Lecture 44:
Residues
Dan Sloughter
Furman University
Mathematics 39
May 21, 2004
44.1
Some terminology
Recall that we say a point
z
0
is a singular point of a function
f
if
f
is
not analytic at
z
0
but is analytic at some point in every neighborhood of
z
0
. We will say that
z
0
is an
isolated
singular point if it is a singular point
and there exists
>
0 such that
f
is analytic in the deleted neighborhood
0
<

z

z
0

<
.
Example 44.1.
Both
z
=
i
and
z
=

i
are isolated singular points of
f
(
z
) =
1
1 +
z
2
.
Example 44.2.
z
= 0 is a singular point, but not an isolated singular point,
of
f
(
z
) = Log(
z
).
If
z
0
is an isolated singular point of
f
, then there exists
R >
0 such that
f
is analytic in
D
=
{
z
∈
C
: 0
<

z

z
0

< R
}
. It follows that
f
(
z
) has a
Laurent series representation for all
z
∈
D
:
f
(
z
) =
∞
n
=0
a
n
(
z

z
0
)
n
+
∞
n
=1
b
n
(
z

z
0
)
n
.
In particular
b
1
=
1
2
πi
C
f
(
z
)
dz,
1
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where
C
is any positively oriented, simple closed contour which lies in
D
and
has
z
0
in its interior. In other words,
C
f
(
z
)
dz
= 2
πib
1
.
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 Spring '04
 DanSloughter
 Math, Z0, Laurent, singular point, isolated singular point

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