# Matematik 2 - Forår 2005

## Reelle og Komplekse Funktioner

14. kursusgang

Tuesday, April 19, 2005,  8:15
Room G5-110

#### Schedule

8:15-8:45
Review in G5-110.  We will give more examples of circuits, after which we will mention Theorem 3.16 (the contour integral of a continuous function which admits a primitive depends only on the endpoints of the circuit), and use it to evaluate contour integrals.
8:45-10:45
Problem session. Work in groups.
10:45-12:00
Lecture in G5-110. We will prove Cauchy's theorem for a starshaped domain (which implies that any holomorphic function in a starshaped domain has a primitive). If time permits, we will also derive Cauchy's integral formula.

#### Problems

From AJ: 3.1.2, 3.1.3, 3.1.4.
• Verify that the chain rule in the real case [Wade, Chap. 4] is valid in the case F(γ(t)), where γ:[a,b]→G is R differentiable and F is holomorphic in G.
• Show that any open and connected set G⊂C is polygonally connected, i.e., any two points can be connected by a polygonal circuit in G.
Hint: Choose z0 ∈ G and define the set: S={z∈ G| z can be connected by a polygonal circuit γz, in G, to z0}. Show that S is both open and closed in G.
• Redo #2.1.7, using Theorem 3.16.

Updated April 15, CD.