1550251515-Classical_Complex_Analysis__Gonzalez_

694 Chapter9

However, in this case the integral on the left of (9.11-11) converges in

the ordinary sense. To see this, consider a rectangle with vertices (-R1, 0),

(R2,0), (R2,Ra), and (-R1,Ra), with Ri > O,.R2 > 0, Ra> 0 large

enough so that all the poles of f in the upper half-plane will lie inside the

rectangle (Fig. 9.15). Because of assumption (3) we have lzf(z)I < K'

for some constant K' > 0 whenever z lies on the vertical sides or on the

upper horizontal side of the rectangle. Thus for the integral along the right
vertical side, we have

< aR

2

(9.11-12)

Similarly, for the integral along the left vertical side, we have

(9-11.13)

As to the integral along the upper side (for which z = x+iR 3 ), we find that

Y,

-R 1 +iR 3 R 2 +iR 3

!

-R, (^0) R2 x
Fig. 9.15