MATH 3160: APPLIED COMPLEX VARIABLES
TEST #2 (VERSION A)
Name:
This test has 5 pages and 5 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. (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
Date: May 2, 2001.
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MATH 3160: APPLIED COMPLEX VARIABLES
TEST #2 (VERSION A)
2. Calculate the integral
Z
|z|=5
1 − cos z
dz .
sin z
(where the circle |z| = 5 is positively oriented).
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #2 (VERSION A)
CR
Cρ
L1
L2
Figure 1. A contour, with ρ (the radius of Cρ ) small, and R (the
radius of CR ) large.
3. Show that
Z
∞
−∞
sin x
dx = π
x
by integration of the function
eiz
z
along a contour like that shown in figure 1.
f (z) =
3
4
MATH 3160: APPLIED COMPLEX VARIABLES
4. Calculate
Z
TEST #2 (VERSION A)
dz
sin(1/z)
|z|=100
where the circle |z| = 100 is positively oriented.
MATH 3160: APPLIED COMPLEX VARIABLES
5. Bonus: Calculate
Z
0
∞
dx
x1/4 (x
+ 4)
TEST #2 (VERSION A)
.
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