1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 487

y*

Fig. 7.26

where 1 is a rectifiable arc, not necessarily closed or simple, z E C - 1*

and f is continuous along 1, are called integrals of-Cauchy type.

It will be assumed that 1 intersects itself only a finite nl}mber of times.

Under this assumption the graph of 1 divides the plane into a finite num-

ber of regions Ro, Ri, ... , Rm, where Ro denotes the unbounded region

(Fig. 7.26). Since the integrand J(()/(( - z) is continuous for z EC-1,
an integral of Cauchy type defines a function F( z) on C - 1. As the next
theorem shows, F( z) is locally analytic, i.e., analytic on each of the regions

Rk (k = O, 1, ... , m), but usually F(z) represents different analytic func-

tions of different regions. For instance, in the case of the classical Cauchy's
formula, where 1 is a simple closed contour, we have F(z) = 27rif(z) if

z E Int 1, while F( z) = 0 if z E Ext 1, assuming f to be analytic on and

within 1. The two results coincide only if f = 0 on 1 U Int 1*.

Theorem 7.28 Let F(z) = J,, J(() d(/(( - z). Then F(z) is analytic on
each of the regions Rk, and

F'(z) = j J(()d( (7.20-1)

(( -z)2

'Y

for z E Rk. More generally, if we let Fn(z) = J J(() d(/(( - z)n (n =

1,2, ... ), we have F~(z) = nFn+i(z). 'Y

Proof Let r = d(z,1) and choose h such that lhl < %r. Then, for ( E 1
we have I( -zl 2'.: rand I( -z - hi 2'.: I( -zl -lhl > r -^1 / 2 r =^1 / 2 r. From

F(z + h) - F(z) = j f(() d( -1 f(() d(

(-z-h 'Y (-z

'Y

J

hf(() d( (7.20-2)

= ((-z-h)((-z)

'Y

we get

F(z+h)-F(z) _ j f(()d(

h - ((-z-h)((-z)

'Y