limit of injective holomorphic maps
Proof of the Riemann Mapping Theorem