Homework 10
Stephen Taylor
June 1, 2005
Page 110

111
1.
Each of the following functions
f
has an isolated singularity at
z
= 0.
Determine its nature; if it is a removable singularity defined
f
(0) so that
f
is analytic at
z
= 0; if it is a pole find the singular part; if it is an essential
singularity determine
f
(
{
z
: 0
<

z

< δ
}
) for arbitrarily small values of
δ
.
(a)
f
(
z
) =
sin
z
z
This function has a removable singularity at
z
= 0 and
f
(0) = 1.
(b)
f
(
z
) =
cos
z
z
This function has a pole at
z
= 0 with singular part 1
/z
.
(c)
f
(
z
) =
cos
z

1
z
This function has a removable singularity at
z
= 0 and
f
(0) = 0.
(d)
f
(
z
) = exp(
z

1
)
This function has an essential singularity at
z
= 0.
f
(
{
z
: 0
<

z

< δ
}
) assumes
every value in the set
C
 {
0
}
an infinite number of times by Picard’s Great
Theorem.
(e)
f
(
z
) =
log(
z
+1)
z
2
This function has a pole at
z
= 0 with singular part 1
/z
.
(f)
f
(
z
) =
cos(
z

1
)
z

1
This function has an essential singularity at
z
= 0.
f
(
{
z
: 0
<

z

< δ
}
) assumes
every value in the set
C
an infinite number of times by Picard’s Great Theorem.
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 Three '09
 Cong
 Math, Essential singularity, removable singularity, Stephen Taylor

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