24 I The Complex Plane and Elementary Functions
Exercises for 1.6
1. Find and plot log z for the following complex numbers z. Specify
the principal value. (a) 2, (b) i, (c) 1 + i, (d) (1 + iv3)/2.
2. Sketch the image under the map w = Log z of each of the following
figures.
(a) the right half-plane Rez > 0,
(b) the half-disk Izl < 1, Re z > 0,
(c) the unit circle Izl = 1,
(d) the slit annulus vie < Izl < e2 , z rt- (_e 2 , -vie),
(e) the horizontal line y = e,
(f) the vertical line x = e.
3. Define explicitly a continuous branch of log z in the complex plane
slit along the negative imaginary axis, C\[O, -ioo).
4. How would you make a branch cut to define a single-valued branch
of the function log(z + i-I)? How about log(z – zo)?
