Complex numbers: algebra, topology and geometry.
The infinite point, Riemann sphere.
Series, absolute convergence. Functional series, uniform convergence. Power series, radius of convergence.
Expenential function, sine and cosine.
Analytic functions, CR conditions.
Integral and its properties. Primitives, path independence.
Cauchy theorem and integral formula.
Rational functions.
Elementary entire functions
Linear-fractinal transformations
Local properties of holomorphic and meromorphic functions.
Classification of singularities
Laurent's Theorem
Topological properties, argument Principle, Rouche's theorem
Residue Theory, applications to integrals and series.
Maximum Principle, Weierstrass' Convergence Theorem.
Schwarz's Lemma
Schwarz's Symmetry Principle
Dirichlet Problem and Poisson Integral.
Conformal mapping by elementary functions.
Schwarz--Christoffel Formula