1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/ Applications 729

replace the circle 'Yt of that figure (see Fig. 9.29). Then we have

J

fo(z) dz= 1-S fo(x) dx + J fo(z) dz+ rb-£ fo(x) dx

. a+£ ls
r+ -y~+

+ J fo(z)dz+ l~/o(x)dx+ J fo(z)dz

'Yt 'Yd

l

a+e J

+ _

8

f 0 (x)dx+ f 0 (z)dz (9.11-65)

'Yt

As in case (i), we obtain

J

27rB

fo(z) dz= y'=A

r+

lim e->0 J fo(z) dz= e->0 lim J fo(z) dz= 0

'Yt 'Yt

and

y

r+

R x

Fig. 9.29