1550251515-Classical_Complex_Analysis__Gonzalez_

418 Chapter 7

y ~

zo"

z,

(^0) Tk x
to t, tk - 1 tk In
Fig. 7.3

J f(z)dz = 1: f(z(t))dz(t) (7.6-2)

n

= lim L f(zk)(zk - Zk-1)

IPl->O k=l

Thus the integral of f along 'Y is expressed in terms of a Riemann-Stieltjes

integral over the real interval [a, ,8]. For the existence of the limit on the
right-hand side of (7.6-2), see [2], p. 231, or [33], p. 121.

This limit is to be understood in the following sense: For every e > 0

there exists a S > 0 such that IPI < S a~d every choice of the rk in the

corresponding subintervals implies that

j f(z) dz -t f(z(rk))[z(tk)-z(tk-1)] < e

'Y k=l

(7.6-3)

As in the real case, the sums I:;=l f ( zk)( Zk - Zk-l) are referred to as the

approximating sums of the integral.

The integral of a continuous function along a continuously differentiable

arc, as defined by (7.3-1), is a special case of (7.6-2). In fact, by the

mean-value theorem of differential calculus we have

z(tk) - z(tk-1) = [x(tk)-x(tk-1)] + i[y(tk) - y(tk-1)]

= x^1 (rfJ(tk -tk-1) + iy'(rf)(tk - tk-1)