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