1. Find all values and plot: (a) (l+i)i, (b) (_i)l+i , (c) 2- 1/ 2 , (d) (1+
iV3)(I-i) .
2. Compute and plot log [(1 + i)2i]. 28 I The Complex Plane and Elementary Functions
3. Sketch the image of the sector {O < arg z < 7r16} under the map
w = za for (a) a = ~, (b) a = i, (c) a = i + 2. Use only the
principal branch of za.
4. Show that (zw)a = zawa, where on the right we take all possible
products.
5. Find iii. Show that it does not coincide with ii.i = i-I.
6. Determine the phase factors of the function za (1- z)b at the branch
points z = 0 and z = 1. What conditions on a and b guarantee that
za(1 – z)b can be defined as a (continuous) single-valued function
on C\[O, I]?
7. Let Xl < X2 < … < Xn be n consecutive points on the real axis.
Describe the Riemann surface of y'(z – Xl)··· (z – xn). Show that
for n = 1 and n = 2 the surface is topologically a sphere with certain
punctures corresponding to the branch points and 00. What is it
when n = 3 or n = 4? Can you say anything for general n? (Any
compact Riemann surface is topologically a sphere with handles.
Thus a torus is topologically a sphere with one handle. For a given
n, how many handles are there, and where do they come from?)
8. Show that y’ z 2 -liz can be defined as a (single-valued) continu-
ous function outside the unit disk, that is, for Izl > 1. Draw branch
cuts so that the function can be defined continuously off the branch
cuts. Describe the Riemann surface of the function.
9. Consider the branch of the function y’z(z3 – 1)(z + 1)3 that is pos-
itive at z = 2. Draw branch cuts so that this branch of the function
can be defined continuously off the branch cuts. Describe the Rie-
mann surface of the function. To what value at z = 2 does this
branch return if it is continued continuously once counterclockwise
around the circle {izi = 2}?
10. Consider the branch of the function y’z(z3 -1)(z + 1)3(z -1) that
is positive at z = 2. Draw branch cuts so that this branch of the
function can be defined continuously off the branch cuts. Describe
the Riemann surface of the function. To what value at z = 2 does
this branch return if it is continued continuously once counterclock-
wise around the circle {Izl = 2}?
11. Find the branch points of .vz3 -1 and describe the Riemann
surface of the function
