MATH 3160: APPLIED COMPLEX VARIABLES
TEST #2 (VERSION B)
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. For contour
integrals, all contours are oriented counterclockwise.
1. Find
Z
z cos(z)
dz.
|z|=1 sin(z)
Date: July 19, 2001.
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2
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #2 (VERSION B)
2. Consider the integral
Z
I=
0
∞
xα
2
(1 + x2 )
dx.
(a) Explain how to set up an integration along a branch cut for this problem. Include a picture of the contour you want to use.
(b) Show that as long as −1 < α < 3 and α 6= 1 and α 6= 2 then the
integral I is finite.
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #2 (VERSION B)
3. (a) Calculate the Laurent expansion about z = 0 of
ez
sin z
up to z 2 terms.
(b) Calculate
ez
Res
z=0 sin z
3
4
MATH 3160: APPLIED COMPLEX VARIABLES
4. Calculate the integral
Z
|z|=1
dz
.
z 3 cosh z
TEST #2 (VERSION B)
MATH 3160: APPLIED COMPLEX VARIABLES
5. Calculate the integral
Z
|z|=1
z dz
.
cosh (1/z)
TEST #2 (VERSION B)
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