17 Cauchy’s theorem-II

TABLE OF CONTENTS

1.  Taylor’s Theorem

2.  Zeros of an analytic function

3.  Laurent’s series

4.  Types of singularities

4.1 Isolated singularity

4.2 Removable singularities, poles and essential singularities

5.   Limit point of zeros and poles 5.1 The point at infinity

6.  Analytic continuation

6.1 General definition of an analytic function

6.2 Analytic continuation by power series

LEARNING OBJECTIVES

1. Taylor’s theorem about power series expansion of analytic functions is stated and proved.

2. Zeros of an analytic function are studied.

3. Laurent’s series about functions analytic in an annulus is stated and proved.

4. Singular points of an analytic function are defined and types of singularities discussed.

5. Limit point of zeros and poles are introduced. Zeros and poles of the point at infinity described.

6. Elementary ideas of analytic continuation are provided.

C a u c h y ’ s  t h e o r e m  –  I I

1.  Taylor’s Theorem

In the last module we had begun our study of the Cauchy’s theorem and its consequences and applications. We will continue with this study in the present module as well. Cauchy’s theorem is one of the most important theorems in complex analysis and can be regarded as the basis for study of analytic functions. In this module we begin with Taylor’s theorem that has to do with power series expansion and analytic functions. We have already seen in the module CA-2 that the sum of a power series with nonzero radius of convergence is an analytic function which is regular everywhere within its circle of convergence. We will now prove the converse of this theorem. This is called the Taylor’s theorem with which we are familiar from theory of real variables as well. The theorem states the following:

If f(z) is an analytic function, regular in the neighbourhood of z = a, then it can be expanded in a power series of the form

with a nonzero radius of convergence.

Proof

Example

2. Zeros of an analytic function

The function g(z) is regular for |z – a| < R and does not vanish at z = a. So let g(a) = c. We will now show that there is a certain neighbourhood of a which does not contain any other zero of f(z). In other words, the zeros of f(z) are isolated.

3.  Laurent’s series

Proof

Following the same steps as for the Taylor’s theorem, we can easily see that

Example-1

Laurent’s series expansion of

Example-2

4. Types of singularities

4.1 Isolated singularity

4.2 Removable singularities, poles and essential singularities

Depending on the nature of the Laurent’s series, there are three cases to be distinguished.

5. Limit point of zeros and poles

5.1 The point at infinity

Theorem

In this connection we have an important result that if the only singularities of an analytic function, including possibly at infinity, are poles, the function must be a rational function.

Proof

6. Analytic continuation

6.1 General definition of an analytic function

The above discussion about analytic continuation provides us with a general definition of an analytic function. Let us suppose that we make all possible analytic continuations of a given function outside the original domain of analyticity, and then all possible continuations of these in every possible way; the complete function is defined as the original and all these continuations so obtained. This complete analytic function may turn out to be a many valued function.

If the function f(z) is not an entire function, there will be certain points which do not belong to any of the domains to which the function has been analytically continued. These exceptional points are the singularities of the complete function.

Another interesting possibility in this process of continuation is that we may reach a boundary beyond which it may be impossible to continue the function in any manner. Such a closed curve is called natural boundary of the complete analytic function.

6.2 Analytic continuation by power series

Let a function f(z) be defined initially by a power series of the form

Example

Let us find analytic continuation of the series

SUMMARY