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19212801 Funktionentheorie

Summer Term 2025

Lecturer: Prof. Dr. Nicolas Perkowski 
Assistant: Dr. Julian Kern

Time and place

  •  Lecture:  Tuesdays, 14:00 - 16:00h, HS 001, Arnimallee 3/5, 
                     Thursdays, 12:00 - 14:00h, HS 001, Arnimallee 3/5.

  • Exercise Session: Tuesdays, 16:00 - 18:00h, HS 001, Arnimallee 3/5.

  • Final Exam:  TBA
  • Follow-up Exam:  TBA
  • Office hours: Tuesdays, 10:00-11:00, Arnimallee 7, room 204.

Prerequisits:  Analysis I - II

Assessment

To receive credits fo the course you need to

  • actively participate in the exercise session 
  • work on and successfully solve the weekly exercises 
  • pass the final exam (see above)

If you are an FU student you only need to register for the course via CM (Campus Management).
If you are not an FU student, you are required to register via MyCampus/Whiteboard.

Exercises 

Problem sets will be put online every Tuesday and can be found under Assignements in Whiteboard. Solutions (in groups of two!) are due by 16:15h on Tuesday of the following week  –  solutions must be submitted in the tutorial or uploaded through the Whiteboard portal.

Course Overview/ Content:

Function theory is a classical field of mathematics, which deals with the properties of complex-differentiable functions on the complex number plane and has connections to algebra, analysis, number theory and geometry.

The concept of complex differentiability restricts real-differentiable functions from R2 to R2 to angle-preserving images. We will discover that complex-differentiable functions are quite rigid objects, but they are endowed with many amazing analytical, geometric, and visual properties.

A major result discussed in this lecture is Cauchy's integral theorem which states that the integral of any complexly differentiable function along a closed path in the complex plane is zero. We will see many nice consequences of this result, e.g. Cauchy's integral formula, the residual theorem and a proof of the fundamental theorem of algebra, as well as modern graphical representation methods.

References

  • E. Freitag and R. Busam 'Complex analysis', (Springer) 2nd Edition 2009 (the original German version is called 'Funktionentheorie')

Further literature will be given during the lecture.