1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 479

Thus if we let p ---+ 0 in (7.17-7), we get

2i ff ( ~ z de dry= f {~~ d( - 27rif(z)

ft r+

or

f(z)= ~ff(() d(-~ ff -1.Ldedry

27l"Z ( - Z 71" j j ( -Z

(7.17-8)

r+ ft

By introducing the winding number !1r(z), we may write (7.17-8) in the

form

nr(z)f(z) = ~ f !(() d( - !1r(z) ff h de dry

27rz ( - z 7r J J ( -z

(7.17-9)

r ft

Formula (7.17-8) is due to D. Pompeiu [27]. If f is analytic in R, then

Ji; = 0 in R, and the formula reduces to the classical Cauchy's formula.

Analogous to (7.17-9), we have

nr(z)f(z) = - ~ f f(() ~( - !1r(z) ff _1, -de dry

27rz ( - z 7r J J ( -z

(7.17-10)

r ft

7.18 Cauchy's Formula for z on the Contour

Cauchy's integral (7.17-1), as well as any of its generalizations, is not de-

fined for z 0 E 0. However, this case may be interpreted in various ways as

a restricted limit, the result depending on the manner in which the limit is

taken. For instance, for the case referred to in Theorem 7.23, we have

lim -2

1

. f !~() d( = nc(z)f(zo)
z--no 7rZ - z
c

for z E Int 0, z 0 E 0, while

lim -^1 f f(()d( = O

z-+zo 27ri ( - z

c

for z E Ext O, z 0 E O.
On the other hand, if the limit is taken along the contour 0 some ad-
ditional assumptions are needed concerning 0, as well as to the manner in
which the limit is taken. Suppose that 0 is simple, closed, and piecewise

smooth with a finite number of corner points (i, ... , (n, and described

once in the positive direction. If zo E 0 is such that zo f/. (k ( k = 1, ... , n)