1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

6.3 • THE CAUCHY-00URSAT THEOREM 225

Figure 6.23. Then the contour C = C 2 - C 1 is a parametrization of the boundary
of the region R that lies between C1 and C2 so that the points of R lie to the
left of C as a point z (t) moves around C. Hence C is a positive orientation of
the boundary of R, and Theorem 6.6 implies that fc f (z) dz= 0.
We can extend Theorem 6.6 to multiply connected domains with more than
one "hole." The proof, which we leave for you, involves the introduction of
several cuts and is similar to the proof of Theorem 6.6.

• EXAMPLE 6.14 Show that f ct "f~ 2 dz= 47ri.

Solut ion Recall that Ct (0) is the circle {z: lz l = 2} with positive orientation.

Using partial fraction decomposition gives

2z 2z 1 1

--- - + so

z^2 + 2 - (z+ iJ2) (z- i./2) - z +iJ2 z -iJ2'

--dz = dz+ dz.

1

2z 1 1 1 1

ct<o> z^2 + 2 ct<o> z + iJ2 ct(o) z -iJ2

(6- 38 )

y

Figure 6. 26 T he mult iply connected domain D and the contours C and C1, C2, ... , Cn

in the statement of the extended Cauchy-:Goursat theorem.