Analytic functions

Here, you can find some materials on complex analysis. First, there is a collection of quite challenging problems taken mainly, but not exclusively, from various mathematical olympiads and competitions such as the Vojtěch Jarník Mathematical Competition organized annually in Ostrava. Solutions of selected problems are also provided.

Below, there are problem sets and some supplementary materials (in Polish) for classes in the Analytic functions course that I taught at the University in Warsaw (2018/19). Those problem sets are by no means complete for that course. They are rather meant to be composed of slightly more difficult problems which can broaden the knowledge gained by solving some standard exercises in complex analysis.

Challenging problems in complex analysis

• Problems
• Solutions

Problem sets (in Polish)

• Part 0
  Basic properties of complex numbers and complex geometry, and a few applications to some (geometric) combinatorial problems.
• Part 1
  Geometry and topology of the complex plane; a few math olympiad-style problems in geometry.
• Part 2
  The stereographic projection. Möbius transformations, cross-ratio and conformality. Classification of homographies. Groups of automorphisms; Blaschke factors. Hyperbolic geometry.
• Part 3
  Holomorphic functions; the uniqueness theorem; Morera's theorem. The maximum modulus principle and Schwarz's lemma.

Some instructive exercises (in Polish)

• Test 1 (exemplary problems)
  
• When is there a homography...?
• Solutions to Test 1
• Test 2 (exemplary problems)
• Complex integration by parts
• Contour integration and exact differential forms
  
• Three problems: Hurwitz, Aut(Ω) and Schwarz