Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

A

B

R

y

x

C 1

C 2

Figure 24.10 Two pathsC 1 andC 2 enclosing a regionR.

analyticity off(z) within and onCbeing sufficient. However, the proof then

becomes more complicated and is too long to be given here.§

The connection between Cauchy’s theorem and the zero value of the integral

ofz−^1 around the composite pathC 4 discussed towards the end of the previous

section is apparent: the functionz−^1 is analytic in the two regions of thez-plane

enclosed by contours (C 2 andC 3 a)and(C 2 andC 3 b).

Suppose two pointsAandBin the complex plane are joined by two different pathsC 1

andC 2. Show that iff(z)is an analytic function on each path and in the region enclosed

by the two paths, then the integral off(z)is the same alongC 1 andC 2.

The situation is shown in figure 24.10. Sincef(z)isanalyticinR, it follows from Cauchy’s
theorem that we have
∫

C 1

f(z)dz−

∫

C 2

f(z)dz=

∮

C 1 −C 2

f(z)dz=0,

sinceC 1 −C 2 forms a closed contour enclosingR. Thus we immediately obtain
∫

C 1

f(z)dz=

∫

C 2

f(z)dz,

and so the values of the integrals alongC 1 andC 2 are equal.

An important application of Cauchy’s theorem is in proving that, in some

cases, it is possible to deform a closed contourCinto another contourγin such

a way that the integrals of a functionf(z) around each of the contours have the

same value.

§The reader may refer to almost any book that is devoted to complex variables and the theory of

functions.