Matematik 2 - Forår 2005

Reelle og Komplekse Funktioner

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 γ_z, in G, to z₀}. Show that S is both open and closed in G.
• Redo #2.1.7, using Theorem 3.16.