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Mathematics 185 – Intro to Complex Analysis
Fall 2005 – M. Christ
Problem Set 11
Please ﬁnish reading Chapter 12. Our emphasis in this chapter is on applying the
theory we’ve developed to concrete calculations and problems. We’ll do still more
problems from this chapter in a future problem set.
Due
Tuesday November 22:
Solve these problems from Chapter 12: 5(iv,v), 11,12,13, 16(iii), 17(i), 19.
Hints
:
5(v): You may use without proof the related fact that
R
∞
0
log(
x
)
1+
x
2
dx
= 0 (the
substitution
x
= 1
/y
expresses this integral as minus one times itself!).
Method (a): Substitute
x
=
e
t
to get an integral over the whole real line. Then apply
the method we used in class to compute
R
∞
∞
e
ax
1+
e
2
x
dx
. Note that the “related fact”
mentioned above translates to
R
∞
∞
te
t
1+
e
2
t
dt
= 0.
Method (b): Let Log denote the branch of the complex log function deﬁned by
Log(
re
iθ
) = ln(
r
) +
iθ
for

π
2
< θ <
3
π
2
. Express
R
0
∞
Log(
x
)
2
1+
x
2
dx
in terms of the
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This note was uploaded on 07/31/2009 for the course MATH 185 taught by Professor Lim during the Spring '07 term at University of California, Berkeley.
 Spring '07
 Lim
 Math

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