**for**

**7.2 Taylor Series
Representations**

In Section
4.4 we showed that functions defined by power series have
derivatives of all orders (Theorem 4.16). In Section
6.5 we demonstrated that analytic functions also have derivatives
of all orders (Corollary 6.2). It seems natural,
therefore, that there would be some connection between analytic
functions and power series. As you might guess, the
connection exists via the __Taylor__
and __Maclaurin__
series of analytic functions.

**Definition 7.2 (**__Taylor
Series__**).** If is
analytic at ,
then the series

is called the Taylor series for f(z)
centered at . When
the center is ,
the series is called the Maclaurin series for f(z).

To investigate when these series converge we will need the following lemma.

**Lemma
7.1.** If are
complex numbers with , and , then

where n is a positive integer.

**Proof of Lemma 7.1 is in the book.
**

We are now ready for the main result of this section.

**Theorem 7.4 (**__Taylor's
Theorem__**).**
Suppose f(z) is analytic in a domain
G, and that
is any disk contained in G. Then
the Taylor series for f(z) converges
to f(z) for all z in ; that
is,

for
all .

Furthermore, for any r, 0<r<R, the convergence is uniform on
the closed subdisk .

**Proof of Theorem 7.4 is in the book.
**

A singular point of a function is a point at which the function fails to be analytic. You will see in Section 7.4 that singular points of a function can be classified according to how badly the function behaves at those points. Loosely speaking, a nonremovable singular point of a function has the property that it is impossible to redefine the value of the function at that point so as to make it analytic there. For example, the function has a nonremovable singularity at z=1. We give a formal definition of this concept in Section 7.4, but with this language we can nuance Taylor's theorem a bit.

**Corollary
7.3.** Suppose that f(z) is
analytic in the domain G that
contains the point . Let be
a nonremovable singular point of minimum distance to the
point . If , then

(i)** **the
Taylor series converges
to
on all of ,

and

(ii)** **if
, the
series does
not converges to .

**Proof of Corollary 7.3 is in the
book.
**

**Example 7.3.** Show
that is
valid for all .

Solution. In Example 4.24 (see Section
4.4) we established this identity with the use of Theorem
4.17. We now do so via Theorem
7.4. If , then
a standard induction argument (which we leave as an exercise) will
show that for . Thus , and
Taylor's theorem gives

,

and since f(z) is analytic in
,
this series expansion is valid for all .

The
disk and
it's images under the mappings:

, , and .

Remark.
The accuracy of the image points for the
approximation is

.

**Example 7.4**. Show
that, for ,

(7-12) (a)
and (b)
.

Solution. For ,

(7-13) .

If we let
take the role of z in (7-13), we get
that

,

for . But iff , thus
we have proven that for .

Next, let take
the role of z in Equation (7-13), we get
that

gives the second part of Equations
(7-12).

The
disk and
it's images under the mappings:

, , and .

Remark
1. The accuracy of the image points for the
approximation is

.

Remark
2. The images of under
the mappings: ,

, and will
appear like those shown above,

because rotates
the plane about the origin and .

Also,
the accuracy of the image points for the
approximation will
be

.

**Explore
Real Solution 7.4 (a).**

**Explore
Complex Solution 7.4 (a).**

**Explore
Real Solution 7.4 (b).**

**Explore
Complex Solution 7.4 (b).**

**Remark
7.1** Corollary 7.3 clears up what often seems to
be a mystery when series are first introduced in
calculus. The calculus analog of Equations
(7-12) is

(7-14) and for .

For many students, it makes sense that the first series in Equations
(7-14) converges only on the
interval because
is undefined at the points . It
seems unclear as to why this should also be the case for the series
representing ,
since the real-valued function
is defined everywhere. The explanation, of course, comes
from the complex domain. The complex function
is not defined everywhere. In fact, the singularities
of are
at the points , and
the distance between them and the point equals
1. According to Corollary 7.3,
therefore, Equations (7-14) are valid
only for ,
and thus Equations (7-14) are valid only
for the real numbers .

Alas, there is a potential fly in this ointment: Corollary 7.3 applies to Taylor series. To form the Taylor series of a function, we must compute its derivatives. We didn't get the series in Equations (7-12) by computing derivatives, so how do we know that they are indeed the Taylor series centered at ? Perhaps the Taylor series would give completely different expressions from those given by Equations (7-12). Fortunately, Theorem 7.5 removes this possibility.

**Theorem 7.5 (Uniqueness of
Power Series).** Suppose that in some disk
we have

.

Then .

**Example 7.5.** Find
the Maclaurin series for .

Solution. Computing derivatives for f(z)
would be an onerous task. Fortunately, we can make use of the
trigonometric identity

.

Recall that the series for sin z (valid for all z)
is . Using
the identity for , we
obtain

By the uniqueness of power series, this last expression is the
Maclaurin series for .

In the preceding argument we used some obvious results of power series representations that we haven't yet formally stated. The requisite results are part of Theorem 7.6.

**Theorem 7.6.** Let
f(z) and g(z)
have the power series representations

,

and

.

If is
any complex constant, then

(7-15)** **,

(7-16)** **, and

(7-17)** **, where

(7-18)** **.

Identity (7-17) is known as the
Cauchy
product of the series for f(z) and
g(z).

**Proof of Theorem 7.6 is in the book.
**

**Example 7.6.** Use
the Cauchy product of series to show that

for .

Solution. We let , for
. In
terms of Theorem 7.6, we have , for
all n, and thus Equation (7-17)
gives

**Extra Example
1.** Use the Cauchy product of series to show
that

for .

**Explore
Solution for Extra Example 1.**

**Extra Example
2.** Show that for .

Solution. Use the result of Example 7.6 and
(7-16) and obtain

**Explore
Solution for Extra Example 2.**

**Exercises
for Section 7.2. Taylor Series
Representations****
**

**The Next Module is
**

**Return to the Complex
Analysis Modules**

__Return
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