Homework 10
Stephen Taylor
June 9, 2005
Page 83
3. Let
p
(
z
) be a polynomial of degree
n
and let
R >
0 be sufficiently large so
that
p
never vanishes in
{
z
:

z
 ≥
R
}
. If
γ
(
t
) =
Re
it
, 0
≤
t
≤
2
π
, show that
γ
p
(
z
)
p
(
z
)
dz
= 2
πin
We represent the partial fractions decomposition of
p /p
as
a
1
z

z
1
+
· · ·
+
a
n
z

z
n
Since we may choose
p
to be monic, we multiply through by it to find
p
(
z
) =
a
1
(
z

z
2
)
· · ·
(
z

z
n
) +
· · ·
+
a
n
(
z

z
1
)
· · ·
(
z

z
n

1
)
Equating leading coefficients we find that the sum of the
a
i
is
n
. So distributing
the line integrals over each term we find that the value of the desired integral is
2
πi
a
i
= 2
πin
Page 87
1.
Suppose
f
:
G
→
C
is analytic and define
φ
:
G
×
G
→
C
by
φ
(
z, w
) =
[
f
(
z
)

f
(
w
)](
z

w
)

1
if
z
=
w
and
φ
(
z, z
) =
f
(
z
). Prove that
φ
is continuous
and for each fixed
w
,
z
→
φ
(
z, w
) is analytic.
As we take the limit as
z
→
w
, we find that
φ
→
f
(
z
) by definition of the
derivative, so function is indeed continuous.
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 Three '09
 Cong
 Math, Calculus, Derivative, Continuous function, partial fractions decomposition, e−z dz z4

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