General topology and complex analysis 1994/95.

For 2nd year mathematics students. Official name: Geometria II.
PRELIMINARIES

Trigonometric functions
Polar coordinates in plane and space
Complex numbers
n-th roots of complex numbers
Lines in the plane
Lines and planes in 3-space
The projective line
The projective plane
Projective coordinates

CONIC SECTIONS

Ellipse, parabola, hyperbola
Diagonalization of symmetric operators
Principal axes of a conic section

TOPOLOGY

Topological spaces
Non-archimedean metrics
Continuous mappings
Quotient topology
Uryson's lemma
Generalized sequences or nets
Filters and direct products
Ultrafilters and compactness
Connectedness
The fundamental group

COMPLEX ANALYSIS

Holomorphic functions
Curve integrals
Stokes' theorem in the plane
Cauchy's integral theorem
Jordan curves
Cauchy's integral formula
Power series in the complex domain
Every holomorphic function is analytic
Morera's theorem
The theorem of Weierstrass
Cauchy inequalities and Liouville's theorem
The identity theorem
The open mapping theorem and the maximum principle
Riemann's extension theorem
Biholomorphic functions
Isolated singular points and Laurent series
Meromorphic functions
The theorem of residues
Calculating integrals with the theorem of residues