Credits: 7.5

Type: bridging

Objectives:

• To introduce the basic properties of local and global holomorphic functions.
• To introduce a key result, the conformal equivalence of simply connected domains in the complex plane.

Knowledge:

• Local Cauchy theory. Zeros of holomorphic functions.
• Laurent series and isolated singularities. Calculating residues.
• Elementary conformal transformations. Schwarz's lemma. Automorphisms.
• Normal families and Riemann's theorem.

Subject-specific competences:

• To understand and describe holomorphic functions and be able to differentiate them from real differentiable functions.
• To gain in-depth knowledge of isolated singularities and be able to apply them.
• To calculate the conformal representations between simply connected regions in the plane.
• To be able to describe the convergence of holomorphic functions over a region.