1550251515-Classical_Complex_Analysis__Gonzalez_

416 Chapter^7

The following additional types of complex integrals are defined by the

corresponding right-hand sides:

J f(z)ds = j f(z)ldzl = 1p f(z(t))lz'(t)ldt (7.5-1)

j f(z)dz = 1: f(z(t))z'(t)dt (7.5-2)

'Y

J f(;: dx = 1: f(z(t))x'(t) d~. (7.5-3)

'Y c··:

J f(z) dy = 1: f(z(t))y'(t) ~t (7.5-4)

'Y

The definitions ab'ove can be extended to piecewise differentiable arcs as

in Section' 7.4.

7 .6 COMPLEX INTEGRALS ALONG A RECTIFIABLE

ARC. THE INTEGRAL AS THE LIMIT OF A SUM

The integrals defined in Sections 7.3, 7.4, and 7.5 suffice for most practical

purposes. However, those definitions can be easily generalized to integrals

along any rectifiable arc in the plane.

Let 'Y: z = z(t) = x(t) + iy(t), a :::; t :::; /3, be an arc defined by the

continuous function z = z(t) over [a,/3]; and let P = {t 0 ,t 1 ,t 2 , ... ,tn},

wl:;tere

ll' = to < ti < t2 < · · · < tn = /3,

be any partition of the interval [a,/3]. The finite sum

n

Sp= L lz(tk) - z(tk-1)1

k=l

represents the length of a polygonal line inscribed in the graph of 'Y· The arc

'Y is said to be rectifiable, or to have finite length, if the set of the nonnegative

real numbers Sp is bounded from above for all possible partitions, and the

length of ';', denoted L(7), is defined by

L('Y) =sup Sp (7.6-1)

p