This is an introductory course in complex analysis.
We will cover the basic theorems of this beautiful subject, with detailed proofs and some illustrating examples.
- Basics: Complex plane (arithmetic, geometry, topology), holomorphic functions, power series, integration along curves.
- Cauchy's theorems: Antiderivatives, Goursat's lemma, free homotopy, simply connected regions, Cauchy's formula.
- Applications of Cauchy's theorems: Liouville's theorem, fundamental theorem of algebra, uniqueness principle, Morera's theorem, uniform limits.
- Laurent series: Classification of isolated singularities, theorems of Riemann and Casorati-Weierstrass, meromorphic functions.
- Residue theorem and its applications: The argument principle, Rouché's theorem, Bernstein's theorem, open mapping theorem, maximum modulus principle.