MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION B)
Name:
This test has 6 pages and 6 problems.
No calculators are allowed. You are permitted only a pencil or pen.
Work out everything as far as you can before making decimal approximations.
The Cauchy–Riemann equations are
∂u
∂v
=
∂x
∂y
∂u
∂v
=−
∂y
∂x
1. What are all of the values of i2i in the form x + iy?
Date: June 8, 2001.
1
2
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION B)
2. Calculate
in the form x + iy.
2+i
3−i
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION B)
3
2
1
–1
–0.5
0.5
1
–1
–2
Figure 1. The images of vertical lines under the transformation w = z 2
3. Show that under the map
w = f (z) = z 2
vertical lines in the z plane become parabolas in the w plane,
as in figure 1.
4
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION B)
4. At which points z does the function
f (z) = |z|2
satisfy the Cauchy–Riemann equations? Explain your answer.
MATH 3160: APPLIED COMPLEX VARIABLES
5. (a) What are all of the values of
log i3 ?
(b) What are all of the values of
3 log i ?
TEST #1 (VERSION B)
5
6
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION B)
6. At what points z does the function
f (x) = sin z̄
satisfy the Cauchy–Riemann equations?