Matematik 2  Forår 2005
Reelle og Komplekse Funktioner
13. kursusgang
Thursday, April 14, 2005, 8:15
Room G5112
Schedule

8:158:45

Review in G5112. We will review results on power series such as
differentiability and radius of convergence, and give examples. We will also
focus on the relationship between trigonometric functions and the exponential
function.

8:4510:45

Problem session. Work in groups.

10:4512:00

Lecture in G5112. We will define contour integrals and discuss some of their
properties.
Problems
 1. Prove Euler's formula:
 exp(iz)=cos(z) + i sin(z), for all z∈C.

 2. (a) Show that exp(z) has no zeros in C.
 (b) Find the solutions to the equation: exp(z)=1.

 3. Find the radius of convergence for the series
.
 4.Suppose
we know that the series
converges at z=1i and diverges at z=5. What can you say about its radius of
convergence? (i.e., find inequalities of the type r≤a,
b≤r).
 5. Find the zeros of the function sin(z).
Updated April 10, CD.