Stephen Taylor
June 2, 2005
1. For each of the following functions, classify the behavior at
∞
(i.e., analytic,
a zero of order
n
, pole of order
m
, or an essential singularity)
(a)
f
(
z
) =
z

1
z
+1
. We consider
f
(
z

1
) =
1

z
1 +
z
which is analytic at
z
= 0. So
f
is analytic at
∞
.
(b)
f
(
z
) =
z
3
+
i
z
We consider
f
(
z

1
) =
1
z
2
+
iz
=
1 +
iz
3
z
2
which has a pole of order two at 0. So we conclude
f
has a pole of order 2 at
∞
.
(c)
f
(
z
) =
z
z
3
+
i
. We consider
f
(
z

1
) =
z
2
1 +
iz
3
which has a zero of order 2 at 0. So we conclude
f
has a zero of order 2 at 0.
(d)
f
(
z
) =
e
z
. Since
f
(
z

1
) =
e
1
/z
, we ﬁnd from the Laurant series expan
sion that this function has an essential singularity at 0. So
f
has an essential
singularity at
∞
.
(e)
f
(
z
) =
sin
z
z
2
We note the Laurant series of
f
(
z

1
) has an essential singularity
at 0, so
f
has an essential singularity at
∞
.
(f)
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 Three '09
 Cong
 Math, Essential singularity, Stephen Taylor

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