1550251515-Classical_Complex_Analysis__Gonzalez_

Integration

Fig. 7.22

But

c

J

f(()d( = 0

(-z

'Yl

(r = l, ... ,p)

by Theorem 7.23, and

~ J f(()d( = f(z)

27ri ( - z

-y+

477

by formula (7.17-3). Combining the last sum on the right-hand side with

the left-hand side, we get

or

2~i J

G-Gi-···-Gn

!CO d( = nr(z)f(z)

(-z

_21. j f~O d( = nr(z)f(z)

7l"Z - Z

r

which is the desired formula.

Theorem 7.25 (Cauchy's Formula for Functions of Class C^1 (R)). Let
R be a multiply connected region bounded by simple closed contours C,
C 1 , ... , Cn with the same properties as in Theorem 7.24, and let r =