Let C1 denote the positively oriented boundary of the square whose sides lie along the lines x = 1, y = +1 and let C2 be the positively oriented circle |z| = 4 (Fig. 63). With the aid of the corollary in Sec. 49, point out why when z+2 (a) f (2) = 3-2+ 1' (b) f (2) = - sin(z/2)' (0) f (2) = _ ez CI FIGURE 63 Need to use following Corollary. Corollary. Let C1 and C2 denote positively oriented simple closed contours, where C1 is interior to C2 (Fig. 61). If a function f is analytic in the closed region consisting of those contours and all points between them, then J., f(2 ) dz = [ f (2)dz. C2 CI O FIGURE 612. Now, we can apply the corollary in Sec. 49, which states that if a function is analytic in a simply connected domain except for a finite number of singularities, then the integral of the function over any two simple closed curves in the domain that enclose the same singularities is equal. Since all the singularities of each function f(z) are inside both C1 and C2 we can conclude that, Corollary. Let Cy and C2 denote positively oriented simple closed contours, where C is interior to C2 (Fig. 61). If a function f is analytic in the closed region consisting of those contours and all points between them, then f(z) dz = f (z) dz. JC