MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION A)
Name:
This test has 7 pages and 7 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. (10 points) Calculate
3+i
√
√
2 − 2i + |1/ 2 + i/ 2|
Date: May 2, 2001.
1
2
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION A)
2. (10 points) Prove that
Re z ≤ |z|
for any complex number z.
MATH 3160: APPLIED COMPLEX VARIABLES
3. (10 points) Find the roots z of
z4 + 9 = 0
TEST #1 (VERSION A)
3
4
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION A)
4. (10 points) What is the value of the limit
z2
?
z→0 |z|2
lim
Explain your answer.
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION A)
5
5. (10 points) Find all complex numbers z which solve the equation
√
2 = cos z .
6
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION A)
6. (10 points) What are all of the values of ii ?
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION A)
7
7. (10 points) Bonus: Derive the polar coordinate expressions
of the Cauchy–Riemann equations from the rectangular ones
given on page 1 above.