Topics in complex analysis

In this course we treat selected topics in complex analysis. We shall mostly focus on functions in one complex variable. For further organizational information see also the course book. Check out the supplementary lecture notes as well for more details.

Topics.
§1. Sequences of holomorphic functions.

• What are reasonable notions of convergence of complex functions? How are they related? What are possible examples and counterexamples?
• Are specific properties (holomorphy, number of zeros, etc.) stable? Under which convergence? What goes wrong in cases of instability?
• Does the convergence of functions imply anything about the convergence of their derivatives?
• What are sufficient conditions guaranteeing either notion of convergence?
  

§2. The Mittag-Leffler theorem.

• How does one construct holomorphic functions with a prescribed principal parts at a discrete set of singularities?
• How can these functions be constructed explicitly using Taylor expansion?
• How can this construction be extended to subdomains of the complex plane? In particular, how does one deal with possible accumulation points at the boundary?
  

§3. Infinite products.

• What is a reasonable notion of convergence of infinite products of complex numbers?
• How do we deal with the occurrence of zeros in these products?
• What are necessary conditions for the convergence of infinite products?
• What are sufficient conditions for the convergence of infinite products?
• How does the convergence of products relate to the known concept of convergence of series by taking exponentials and (principal branches of) logarithms?
• What is a reasonable notion of absolute convergence of infinite products? Why is the naive attempt inspired by absolute convergence of series not reasonable?
• How does one define locally uniform convergence of products of continuous functions?
• How does one define locally normal convergence of products of continuous functions? What specific form do the factors take?
• How does the number of zeros and their order of the factors transfer to their infinite product?
• How does the logarithmic derivative of an infinite product of functions look like?
  

§4. The Weierstraß product theorem.

• Given a finite set S of "zeros", how can one construct a function whose zeros are precisely the elements of S? How to ensure these zeros have prescribed multiplicities?
• Given an infinite set S of "zeros", how can one construct a function whose zeros are precisely the elements of S? How to ensure these zeros have prescribed multiplicities?
• How does the Weierstraß product theorem entail a representation formula for entire functions?
  

§5. Picard's little and great theorem.

• Is a nonconstant entire function "almost" surjective (Picard's little theorem)?
• What are examples of surjective nonconstant entire functions? What are examples of nonconstant entire functions that omit precisely one value?
• How does a holomorphic function behave around an essential singularity (Picard's great theorem)?
• How can one quantify the "size" of the image of certain holomorphic functions (Bloch's theorem)?
• How to find logarithms and primitives on simply connected domains? Give examples of non-simply connected domains which do not admit a unique logarithm and a unique n-th root.
• Which representation formulas hold for for holomorphic functions omitting precisely two values?
• Which compactness properties hold for for holomorphic functions omitting precisely two values (sharpened Montel theorem)?
  

§6. The Riemann mapping theorem.
§7. Holomorphic functions on the Riemann sphere.
§8. An introduction to complex analysis in several variables.

References.
R. Remmert. Classical topics in complex function theory. Springer, New York, 1998.
C. Laurent-Thiébaut. Holomorphic function theory in several variables: an introduction. Springer, London, 2011.