Complex Analysis

(08 030400, Winter term 2018/19)

Instructor: Oliver Roth

Language of Instruction: English or German (depending on the students' preferences)

Registration: SB@home

Class Hours: Tue 12-14 (SE 40) & Wed 16-18 (S0.107), Problem Session: Mo 14-16 (SE 40)

We start on Tue, Oct. 16, 2018, 12:15 (SE 40).

Course material: The course ( in combination with the course Real methods in Complex Analysis, which will be offered in the summer term 2019) covers advanced topics in complex analysis such as

  1. the inhomogenous Cauchy-Riemann equation
  2. Runge's theorem
  3. Analytic continuation
  4. Harmonic functions and the Dirichlet problem
  5. Potential theory in the complex plane
  6. Covering properties of holomorphic maps
  7. Conformal metrics
  8. Holomorphic functional calculus and Banach algebras
  9. Spectral projections
  10. Spectral theorem for unbounded operators
  11. Complex dynamics
  12. Riemann surfaces
  13. Uniformisation
  14. Banach spaces of holomorphic functions
  15. Univalent functions
  16. Loewner theory
  17. Complex ODEs

There are numerous connections to Functional Analysis (Topics 1, 2, 8, 9, 10, 14), PDE Theory (Topics 1, 2, 4, 5, 7), Differential Geometry (Topics 4, 7, 12, 13), Algebra (Topics 6, 17) and Dynamical Systems (Topics 11, 16, 17) and there will be plenty of applications to Mathematical Physics. However, no prior knowledge of these fields is required.

A solid undergraduate course on basic complex analysis.

L. Ahlfors, Conformal Invariants - Topics in Geometric Function Theory, Amer. Math. Soc. 2010
M. Andersson, Topics in Complex Analysis, Springer 1996
C. Berenstein, R. Gay, Complex Variables, Springer 1997
J. Bruna, J. Cufi, Complex Analysis, Europ. Math. Soc. 2013
O. Forster, Lectures on Riemann Surfaces, Springer 1999.
J. Garnett, Bounded Analytic Functions, Springer 2007
I. Gohberg, J. Leiterer, Holomorphic Operator Functions of One Variable and Applications, Springer, 2009
B.C. Hall, Quantum Theory for Mathematicians, Springer 2013
H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer 2000
W. Kaballo, Aufbaukurs Funktionalanalysis und Operatortheorie, Springer 2014
P. Lax, Functional Analysis, Wiley 2002
R. Narasimhan, Y. Nievergelt, Complex Analysis in One Variable, Birkhäuser 2001
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer 1992
T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. 1995
S. Segal, Nine Introductions in Complex Analysis, Elsevier 2007
B. Simon, A Comprehensive Course in Analysis, Part 1-4, Amer. Math. Soc. 2015.

The Mandelbrot Set: Poster

Bloch's constant:

Last Updated: 1-Jun-18