April 20-24, 2009 Week #12 Practice Quiz Name:
Math 412
SOLUTIONS
Problem 1. (4 points each) Evaluate each contour integral below:
(a):
ZC
cos z
2dz,
where Cis the line segment from z= 0 to z=π+ 2i.
SOLUTION: Since fis entire, we can apply Theorem 6.9: We have
ZC
cos z
2dz = 2 sin z
2
π+2i
0
= 2 sin π
2+i= 2 cosh 1,
where we have used identity (5-33) on p. 178 of the text. 2
(b):
ZC
cos z
z(z2+ 8)dz,
where Cis the square with vertices (±2,±2), with counterclockwise orientation.
SOLUTION: The function f(z) = cos z
z2+ 8 is analytic in and on C, so we apply Cauchy’s Integral
Formula with z0= 0 to get
2πif (0) ·1
8=iπ
4.
2
(c):
ZC
4ez
z2−2z−3dx,
where Cis a circle of radius 5centered at the origin and with clockwise orientation.
SOLUTION: We write 1
z2−2z−3=A
z−3+B
z+ 1.
Cross-multiplying, we find that A(z+ 1) + B(z−3) = 1, so that A+B= 0 and A−3B= 1. We
quickly find that
A=1
4and B=−1
4.
Thus, we have
ZC
4ez
z2−2z−3dz =ZC
ez
z−3dz −ZC
ez
z+ 1dz =−Z−C
ez
z−3dz +Z−C
ez
z+ 1dz,
