454 Chapter^7
approaches fc f(z) dz as the norm of P tends to zero. Because C and Cr
are homotetic curves, the sum
n
L J[b + rp(Bk)][rp(Bk) -rp(Bk-1)]
k=lapproaches fer f(z) dz as IPI ~ 0. Hence, by (7.11-1),
J rf[b+rp(B)]dz = J f(z)dz = 0
C Cr '
and it follows thatJ f(z) dz= J J[b + p(B)] dz
c c= J {f[b+p(B)]-rf[b+rp(B)]} dz
c= (1-r) J f[b + p(B)] dz+ r J {f[b + p(B)] - f[b + rp(B)]} dz
c c
(7.11-2)By the uniform continuity of f on the compact set R, for any given
€ > 0 there is a S > 0 such that
lf(z) - f(z')I < €for any pair of points z, z' E R satisfying iz -z'I < S. Letting K =
maxo::;e9" IP( B) I, and choosing r such that 1 - r < S / K, we have
l(b + p(B)) - (b + rp(B))I = (1-r)lp(B)I < S
implying that
lf[b + p(B)] -J[b + rp(B)]I < €
Thus from (7.11-2) we obtain[ J(z) dzl :::; [(1-r)M + rE]L(C)
where M = maxzEC IJ(z )I. Letting r ~ 1-we conclude that
I [ f(z) dzl:::; EL(C)