1550251515-Classical_Complex_Analysis__Gonzalez_

424 Chapter^7

Proof ( a)Rectifiable arc. A partition P = { t 0 , t 1 , ••• , tn} of the interval

[a, ,8] induces a subdivision { ( 0 , (1 ... , (n} of the arc l'': ( = ((t), a ::::; t ::::;

,8, by letting (k = ((tk), as well as a subdivision {zo, z1, ... Zn} of the arc

l': z = g(((t)), a :S t :S ,8, by letting Zk = g((k) (Fig. 7.4). Of course, it

is being assumed that g is nonconstant.

We have, by the definition of the derivative,

Zk - Zk-1 '(1" )

;- ;- = 9 '>k-1 + Ek

'>k - '>k-1

(7.8-1)

where Ek ~ 0 as A(k = (k - (k-l ~ 0. Since g'(() is continuous on the

compact set l'', there is some M > 0 such that 19'(()1 :SM for all (El'',

and if 'f/P =max kkl for the partition P, we obtain

lzk - Zk-1 I ::::; (M + 'f/P )l(k - (k-1 I

Hence

n n

L lzk - Zk-11 :S (M + 'f/P) L ICk - (k-11 (7.8-2)

k=l k=l

Since the function ( = ((t) is of bounded variation over [a, ,BJ, there exists
a I< > IO such that
n
I: 1ck - ck-11 <I< (7.8-3)
k=l

whatever the partition P.

~-plane z-plane

Zn

/'"

.

y*

),

~o Z1

zo

(^0 0)
to 11 tn
0 Ci 13
Fig. 7.4