MATH 3160: APPLIED COMPLEX VARIABLES
FINAL EXAM (VERSION C)
Name:
This test has 8 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.
1. Find all solutions z to the equation
tanh z = 1 .
Date: May 6, 2002.
1
2 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION C)
2. (a) Calculate the Laurent expansion about z = 0 of
1
2
z sin
z2
up to 1/z 4 terms.
(b) Calculate
2
Res z sin
z=0
1
z2
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION C) 3
3. Calculate the integral
Z
cos z
dz
2
C z (z + 8)
where C is the square with corners at the four points ±2 ± 2i.
4 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION C)
4. Calculate the integral
Z
|z|=1
cos (1/z) dz
.
z 2 sin (1/z)
Hint: why can’t you just use residues to get it? What happens
if you use the coordinate change w = 1/z?
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION C) 5
5. Show that the map
w=
z−1
z+1
1/2
(with the principal branch of the square root) takes the z plane
slit along the segment −1 ≤ x ≤ 1 onto the right half plane
u > 0 (where z = x+iy and w = u+iv). Hint: it is a composite
function. Figure out where the linear fractional transformation
Z = (z − 1)/(z + 1) takes the upper and lower half planes, and
where it maps the segment −1 ≤ x ≤ 1.
6 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION C)
1
0.5
–1
–0.5
0.5
1
1.5
2
–0.5
–1
6. Using conformal mapping, find the harmonic function f which
is defined outside of the two circles drawn in figure 6 (unfortunately they don’t quite look like circles, because of some problems with graphics), with f = 0 on the little circle, and f = 1
on the big circle. (The answer looks like figure 6 on the next
page.) Show that the isotherms and flow lines are circles, and
explain why they are perpendicular to each other.
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION C) 7
1
0.5
0
2
2
1
1
y
0
0
–1
–1
–2
–2
x
8 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION C)
7. Suppose that f (z) is an entire analytic function, and that the
real part of f (z), which we will call u(x, y), has an upper bound:
u(x, y) ≤ u0
for any x, y. Show that f (z) constant. Hint: consider g(z) =
ef (z) .