Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.10 Cauchy’s integral formula

y

x

C 1 γ

C 2

C

Figure 24.11 The contour used to prove the result (24.43).

Consider two closed contoursCandγin the Argand diagram,γbeing sufficiently small

that it lies completely withinC. Show that if the functionf(z)is analytic in the region

between the two contours then

∮

C

f(z)dz=

∮

γ

f(z)dz. (24.43)

To prove this result we consider a contour as shown in figure 24.11. The two close parallel
linesC 1 andC 2 joinγandC, which are ‘cut’ to accommodate them. The new contour Γ
so formed consists ofC,C 1 ,γandC 2.
Within the area bounded by Γ, the functionf(z) is analytic, and therefore, by Cauchy’s
theorem (24.40),
∮

Γ

f(z)dz=0. (24.44)

Now the partsC 1 andC 2 of Γ are traversed in opposite directions, and in the limit lie on
top of each other, and so their contributions to (24.44) cancel. Thus
∮

C

f(z)dz+

∮

γ

f(z)dz=0. (24.45)

The sense of the integral roundγis opposite to the conventional (anticlockwise) one, and
so by traversingγin the usual sense, we establish the result (24.43).

A sort of converse of Cauchy’s theorem is known asMorera’s theorem,which

states that iff(z) is a continuous function ofzin a closed domainRbounded by

acurveCand, further,

∮

Cf(z)dz=0,thenf(z) is analytic inR.

24.10 Cauchy’s integral formula

Another very important theorem in the theory of complex variables isCauchy’s

integral formula, which states that iff(z) is analytic within and on a closed