Matematik 2 - Forår 2005

Reelle og Komplekse Funktioner

13. kursusgang

Thursday, April 14, 2005,  8:15
Room G5-112


Review in G5-112.  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.
Problem session. Work in groups.
Lecture in G5-112. We will define contour integrals and discuss some of their properties.


1. Prove Euler's formula:
exp(iz)=cos(z) + i sin(z), for all zC.
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=1-i and diverges at z=5. What can you say about its radius of convergence? (i.e., find inequalities of the type ra, br).
5. Find the zeros of the function sin(z).

Updated April 10, CD.