{"id":1071,"date":"2018-11-26T05:57:53","date_gmt":"2018-11-26T05:57:53","guid":{"rendered":"http:\/\/msp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=1071"},"modified":"2019-04-30T12:06:15","modified_gmt":"2019-04-30T12:06:15","slug":"cauchys-theorem-i","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp01\/chapter\/cauchys-theorem-i\/","title":{"rendered":"Cauchy\u2019s theorem-I"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\nTABLE OF CONTENTS<\/strong>\r\n

1. Introduction<\/p>\r\n

1.1 Rectifiable arcs<\/p>\r\n

1.2 Contours<\/p>\r\n

2. Integration<\/p>\r\n

2.1 Integration along a regular arc<\/p>\r\n

2.2 Absolute value of a complex integral<\/p>\r\n

3.\u00a0 Cauchy\u2019s theorem<\/p>\r\n

3.1 Elementary proof of Cauchy\u2019s theorem<\/p>\r\n

3.2 Deformation of contours<\/p>\r\n

4.\u00a0 Cauchy\u2019s integral formula<\/p>\r\n

5.\u00a0 Derivative of an analytic function<\/p>\r\n

5.1 Higher order derivatives<\/p>\r\n

5.2 Cauchy\u2019s inequalities<\/p>\r\n

5.3 Liouville\u2019s theorem<\/p>\r\n

6.\u00a0 The converse of Cauchy\u2019s theorem \u2013 Morera\u2019s theorem<\/p>\r\n\r\n<\/div>\r\n \r\n\r\nLEARNING OBJECTIVES<\/strong>\r\n

1. As a prelude to complex integration, rectifiable arcs and contours are defined.<\/span><\/p>\r\n

2. Complex integration is introduced; integration along a regular arc and absolute value of a complex integral are described.<\/span><\/p>\r\n

3. Cauchy\u2019s theorem is stated and elementary proof of Cauchy\u2019s theorem given. Deformation of contours is introduced.<\/span><\/p>\r\n

4. Cauchy\u2019s integral formula is derived.<\/span><\/p>\r\n

5. Using Cauchy\u2019s integral formula, derivatives of analytic functions and Cauchy\u2019s inequalities are obtained, and Liouville\u2019s theorem stated.<\/span><\/p>\r\n

6. The converse of Cauchy\u2019s theorem \u2013 Morera\u2019s theorem is proved.<\/span><\/p>\r\n\r\n

\r\n\r\n \r\n

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

\r\n\r\n \r\n\r\n1. Introduction<\/strong><\/span>\r\n\r\n \r\n\r\n1.1 Rectifiable arcs<\/span>\r\n

Integration is one of the central ideas in the study of analytic functions of a complex variable. Cauchy\u2019s theorem is the most important result in this context and is the basis of many of its applications, particularly practical applications in the realm of physics and geometry. But before discussing the theorem it is useful to first introduce the idea of rectifiable arcs, contours and Riemann integration for a proper understanding of the conditions under which the theorem applies.<\/p>\r\n

We first consider briefly how the length of a curve in the complex plane over which integration is to be performed can be defined. Let the equation of a Jordan arc be given in terms of a parameter t<\/em>, so that<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

The necessary and sufficient condition for a Jordan arc to be rectifiable is that the sum <\/span>L<\/em> should be bounded for all possible subdivisions of the parameter in the given range. We are by and large concerned not with the Jordan arcs in general, but with <\/span>regular arcs<\/em>; arcs that have continuous tangents. For such arcs the derivatives\u00a0\"\"\u00a0exist and are continuous in the entire range of the parameter <\/span>t<\/em>. Such an arc is definitely rectifiable and its length is<\/span><\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n1.2 Contours<\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\n2. Integration<\/span><\/strong>\r\n

We are very familiar with the concept of integration. Usually integration is introduced as an operation that is the inverse of differentiation. That is, if the derivative of the function F<\/em> is f<\/em> then F<\/em> is the integral of f<\/em>. But in its applications to physics etc., it is the definite integral which is of greater importance. In the theory of functions of a complex variable also we begin integration as the limit of a sum over a certain path and later on find its connection to differentiation. So let us now describe this definition, the Riemann integration<\/em>, in the context of complex functions.<\/p>\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 the integral<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n

In this case the result is independent of the path chosen.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n2.1 Integration along a regular arc<\/span>\r\n

Let f<\/em>(z<\/em>) be continuous along a regular arc L<\/em>, whose equation is given by<\/p>\r\n\"\"\r\n

Then it can be shown that the function <\/span>f<\/em>(<\/span>z<\/em>) is integrable along <\/span>L<\/em> and the integral is<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nExample-1<\/span><\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\nExample-2<\/span><\/span>\r\n\r\nEvaluate<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n2.2 Absolute value of a complex integral<\/span><\/span>\r\n

We have a useful result about the absolute value of the integral of a complex function over a contour. It is given by the following theorem:<\/span><\/p>\r\n

If a function <\/span>f<\/em>(<\/span>z<\/em>) is continuous on a contour <\/span>L<\/em> of length <\/span>l<\/em>, and on the contour<\/span><\/p>\r\n\"\"\r\n\r\n \r\n\r\nProof<\/span><\/span>\r\n

If the contour <\/span>L<\/em> consists of a sum of regular arcs, then the result for the contour can be obtained by adding the result for each regular arc and using the triangle inequality. Hence we assume from the outset that the contour <\/span>L<\/em> is a regular arc.<\/span><\/p>\r\n

If <\/span>g<\/em>(<\/span>t<\/em>) is any complex continuous function of a real variable <\/span>t<\/em>, then from triangle inequality<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3.\u00a0 Cauchy\u2019s theorem<\/strong><\/span>\r\n

Cauchy\u2019s theorem is perhaps the most important result in the study of complex analysis. We have remarked above that in general the result of complex integration depends not only on the end points but also on the path chosen between the end points. Cauchy\u2019s theorem delineates the conditions under which the integral is independent of the path chosen. It says that if f<\/em>(z<\/em>) is an analytic function, regular in some domain D<\/em> of the complex plane, z0\u00a0<\/sub><\/em>and z1<\/sub><\/em> are two point of D<\/em>, and L<\/em> is a rectifiable arc joining the two points that lies wholly in D<\/em>, then the integral of f<\/em>(z<\/em>) is independent of the path chosen and depends only on the points z0\u00a0<\/sub><\/em>and z1<\/sub><\/em>. This is a fundamental property of analytic functions and may as well be regarded as the definition of analyticity.<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n3.1 Elementary proof of Cauchy\u2019s theorem<\/span>\r\n\r\n\"\"\r\n

This is more than adequate for us, since the examples we cite and most cases of interest, all fall within this category. Let <\/span>D<\/em> be the closed domain which consists of all points within and on the closed contour <\/span>C<\/em>. Separating\u00a0<\/span>z\u00a0 <\/em>and <\/span>f<\/em>(<\/span>z<\/em>) into their real and imaginary parts, we write<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n

This in fact means that <\/span>u<\/em> and <\/span>v<\/em> and their partial derivatives are continuous. Hence the conditions of the Green\u2019s theorem are satisfied, so that<\/span><\/p>\r\n\"\"\r\n\r\n<\/div>\r\n \r\n\r\n3.2 Deformation of contours<\/span><\/span>\r\n

If a function is not regular within a closed contour then of course its integral in general is not zero. The calculation of integral of an analytic function, not necessarily regular, around a closed contour is often simplified by deformation of the contour. In this regard we have the following theorem<\/span><\/p>\r\n\r\n

\r\n\r\n \r\n\r\nTheorem<\/span>\r\n

The value of the contour integral of an analytic function is unaltered by any deformation of the contour provided in doing so no singularities of the function are crossed.<\/p>\r\n \r\n

Proof<\/span><\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n4.\u00a0 Cauchy\u2019s integral formula<\/span><\/strong><\/span>\r\n

We next discuss Cauchy\u2019s integral formula which is of great practical importance as well. The formula states that if f(z)<\/em> is an analytic function regular within a closed contour <\/span>C<\/em>, continuous within and on <\/span>C<\/em>, and if <\/span>a<\/em> is any point within <\/span>C<\/em>, then<\/span><\/p>\r\n\"\"\r\n\r\n \r\n\r\nProof<\/span><\/span>\r\n\r\n\"\"\r\n

Here <\/span>\u03b7<\/em> is a function of <\/span>z<\/em> and <\/span>a<\/em>, which tends to zero as <\/span>z<\/em> tends to <\/span>a<\/em>. Hence, given ant number <\/span>\u03b5<\/em>, we can find a neighbourhood |<\/span>z<\/em> \u2013 <\/span>a<\/em>| < <\/span>\u03b4<\/em> in which the inequality |<\/span>\u03b7<\/em>| < <\/span>\u03b5<\/em> holds. Now draw a circle <\/span>\u0393<\/em> with centre <\/span>a<\/em> and radius <\/span>r<\/em>,\u00a0<\/span>where <\/span>r<\/em> < <\/span>\u03b4<\/em> and also so small that the circle <\/span>\u0393<\/em> lies entirely within the contour <\/span>C<\/em>. Then\u00a0f(z)<\/em>is regular in the annulus bounded by <\/span>C<\/em> and <\/span>\u0393<\/em>. Hence by the theorem on deformation on contours<\/span><\/p>\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\n5. Derivative of an analytic function<\/span><\/strong><\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n5.1 Higher order derivatives<\/span><\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nExample<\/span><\/span>\r\n\r\nEvaluate<\/span>\r\n\r\n\"\"\r\n

Here <\/span>C<\/em> is a circle of radius 4 with centre at the origin and is described in the counter clockwise direction.<\/span><\/p>\r\n\"\"\r\n\r\n \r\n\r\n5.2 Cauchy\u2019s inequalities<\/span><\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\n5.3 Liouville\u2019s theorem<\/span><\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\n6.\u00a0 The converse of Cauchy\u2019s theorem \u2013 Morera\u2019s theorem<\/span><\/strong><\/span>\r\n

Let <\/span>f<\/em>(<\/span>z<\/em>) be a single-valued function continuous within a closed contour <\/span>C<\/em>. The necessary and sufficient condition that the integral of <\/span>f<\/em>(<\/span>z<\/em>) along any contour within <\/span>C<\/em> may depend only on the end points is that <\/span>f<\/em>(<\/span>z<\/em>) be an analytic function, regular within <\/span>C<\/em>.<\/span><\/p>\r\n \r\n\r\nProof<\/span><\/span>\r\n

The sufficiency of the condition is straightforward. The difference between the integrals of <\/span>f<\/em>(<\/span>z<\/em>) along two different contours which lie within <\/span>C<\/em> and have the same end points, is equal to the integral of <\/span>f<\/em>(<\/span>z<\/em>) around a closed contour in <\/span>C<\/em>, and that is zero if <\/span>f<\/em>(<\/span>z<\/em>) is regular within <\/span>C<\/em>.<\/span><\/p>\r\n

To show that the condition is also necessary, we consider the integral of <\/span>f<\/em>(<\/span>z<\/em>) along a path from a fixed point <\/span>a<\/em> to a variable point <\/span>z<\/em>. Since by assumption the integral of <\/span>f<\/em>(<\/span>z<\/em>) is independent of the path, it is a single valued function of <\/span>z<\/em>. Denote this function by <\/span>g<\/em>(<\/span>z<\/em>):<\/span><\/p>\r\n\"\"\r\n\r\nWe have to demonstrate that <\/span>g<\/em>(<\/span>z<\/em>) is an analytic function, regular within <\/span>C<\/em>.<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n \r\n\r\nSUMMARY<\/strong>\r\n