Graphics Reference
In-Depth Information
Definition.
The integer m is called the
order
of the pole of f at
a
.
Since
1
f
()
=
z
,
()
g
z
it follows that
m
zza
-
()
=-
(
)
( )
f
h
z
,
where h(
z
) is analytic in a neighborhood of
a
and h(
a
) π
0
.
Definition.
A
rational
function is a quotient of two polynomials that have no
common root.
Definition.
A
meromorphic
function is a function that is analytic on an open set
except possibly for poles.
E.4.2. Example.
Rational functions are a special case of meromorphic functions.
E.4.3. Example.
If f(
z
) and g(
z
) are analytic functions and if g(
z
) is not identically
0
, then f(
z
)/g(
z
) is a meromorphic function.
Basic facts about series can be applied to complex series with negative powers of
z
such as
•
Â
-
n
az
.
(E.5)
n
n
=
0
By replacing
z
with 1/
z
one concludes that the series (E.5) converges for all
z
with
z
> |R| for some R and is an analytic function there. Therefore, we can talk about
series of the form
•
Â
n
(E.6)
az
,
n
n
=-•
where we say that it is convergent if the two parts, the part with positive powers of
z
and the part with negative powers of
z
, are separately convergent. More generally, we
can consider series of the form (E.6) where we expand about some arbitrary point
z
0
.
Such series, if they converge, will then converge in some annulus about
z
0
and define
an analytic function there. The following is a converse to this.
E.4.4. Theorem.
Let 0 £ R
1
< R
2
and assume that f(
z
) is analytic for all
z
in the
annulus R
1
< |
z
-
z
0
| < R
2
. Then f(
z
) has a unique representation in that region of the
form
•
•
a
n
Â
n
Â
()
=
(
)
f
z
+
bz z
-
,
(E.7)
n
0
n
(
)
zz
-
n
=
1
n
=
0
0
