Module 2: Complex Variables

Lesson 17

Taylor Series and Laurent Series

17.1 Taylor Series

The following theorem shows that a Taylor series can be found for an analytic
function.

17.1.1 Theorem: Let be analytic in a domain and let be any

point in . Then there is a unique Taylor series

where

and contains . This representation is valid in the largest open disk with
center in which is analytic. The remainder of (17.1.1) can be
represented as:

The coefficient satisfy the inequality , where is the maximum of

on a circle in whose interior is also in .

Proof: By Cauchy’s integral formula, we have
 Taylor Series and Laurent Series

for lying inside . Now

This expansion is valid for , we can do so as in on and we

choose inside the circle of radius with center , so that .

Thus

Using (17.1.5) in (17.1.4), we get

 Taylor Series and Laurent Series

This is Taylor’s formula with remainder term.

Since analytic functions have derivatives of all orders, we can take in (17.1.6)
as large as possible. If we let , we get (17.1.1). Clearly, (17.1.1) will
convergence and represent if and only if

Since is on and is inside Since is analytic inside

and on it is bounded, and so is the function , i.e.,

Also has the radius and the length

Hence by the ML-inequality, we get from (17.1.3)
 Taylor Series and Laurent Series

Now as lies inside . Hence the term on the right as .

Hence the convergence of the Taylor series is proved. Uniqueness follows since
power series have unique representation of functions.

Finally

17.1.3 Maclaurin’s Series

A Maclaurin’s series is a Taylor series with center That is,

A point at which is not differentiable but such that every disk with
center contains points at which is differentiable. We say that is
singular at or has a singularity at .

17.1.3 Theorem: A power series with nonzero radius of convergence is the

Taylor series of its sum.

Proof: Given the power series

 Taylor Series and Laurent Series

Then Now

Thus Further,

Thus

In general, With these coefficients the given series becomes

the Taylor’s series of .

17.1.4 Remark: Complex analytic functions have derivatives of all orders and
they can always be represented by power series of the from (17.1.1). This is not
true in general for real valued functions. In fact, there are real functions for
which derivatives of all orders exist but it cannot be represented by a power
series.

Consider for example, ,

This function cannot be represented by a Maclaurin’s series since all its

derivatives vanish at zero.

17.1.5 Examples:

1. . Then . Hence the Maclaurin’s

expansion of is the geometric series

 Taylor Series and Laurent Series

is singular at This point lies on the circle of convergence.

2.

3.

4.

5.

6.

7.

8.

9. =

10. To find Maclaurin’s series for ,

Integrating the power series term by term:

representing the principal value of

 Taylor Series and Laurent Series

17.2 Laurent Series

The following theorem gives the conditions for the existence of a Laurent’s
series.

17.2.1 Theorem: If is analytic on two concentric circles and with

center and in the annulus between them, then can be represented by the
Laurent series

consisting of nonnegative powers and the principal part (the negative powers).
The coefficients of this Laurent series are given by the integrals

taken counter clockwise around any simple closed path that lies in the annulus
and encircles the inner circle.

This series converges and represents in the open annulus obtained from the
given annulus by continuously increasing the outer circle and decreasing
until each of the circles reaches a point where is singular.
 Taylor Series and Laurent Series

In the special case that is the only singular point of inside , this
circle can be shrunk to the point , giving convergence in a disk except at the
center.

Proof: By Cauchy’s integral formula for multiply connected domains, we get

where is any point in the given annulus and both and are counter-
clockwise. Now integral is exactly the Taylor series so that

with coefficients

Here can be replaced by by the principal of deformation of path as is a

point not in the annulus.

To get the expansion for we note that for on and is the

annulus.
Now
 Taylor Series and Laurent Series

Multiplying by and integrating over on both the sides, we get

where,

The integral over can be replaced by integrals over .

We see that on the right, the power is multiplied by as given in

(17.2.2). This proves Laurent’s theorem provided

Now if the principal part consists of finitely many terms only, then there is

nothing to prove. Otherwise, we note that in is bounded in the

absolute value, say on because is analytic in the

 Taylor Series and Laurent Series

annulus and on , and lies on and outside, so that From

this and the ML-inequality, we get

The first series in (17.2.1) is a Taylor series and hence it converges in

the disk with center whose radius equals the distance of that singularity of
which is closet to Also, must be singular at all points outside
where is singular.

The second series in (17.2.1) representing is a power series in .

Let the given annulus be where and are radii of and

respectively. Then . Hence this power series in must converge

at least in the disk . This corresponds to the exterior of

, so that is analytic for all in the exterior E of the circle with center
and radius equal to the maximum distance from to the singularities of
inside The domain common to and is the open annulus.

17.2.2 Remark: The Laurent series of a given analytic function is unique

in its annulus of existence. However, may have different Laurent series in
two annulus with the same center.
 Taylor Series and Laurent Series

17.2.3 Examples:

1. with center 0.

for | z | > 0 . Hence the annulus is the whole complex plane except the
origin.

2.

3.

and

valid for

4. center 0

From the previous geometric series, we get by multiplying ,

5. center 0
 Taylor Series and Laurent Series

for (first for and second for

for

We can also write

for (first for and second for )

for

Suggested Readings

Ahlfors, L.V. (1979). Complex Analysis, McGraw-Hill, Inc., New York.

Boas, R.P. (1987). Invitation to Complex Analysis, McGraw-Hill, Inc., New

York.

Brown, J.W. and Churchill, R.V. (1996). Complex Variables and Applications.
McGraw-Hill, Inc., New York.

Conway, J.B. (1993). Functions of One Complex Variable, Springer-Verlag,

New York.
 Taylor Series and Laurent Series

Fisher, S.D. (1986). Complex Variables, Wadsworth, Inc., Belmont, CA.

Jain, R.K. and Iyengar, S.R.K. (2002). Advanced Engineering Mathematics,

Narosa Publishing House, New Delhi.

Ponnusamy, S. (2006). Foundations of Complex Analysis, Alpha Science

International Ltd, United Kingdom.