1. Calculate and plot eZ for the following points z:
(a) 0 (c) 7r(i -1)/3 (e) 7ri/m, m = 1,2,3, …
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(b) 7ri + 1 (d) 377ri (f) m(i -1), m = 1,2,3, …
2. Sketch each of the following figures and its image under the expo-
nential map w = eZ • Indicate the images of horizontal and vertical
lines in your sketch.
(a) the vertical strip 0 < Rez < 1,
(b) the horizontal strip 57r /3 < 1m z < 87r /3,
(c) the rectangle 0 < x < 1, 0 < y < rr/4,
(d) the disk JzJ ~ 7r/2,
(e) the disk JzJ ~ 7r,
(f) the disk JzJ ~ 37r /2.
3. Show that e Z = e Z •
4. Show that the only periods of e Z are the integral multiples of 2rri,
that is, if eZ+’x = eZ for all z, then>. is an integer times 27ri.
6. The Logarithm Function
For z -lOwe define log z to be the multivalued function
logz 10gJzJ+iargz
= log JzJ + i Arg z + 27rim, m = O,±1,±2, ….
The values of log z are precisely the complex numbers w such that eW = z.
To see this, we plug in and compute. If w = log JzJ + i Arg z + 2rrim, then
