MATH 3160: APPLIED COMPLEX VARIABLES
FINAL EXAM (VERSION B)
Name:
This test has 7 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.
1. Find all solutions z to the equation
tan z = 1 .
Date: June 5, 2001.
1
2 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION B)
2. (a) Calculate the Laurent expansion about z = 0 of
cos z
sin z
up to z 2 terms.
(b) Calculate
cos z
Res
z=0 sin z
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION B) 3
3. Calculate the integral
Z
∞
−∞
x2
dx .
x4 + 1
4 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION B)
4. Consider the integral
Z
|z|=R
√
e−1/z z 2 − 1
dz .
z3
√
where the branch of z 2 − 1 is any branch defined in the whole
plane except on the real line between x = −1 and x = 1, and
the radius R is bigger than 1.
(a) Show that the integral is less than 4πe/R.
(b) Show that the value of the integral with R = 2 is
√
Z
e−1/z z 2 − 1
dz = 0 .
z3
|z|=2
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION B) 5
Figure 1. The potential on the boundary of the unit disk
1
0.8
0.6
0.4
0.2
0
–1
–1
–0.5
–0.5
0x~
y~ 0
0.5
0.5
1
1
Figure 2. The potential inside the unit disk
5. Find the potential U (x, y) inside the unit circle so that U = 1 on
a three quarter arc of the circle, and U = 0 on the remaining one
quarter, as drawn in figure 1. The function U (x, y) is drawn in
figure 2. Hint: you could use a linear fractional transformation
taking 1, i, −1 to 0, ∞, −1.
6 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION B)
(continued)
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION B) 7
6. Calculate the integral
Z
0
2π
dθ
.
2 + cos θ