Math 536 HOMEWORK 8 (due Friday, June 6) Spring 2008
I. Do 16.5: 1, 2 (3+3)
II. Do 5.8 (p.163): 7, 8 (3+3)
III. Consider
f(z) = Zz
0
ew2dw.
(a) (3 points) Prove fis entire and maps the plane onto the plane. (Hint: f0is even)
(b) (3 points) Show that fis locally 1-to-1 but not 1-to-1 in the plane.
(c) (2 points) Explain what is wrong with the following argument:
If γis any curve in the plane, then a local inverse of fcan be defined in a neighborhood of
each point of γ. Since the plane is simply connected, a global inverse f−1of fcan be defined
in the plane by the Monodromy Theorem.
1