1550251515-Classical_Complex_Analysis__Gonzalez_

442 Chapter^7

y

d ---·-- R

R4

r l

R3

• 
  
  
  
  • 
    
  
  
  
  


• A -

B

R1 i! R2 G

c - - -

0 a b x

Fig. 7.6.

where fJR is the boundary of R described once in the positive direction
(Fig. 7.6)

Proof Let

I(R) = J f(z) dz (7.10-1)

8R

for any rectangle R. If R is a degenerated rectangle for which a = b (or,

c = d), the fJR consists of a vertical (horizontal) segment described twice
in opposite directions, so I( R) = 0 trivially.
If R is a proper rectangle, so that a < b, c < d, decompose R into
four congruent rectangles R^1 , R^2 , R^3 , R^4 by bisecting its sides, as shown

in Fig. 7.6. If the boundaries of the rectangles Ri (j = 1, ... , 4) are also

positively oriented, then each common side, as AB in Fig. 7.6, is described

----+

twice in opposite directions since the orientation AB leaves the rectangle

R^3 on the left half-plane, while the reverse orientation BA leayes R^2 on

the left half-pla;ne. Hence we have ·

and it follows that

Consequently, ~t least for one rectangle Ri the inequality