**for**

**7.3 Laurent Series
Representations**

Suppose f(z)
is not analytic in , but
is analytic in . For
example, the function is
not analytic when but
is analytic for . Clearly,
this function does not have a __Maclaurin
series__ representation. If we use the
__Maclaurin__
series for , however,
and formally divide each term in that series by ,
we obtain the representation

that is valid for all z such
that .

**Extra Example
1.** Use *Mathematica* to find the series
for .

**Explore
Solution for Extra Example 1.**

This example raises the question as to
whether it might be possible to generalize the __Taylor__
series method to functions analytic in an annulus

.

Perhaps we can represent these functions with a series that employs
negative powers of z in some way as
we did with . As
you will see shortly, we can indeed. We begin by defining
a series that allows for negative powers of z.

**Definition 7.3 (**__Laurent
Series__**).** Let be
a complex number for . The
doubly infinite series , called
a __Laurent__
series, is defined by

(7-21) ,

provided the series on the right-hand side of this equation
converge.

**Remark 7.2.** Recall
that is
a simplified expression for the sum . At
times it will be convenient to write as

,

rather than using the expression given in Equation
(7-21).

**Definition
7.4.** Given , we
define the annulus centered at with
radii r and R
by

.

The closed annulus centered at with radii r
and R is denoted by

.

Figure 7.3 illustrates these terms.

Figure 7.3The closed annulus . The shaded portion is the open annulus .

**Theorem
7.7.** Suppose that the Laurent
series converges
on an annulus . Then
the series converges uniformly on any closed
subannulus
where .

**Proof of Theorem 7.7 is in the book.
**

The main result of this section specifies
how functions analytic in an annulus can be expanded in a __Laurent__
series. In it, we will use symbols of the
form , which
- we remind you - designate the positively oriented circle with
radius and
center . That
is, , oriented
counterclockwise.

**Theorem 7.8 (Laurent's
Theorem).** Suppose , and
that f(z) is analytic in the annulus
. If is
any number such that ,
then for all
the function value
has the Laurent series representation

(7-22) ,

where for ,
the coefficients
are given by

(7-23) and .

**Proof of Theorem 7.8 is in the book.
**

**Remark.** What
happens to the Laurent series if f(z)
is analytic in the disk ? If
we look at equation (7-23), we see that
the coefficient
for the positive power
equals
by using Cauchy's integral formula for derivatives. Hence,
the series in equation (7-22) involving
the positive powers of
is actually the

Taylor series for f(z). The
__Cauchy____-____Goursat__
theorem shows us that the coefficients for the negative powers of
equal zero. In this case, therefore, there are no negative
powers involved, and the Laurent series reduces to the Taylor
series.

Theorem 7.9 delineates two important aspects of the Laurent series.

**Theorem 7.9 (Uniqueness and
differentiation of Laurent Series).** Suppose
that is
analytic in the annulus , and
has the Laurent series representation

for
all .

(i)
If for
all , then
for all n.

(In
other words, the Laurent series for f(z) in
a given annulus is unique.)

(ii)** **For
all ,
the derivatives for may
be obtained by termwise differentiation of its Laurent series.

**Proof of Theorem 7.9 is in the book.
**

The uniqueness of the Laurent series is an important property because the coefficients in the Laurent expansion of a function are seldom found by using Equation (7-23). The following examples illustrate some methods for finding Laurent series coefficients.

**Example 7.7.** Find
three different Laurent series representations for the
function involving
powers of z.

Solution. The function f(z)
has singularities at and
is analytic in the disk , in
the annulus , and
in the region . We
want to find a different Laurent series for f(z)
in each of the three domains D,
A, and R. We
start by writing f(z) in its partial
fraction form:

(7-30) .

We use Theorem 4.12 and Corollary 4.1 to obtain the following
representations for the terms on the right side of Equation
(7-30):

Representations (7-31) and
(7-33) are both valid in the
disk , and
thus we have

valid
for ,

which is a Laurent series that reduces to a Maclaurin series.

In the annulus , representations
(7-32) and
(7-33) are valid; hence we
get

valid for

Finally, in the region we
use Representations (7-32) and
(7-34) to obtain

valid
for .

**Example 7.8.** Find
the Laurent series representation for that
involves powers of z.

Solution. We know that , and
hence the Maclaurin series for is

,

then we can write

or in another way we can write

We formally divide each term by
to obtain the Laurent series

**Example 7.9.** Find
the Laurent series for centered
at .

Solution. The Maclaurin series
for exp
z is ,
which is valid for all z. We
let take
the role of z in this equation to get

,

which is valid for .

**Exercises
for Section 7.3. Laurent Series
Representations****
**

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**

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Analysis Modules **

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to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell