MATH 3160: APPLIED COMPLEX VARIABLES
FINAL EXAM (VERSION A)
1. Calculate
Arg
1
1+i
Solution: −π/4
2. Is u(x, y) = e−y cos x the real part of an analytic function? If
so, find the imaginary part of that function.
Solution:
v(x, y) = e−y sin x + c
where c is any real constant.
3. (a) Give two nonzero terms in the Laurent expansion of
cos (1/z) sin (1/z)
about z = 0.
(b) What sort of singular point does this function have at z =
0?
(c) Find
Z
cos(1/z) sin(1/z)dz .
|z|=1
Solution:
(a)
2 1
4 1
1 2 1
−
+
−
+ ···
3
5
z 3z
15 z
315 z 7
(b) It has an isolated essential singular point.
(c) The residue is 1 so
Z
cos(1/z) sin(1/z)dz = 2πi .
cos (1/z) sin (1/z) =
|z|=1
4. Show that
Z
∞
sin x
dx = π
−∞ x
by integration of the function
f (z) =
Date: April 24, 2001.
1
eiz
z
2 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A)
CR
L1
Cρ
L2
Figure 1. A contour, with ρ (the radius of Cρ ) small,
and R (the radius of CR ) large.
0
sin z
π/2
0
z) 1/2 (sin 0
1
Figure 2. The map w =
√
1
sin z
along a contour like that shown in figure 1.
Solution: See page 220 of Churchill & Brown.
5. Using the principal branch of the square root, let
√
Z = sin z .
Where does it map the half strip
0 < x < π/2, y > 0?
Draw the boundaries of this half strip, and show where they go
under this map.
Solution: See figure 2.
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A) 3
15
10
5
–1.5
–1
–0.5
0.5
x~
1
1.5
Figure 3. The stream lines of the flow whose complex
potential is F (z) = sin z
6. (a) Show that F (z) = sin z is the complex potential of a flow
that never leaves the half strip
−π/2 ≤ x ≤ π/2, y ≥ 0 .
(b) Calculate the equations for all of the stream lines. Write
them in the form of y as a function of x. (They should
look like the ones drawn in figure 3.)
(c) Show that the stream function is positive everywhere in
the half strip.
(d) Show from the definition of a stream line (level curves of
the stream function) that the only critical point of y as
a function of x on each stream line is at x = 0. Hint:
implicitly differentiate the equation ψ = c0 , where ψ is the
stream function.
(e) Show that the closest any stream line gets to the x axis is
at the point
q
2
x = 0, y = ln c0 + c0 + 1 .
4 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A)
(f ) Show that the stream line passing through the point
√
π
x = , y = ln 2
4
never goes closer to the x axis than a distance
√ !
1 + 17
.
ln
4
Solution:
(a) The flow never leaves this half strip because the velocity
V (z) = F 0 (z)
is imaginary at the left and right sides, so goes straight up
or down, and is real on the bottom, so travels left or right.
(b) The stream function is
ψ = Im F = sinh y cos x .
The stream lines are therefore
sinh y cos x = c0
or
y = ln
c0
±
cos x
r
!
c20
+1
cos2 x
Clearly we have to take the positive root:
!
r
c20
c0
y = ln
+
+1
cos x
cos2 x
(c) The stream function is sinh y cos x which is positive because
y > 0, so that sinh y > 0, and x is between −π/2 and π/2
so that cos x is positive.
(d) We find
sinh y sin x
y 0 (x) =
.
cosh y cos x
The only critical points are at sinh y = 0 or sin x = 0,
which occur only at y = 0 or x = 0. So the one stream line
along the real axis has critical points everywhere, but all
the others have only one critical point: at x = 0.
(e) The second derivative y 00 (x) at x = 0 is
y 00 (0) = c0 = sinh y > 0
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A) 5
which we see by implicit differentiation. Plugging in x = 0,
we get
q
2
y = ln c0 + c0 + 1 .
(f ) The stream line through x = π/4, y = ln 2 has constant
1
c0 =
4
which gives minimum height of this stream line
√ !
1 + 17
ln
.
4