Matematik 2  Forår 2005
Reelle og Komplekse Funktioner
14. kursusgang
Tuesday, April 19, 2005, 8:15
Room G5110
Schedule

8:158:45

Review in G5110. 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.

8:4510:45

Problem session. Work in groups.

10:4512:00

Lecture in G5110. 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 z_{0} ∈ G and define the set:
S={z∈ G z can be connected by a polygonal circuit γ_{z}, in G, to z_{0}}.
Show that S is both open and closed in G.

Redo #2.1.7, using Theorem 3.16.
Updated April 15, CD.