Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.9 CAUCHY’S THEOREM

namely Cauchy’s theorem, which is the cornerstone of the integral calculus of

complex variables.

Before discussing Cauchy’s theorem, however, we note an important result

concerning complex integrals that will be of some use later. Let us consider the

integral of a functionf(z) along some pathC.IfMis an upper bound on the

value of|f(z)|on the path, i.e.|f(z)|≤MonC,andLis the length of the pathC,

then
∣
∣
∣
∣

∫

C

f(z)dz

∣

∣

∣

∣≤

∫

c

|f(z)||dz|≤M

∫

C

dl=ML. (24.39)

It is straightforward to verify that this result does indeed hold for the complex

integrals considered earlier in this section.

24.9 Cauchy’s theorem

Cauchy’s theoremstates that iff(z) is an analytic function, andf′(z) is continuous

at each point within and on a closed contourC,then
∮

C

f(z)dz=0. (24.40)

In this statement and from now on we denote an integral around a closed contour

by

∮

C.

To prove this theorem we will need the two-dimensional form of the divergence

theorem, known as Green’s theorem in a plane (see section 11.3). This says that

ifpandqare two functions with continuous first derivatives within and on a

closed contourC(bounding a domainR)inthexy-plane, then
∫∫

R

(

∂p

∂x

+

∂q

∂y

)

dxdy=

∮

C

(pdy−qdx). (24.41)

Withf(z)=u+ivanddz=dx+idy, this can be applied to

I=

∮

C

f(z)dz=

∮

C

(udx−vdy)+i

∮

C

(vdx+udy)

to give

I=

∫∫

R

[

∂(−u)

∂y

+

∂(−v)

∂x

]

dx dy+i

∫∫

R

[

∂(−v)

∂y

+

∂u

∂x

]

dx dy. (24.42)

Now, recalling thatf(z) is analytic and therefore that the Cauchy–Riemann

relations (24.5) apply, we see that each integrand is identically zero and thusIis

also zero; this proves Cauchy’s theorem.

In fact, the conditions of the above proof are more stringent than they need

be. The continuity off′(z) is not necessary for the proof of Cauchy’s theorem,