1550251515-Classical_Complex_Analysis__Gonzalez_

724 Chapter9

R x

Fig. 9.2"f

.+ J fo(z)dz+ 1=:+' fo(x)dx+ j fo(z)dz (9.11-62)

'Yt 'Yt

But for z E r+ we have

so that

Also,

lfo(z)I =I (l + z 2 )~ I ::::; (R2! 1)3/2

J

2~R

fo(z)dz::::; (R2-1)3/2 ---tO

r+

as R ---too

J

fo(z) dz= 2~iRe~f 0 = 2~i.

1

V2 = -;

z=i 2z(- 2) v2

'Yt

J

fo(z)dz = 2~i Res.Jo= 2~i ·~y12 = -;:

z=-• -2z 2) v2

'Yt

For z E "It we have z - 1 = f.eie, lz -ll = E, and by choosing f. ::::; 0.4,

ll + zl = 12 + f.ei^8122 - f. > 1

ll + z2I = 12 + 2Eeie + f.2e2iel 2 2 - 2E - f.2 > 1