Math 417, Fall 2011
Exercise Set #5
Turn in: October 5
1.
Let
C
denote the line segment from
z
=
i
to
z
= 1. Show that
C
dz
z
4
≤
4
√
2
.
2.
Let
C
R
denote the upper half circle for

z

=
R
, for
R >
1, parametrized in the
counterclockwise direction. Show that
C
R
Log
(
z
)
z
2
dz
≤
2
π
π
+ ln
R
R
.
Conclude that the integral tends to zero as
R
tends to infinity.
3.
Suppose that
f
(
z
) and
g
(
z
) are analytic functions defined in a domain
Ω
, and that
C
is a contour in
Ω
starting at
z
1
and ending at
z
2
. Show that:
C
f
(
z
)
g
(
z
)
dz
=
f
(
z
2
)
g
(
z
2
)
−
f
(
z
1
)
g
(
z
1
)
−
C
g
(
z
)
f
(
z
)
dz
For the next three problems, all integrals can be done using results from class or the
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 Staff
 Math, dz

Click to edit the document details