1550251515-Classical_Complex_Analysis__Gonzalez_

Integration

:::; 1: Jf(z(t))JJz'(t)J dt

= J Jf(z)JJdzJ

'"'(

7. If maxzE'"'f JJ(z)J :::; Mand L = L('Y) = J'"'f JdzJ, it follows that

J Jf(z)JJdz:::; M J JdzJ =ML

'"'( '"'(

Hence from property 6 we conclude that

J J(z)dz :::; ML

'"'(

423

This inequality is called Darboux's inequality. It is very useful in finding

an upper bound for the absolute value of a complex integral.

Example Consider J'"'f dz/(z^2 + 9), where 1': z = 2eit, 0 :::; t :::; 1r. For
z E 1' we have

Jz^2 + 9J 2: 9 - JzJ^2 = 9 - 4 = 5

so that

I z^2 ~ 91:::; ~

Also, L('Y) = 27r. Hence by Darboux's inequality we have

7.8 Further Properties of the Complex Integral

Theorem 7.3 Let z = g(() be analytic in some open set containing the
rectifiable arc 1'': ( = ((t), a :::; t :::; (3, and let 1' = g(1'): z = g(((t)),

a :::; t :::; (3. Then 1' is also rectifiable,. and if f is continuous along 1',

we have

J f (z) dz = J f [g( ()Jg' ( () d(

"Y "Y'