16 Cauchy’s theorem-I
TABLE OF CONTENTS
1. Introduction
1.1 Rectifiable arcs
1.2 Contours
2. Integration
2.1 Integration along a regular arc
2.2 Absolute value of a complex integral
3. Cauchy’s theorem
3.1 Elementary proof of Cauchy’s theorem
3.2 Deformation of contours
4. Cauchy’s integral formula
5. Derivative of an analytic function
5.1 Higher order derivatives
5.2 Cauchy’s inequalities
5.3 Liouville’s theorem
6. The converse of Cauchy’s theorem – Morera’s theorem
LEARNING OBJECTIVES
1. As a prelude to complex integration, rectifiable arcs and contours are defined.
2. Complex integration is introduced; integration along a regular arc and absolute value of a complex integral are described.
3. Cauchy’s theorem is stated and elementary proof of Cauchy’s theorem given. Deformation of contours is introduced.
4. Cauchy’s integral formula is derived.
5. Using Cauchy’s integral formula, derivatives of analytic functions and Cauchy’s inequalities are obtained, and Liouville’s theorem stated.
6. The converse of Cauchy’s theorem – Morera’s theorem is proved.
C a u c h y ’ s t h e o r e m – I
1. Introduction
1.1 Rectifiable arcs
Integration is one of the central ideas in the study of analytic functions of a complex variable. Cauchy’s 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.
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, so that
The necessary and sufficient condition for a Jordan arc to be rectifiable is that the sum L 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 regular arcs; arcs that have continuous tangents. For such arcs the derivatives 
exist and are continuous in the entire range of the parameter t. Such an arc is definitely rectifiable and its length is
1.2 Contours
2. Integration
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 is f then F is the integral of f. 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, in the context of complex functions.
Example
Let us find the integral
In this case the result is independent of the path chosen.
2.1 Integration along a regular arc
Let f(z) be continuous along a regular arc L, whose equation is given by
Then it can be shown that the function f(z) is integrable along L and the integral is
Example-1
Example-2
Evaluate
2.2 Absolute value of a complex integral
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:
If a function f(z) is continuous on a contour L of length l, and on the contour
Proof
If the contour L 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 L is a regular arc.
If g(t) is any complex continuous function of a real variable t, then from triangle inequality
3. Cauchy’s theorem
Cauchy’s 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’s theorem delineates the conditions under which the integral is independent of the path chosen. It says that if f(z) is an analytic function, regular in some domain D of the complex plane, z0 and z1 are two point of D, and L is a rectifiable arc joining the two points that lies wholly in D, then the integral of f(z) is independent of the path chosen and depends only on the points z0 and z1. This is a fundamental property of analytic functions and may as well be regarded as the definition of analyticity.
3.1 Elementary proof of Cauchy’s theorem
This is more than adequate for us, since the examples we cite and most cases of interest, all fall within this category. Let D be the closed domain which consists of all points within and on the closed contour C. Separating z and f(z) into their real and imaginary parts, we write
This in fact means that u and v and their partial derivatives are continuous. Hence the conditions of the Green’s theorem are satisfied, so that
3.2 Deformation of contours
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
Theorem
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.
Proof
4. Cauchy’s integral formula
We next discuss Cauchy’s integral formula which is of great practical importance as well. The formula states that if f(z) is an analytic function regular within a closed contour C, continuous within and on C, and if a is any point within C, then
Proof
Here η is a function of z and a, which tends to zero as z tends to a. Hence, given ant number ε, we can find a neighbourhood |z – a| < δ in which the inequality |η| < ε holds. Now draw a circle Γ with centre a and radius r, where r < δ and also so small that the circle Γ lies entirely within the contour C. Then f(z)is regular in the annulus bounded by C and Γ. Hence by the theorem on deformation on contours
Example
5. Derivative of an analytic function
5.1 Higher order derivatives
Example
Evaluate
Here C is a circle of radius 4 with centre at the origin and is described in the counter clockwise direction.
5.2 Cauchy’s inequalities
5.3 Liouville’s theorem
6. The converse of Cauchy’s theorem – Morera’s theorem
Let f(z) be a single-valued function continuous within a closed contour C. The necessary and sufficient condition that the integral of f(z) along any contour within C may depend only on the end points is that f(z) be an analytic function, regular within C.
Proof
The sufficiency of the condition is straightforward. The difference between the integrals of f(z) along two different contours which lie within C and have the same end points, is equal to the integral of f(z) around a closed contour in C, and that is zero if f(z) is regular within C.
To show that the condition is also necessary, we consider the integral of f(z) along a path from a fixed point a to a variable point z. Since by assumption the integral of f(z) is independent of the path, it is a single valued function of z. Denote this function by g(z):
We have to demonstrate that g(z) is an analytic function, regular within C.
SUMMARY
- As a prelude to complex integration, we begin with the definition of rectifiable arcs and contours.
- We next introduce integration of functions of a complex variable; describe integration along a regular arc and obtain an inequality about absolute value of a complex integral.
- We state and provide an elementary proof of Cauchy’s theorem, one of the most important results in complex analysis. We introduce deformation of contours and see how it helps in the evaluation of integrals.
- Then we derive Cauchy’s integral formula which is of great practical importance.
- Next we use Cauchy’s integral formula to find derivatives of analytic functions and obtain Cauchy’s inequalities. We also prove Liouville’s theorem about entire functions.
- Finally we prove the converse of Cauchy’s theorem, often called Morera’s theorem.
| you can view video on Cauchy’s theorem-I | |