1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 421

the integral on the left-hand side exists. With the notation of Definition

7.6, we have

J[kd1(z) + kzf2(z)] dz

"{

n

= lim ~)kd1(z(rk)) + kzf2(z(rk))][z(tk)-z(tk-1)]

IPl--o k=l

n

=kl lim """'fi(z(rk))[z(tk) - z(tk- 1 )]

IPJ ..... o~

n

+ kz lim Lfz(z(rk))[z(tk)-z(tk-1)]

IPl--o k=1

=kl j fi(z) dz+ kz 1 f 2 (z) dz

"{

(b) Continuously differentiable arc. Under the stronger assumption that

'Y is of class C^1 ' we may proceed as follows:

J[kd1(z) + kzf2(z)] dz= 1:[kd1(z(t)) + kzf2(z(t))]z'(t) dt

"{

=kl 1fi fi(z(t))z'(t) dt + k 2 1: fz(z(t))z'(t)dt

=kl J fi(z) dz+ kz J fz(z) dz

"{ "{

where we have used Definition 7.3 and Theorem 7.1,(2),(5).

Note Since shorter proofs can be made by assuming 'Y to be continuously
differentiable, in what follows we restrict ourselves, with a few exceptions,
to this special case, which suffices for nearly all practical purposes.

2. If "fl and "( 2 are in juxtaposition the arc 'Yl + "(z is either smooth
   or piecewise continuously differentiable. In either case let r be the end-
   point of the interval over which "fl is defined (equals the initial point over
   which "( 2 is defined). Suppose that "( 1 + 12 is defined over [a, ,8]. Then
   we have

J f(z) dz= 1r f(z1(t))zUt) dt + lfi f(z2(t))z~(t) dt

"11 +"12