{"id":1122,"date":"2018-11-26T07:29:32","date_gmt":"2018-11-26T07:29:32","guid":{"rendered":"http:\/\/msp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=1122"},"modified":"2019-04-30T12:07:44","modified_gmt":"2019-04-30T12:07:44","slug":"cauchys-theorem-ii","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp01\/chapter\/cauchys-theorem-ii\/","title":{"rendered":"Cauchy\u2019s theorem-II"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n

TABLE OF CONTENTS<\/strong><\/p>\r\n

1.\u00a0 Taylor\u2019s Theorem<\/p>\r\n

2.\u00a0 Zeros of an analytic function<\/p>\r\n

3.\u00a0 Laurent\u2019s series<\/p>\r\n

4.\u00a0 Types of singularities<\/p>\r\n

4.1 Isolated singularity<\/p>\r\n

4.2 Removable singularities, poles and essential singularities<\/p>\r\n

5.\u00a0\u00a0 Limit point of zeros and poles 5.1 The point at infinity<\/p>\r\n

6.\u00a0 Analytic continuation<\/p>\r\n

6.1 General definition of an analytic function<\/p>\r\n

6.2 Analytic continuation by power series<\/p>\r\n \r\n\r\n<\/div>\r\n

\r\n

LEARNING OBJECTIVES<\/strong><\/p>\r\n

1. Taylor\u2019s theorem about power series expansion of analytic functions is stated and proved.<\/p>\r\n

2. Zeros of an analytic function are studied.<\/p>\r\n

3. Laurent\u2019s series about functions analytic in an annulus is stated and proved.<\/p>\r\n

4. Singular points of an analytic function are defined and types of singularities discussed.<\/p>\r\n

5. Limit point of zeros and poles are introduced. Zeros and poles of the point at infinity described.<\/p>\r\n

6. Elementary ideas of analytic continuation are provided.<\/p>\r\n \r\n\r\n<\/div>\r\n

\r\n

C a u c h y \u2019 s\u00a0 t h e o r e m\u00a0 -\u00a0 I I<\/strong><\/p>\r\n \r\n\r\n1.\u00a0 Taylor\u2019s Theorem<\/strong><\/span>\r\n

In the last module we had begun our study of the Cauchy\u2019s theorem and its consequences and applications. We will continue with this study in the present module as well. Cauchy\u2019s 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\u2019s 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\u2019s theorem<\/em> with which we are familiar from theory of real variables as well. The theorem states the following:<\/p>\r\nIf f<\/em>(z<\/em>) is an analytic function, regular in the neighbourhood of z<\/em> = a<\/em>, then it can be expanded in a power series of the form\r\n\r\n\"\"\r\n\r\nwith a nonzero radius of convergence.\r\n\r\n \r\n\r\nProof<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nExample<\/span><\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\n2. Zeros of an analytic function<\/span><\/span><\/strong>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

The function g<\/em>(z<\/em>) is regular for |z<\/em> \u2013 a<\/em>| < R<\/em> and does not vanish<\/em> at z<\/em> = a<\/em>. So let g<\/em>(a<\/em>) = c<\/em>. We will now show that there is a certain neighbourhood of a<\/em> which does not contain any other zero of f<\/em>(z<\/em>). In other words, the zeros of f<\/em>(z<\/em>) are isolated<\/em>.<\/p>\r\n

\"\"<\/p>\r\n\"\"\r\n\r\n \r\n\r\n3.\u00a0 Laurent\u2019s series<\/strong><\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\nProof<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nFollowing the same steps as for the Taylor\u2019s theorem, we can easily see that<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nExample-1<\/span><\/span>\r\n\r\nLaurent's series expansion of<\/span>\r\n\r\n\"\"\r\n\r\nExample-2<\/span><\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n4. Types of singularities<\/span><\/strong><\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\n4.1 Isolated singularity<\/span><\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\n4.2 Removable singularities, poles and essential singularities<\/span><\/span>\r\n

Depending on the nature of the Laurent\u2019s series, there are three cases to be distinguished.<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n5. Limit point of zeros and poles<\/span><\/strong><\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n5.1 The point at infinity<\/span><\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\nTheorem<\/span><\/span>\r\n

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.<\/span><\/p>\r\n \r\n\r\nProof<\/span><\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n6. Analytic continuation<\/span><\/strong><\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n6.1 General definition of an analytic function<\/span><\/span>\r\n

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.<\/span><\/p>\r\n

If the function <\/span>f<\/em>(<\/span>z<\/em>) 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.<\/span><\/p>\r\n

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 <\/span>natural boundary<\/em> of the complete analytic function.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n6.2 Analytic continuation by power series<\/span>\r\n\r\nLet a function f<\/em>(z<\/em>) be defined initially by a power series of the form\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nExample<\/span><\/span>\r\n\r\nLet us find analytic continuation of the series<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
<\/div>\r\n

SUMMARY<\/strong><\/p>\r\n\"\"\r\n\r\n\r\n\r\n\r\n
you can view video on Cauchy\u2019s theorem-IIy<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"
\"epgp<\/a>
\n<\/span><\/div>\n
\n

TABLE OF CONTENTS<\/strong><\/p>\n

1.\u00a0 Taylor\u2019s Theorem<\/p>\n

2.\u00a0 Zeros of an analytic function<\/p>\n

3.\u00a0 Laurent\u2019s series<\/p>\n

4.\u00a0 Types of singularities<\/p>\n

4.1 Isolated singularity<\/p>\n

4.2 Removable singularities, poles and essential singularities<\/p>\n

5.\u00a0\u00a0 Limit point of zeros and poles 5.1 The point at infinity<\/p>\n

6.\u00a0 Analytic continuation<\/p>\n

6.1 General definition of an analytic function<\/p>\n

6.2 Analytic continuation by power series<\/p>\n

 <\/p>\n<\/div>\n

\n

LEARNING OBJECTIVES<\/strong><\/p>\n

1. Taylor\u2019s theorem about power series expansion of analytic functions is stated and proved.<\/p>\n

2. Zeros of an analytic function are studied.<\/p>\n

3. Laurent\u2019s series about functions analytic in an annulus is stated and proved.<\/p>\n

4. Singular points of an analytic function are defined and types of singularities discussed.<\/p>\n

5. Limit point of zeros and poles are introduced. Zeros and poles of the point at infinity described.<\/p>\n

6. Elementary ideas of analytic continuation are provided.<\/p>\n

 <\/p>\n<\/div>\n

\n

C a u c h y \u2019 s\u00a0 t h e o r e m\u00a0 –\u00a0 I I<\/strong><\/p>\n

 <\/p>\n

1.\u00a0 Taylor\u2019s Theorem<\/strong><\/span><\/p>\n

In the last module we had begun our study of the Cauchy\u2019s theorem and its consequences and applications. We will continue with this study in the present module as well. Cauchy\u2019s 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\u2019s 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\u2019s theorem<\/em> with which we are familiar from theory of real variables as well. The theorem states the following:<\/p>\n

If f<\/em>(z<\/em>) is an analytic function, regular in the neighbourhood of z<\/em> = a<\/em>, then it can be expanded in a power series of the form<\/p>\n

\"\"<\/p>\n

with a nonzero radius of convergence.<\/p>\n

 <\/p>\n

Proof<\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n

Example<\/span><\/span><\/p>\n

\"\"<\/p>\n

 <\/p>\n

2. Zeros of an analytic function<\/span><\/span><\/strong><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n<\/div>\n

\n

The function g<\/em>(z<\/em>) is regular for |z<\/em> \u2013 a<\/em>| < R<\/em> and does not vanish<\/em> at z<\/em> = a<\/em>. So let g<\/em>(a<\/em>) = c<\/em>. We will now show that there is a certain neighbourhood of a<\/em> which does not contain any other zero of f<\/em>(z<\/em>). In other words, the zeros of f<\/em>(z<\/em>) are isolated<\/em>.<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n

3.\u00a0 Laurent\u2019s series<\/strong><\/span><\/p>\n

\"\"<\/p>\n

 <\/p>\n

Proof<\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Following the same steps as for the Taylor\u2019s theorem, we can easily see that<\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n

Example-1<\/span><\/span><\/p>\n

Laurent’s series expansion of<\/span><\/p>\n

\"\"<\/p>\n

Example-2<\/span><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n

4. Types of singularities<\/span><\/strong><\/span><\/p>\n

\"\"<\/p>\n

 <\/p>\n

4.1 Isolated singularity<\/span><\/span><\/p>\n

\"\"<\/p>\n

 <\/p>\n

4.2 Removable singularities, poles and essential singularities<\/span><\/span><\/p>\n

Depending on the nature of the Laurent\u2019s series, there are three cases to be distinguished.<\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n

 <\/p>\n

5. Limit point of zeros and poles<\/span><\/strong><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n

5.1 The point at infinity<\/span><\/span><\/p>\n

\"\"<\/p>\n

 <\/p>\n

Theorem<\/span><\/span><\/p>\n

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.<\/span><\/p>\n

 <\/p>\n

Proof<\/span><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n

6. Analytic continuation<\/span><\/strong><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n

6.1 General definition of an analytic function<\/span><\/span><\/p>\n

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.<\/span><\/p>\n

If the function <\/span>f<\/em>(<\/span>z<\/em>) 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.<\/span><\/p>\n

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 <\/span>natural boundary<\/em> of the complete analytic function.<\/span><\/p>\n<\/div>\n

\n

 <\/p>\n

6.2 Analytic continuation by power series<\/span><\/p>\n

Let a function f<\/em>(z<\/em>) be defined initially by a power series of the form<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n

Example<\/span><\/span><\/p>\n

Let us find analytic continuation of the series<\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n<\/div>\n

<\/div>\n

SUMMARY<\/strong><\/p>\n

\"\"<\/p>\n\n\n\n
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