420 Chapter 7Definition 7. 7 With the assumptions and notations of Definition 7 .6, we
also define(7.6-4)The limit on the right-hand side exists since f(z(t)) is continuous and Jz(t)J
is also a function of bounded variation over [a, ,8] .'7.7 ELJEMENTARY PROPERTIES OF THE COMPLEX
INTEGRAL '
The following properties of the complex integral along a rectifiable arc fol-
low from (7.6-2) and the corresponding properties of the Riemann-Stieltjes
integral. In the case of a continuously differentiable arc these properties
may be derived from those listed in Theorem 7.1.Theorem 7.2 The integral of a continuous complex function f(z) along
a rectifiable arc 1: z = z(t), a :::; t :::; ,8, has the following properties:1. J,,[kd1(z) :- kd2(z)] dz= ki J,, fi(z) dz+ k2 J,, f2(z) dz wh~re kl an.cl
k 2 are arbitrary complex constants. In other words, the mtegral is
linear with respect to the integrand.2. J,,i+ 12 f.(z)dz.= J,,i f(~~dz +),, 2 f(z)dz where the ~erminal point of
11 comcides with the imtial pomt of 12 , and the notation ')'1 +1 2 stands
for the path consisting of 11 followed by 12 , called the juxtaposition of
11 and 12 •
This property may be stated by saying that the integral is additivewith respect to the path. It generalizes easily to a decomposition of
the path into a finite number of subarcs. Also, the property holds
for a chain of arcs (see Definition 3.26), provided that each integral is
multiplied by the app'ropriate coefficient.- f_,,J(z)dz = - J,,J(z)dz, i.e., if the orientation of the path is
reversed, the integral changes in sign. - The integral along a closed path does not depend on the point of the
path that is taken as the initial point.
5. f,,f(z)dz = J,,J(z)dz
- IJ,,J(z)dzj:::; J,,lf(z)JJdzJ
7. IJ,,J(z) dzj:::; ML where max Jf(z)J:::; Mand L = L(1) =length of 1·
Proofs 1.( a) Rectifiable arc. The functions Ji and f2 are supposed to be
continuous along 1· Hence kif 1 + k 2 f2 is also continuous along 1, and