Math 120A Homework
IX Due July 31, page 149 #2abc, 3.
page 160 #1abcef (not d), 2abc, 3.
page 170 #1abc, 6, 10.
This problem is to do what we can towards evaluating the integral on page 267 #8 with the
tools that we now have. Let I denote the value of the integral of the problem. The integral is real.
Our first step is to change the contour to the contours CR depending on R shown on page 268 in
Figure 95. For the integrand, we substitute z for x ; we could say we analytically continue the
integrand from the positive real axis to the upper half plane except for the singularity at
z
1

3

i.

2
2

(a) Show that as R   , the integral from 0 to R tends to
 1
3 
2 i / 3
I . (Take
ito 0 , tends to e
2
2


as small
necessary. Be sure to parameterize the straight line toward the
 and numerous steps as 
origin.)
 circular arc tends to 0 . (Use the ML estimate
(c) Show

 that as R   , the integral along the 
on the magnitude of the integral.)
(d) On the other hand, from corollary on page 159, show that each integral over a CR
with R  1 has the same value as the integral over a small positively oriented circle centered
i / 3

e
at z 
.
(e) Conclude that we could easily find I , if we just knew what number was obtained by
integrating about the singularity as we indicated in part (d).

(b) Show that as R   , the integral from R




I.
page 179 #3, 4.


X. Due August 2, 2012 page 195 #2, 3, 4, 7, 11ab, 13.
page 205 #2, 3, 4, 7.
page 239 #1, 3.
For the final, study page 267 #1, 2, 8, and page 275 #2, 3, 4.
MATH 120A FINAL EXAMINATION FRIDAY, AUGUST 3, 2012
7:00 – 9;59 PM AP&M B412