Singularities/Residues/ Applications 727yf+Fig. 9.281
a+E J
+ b-, fo(x)dx + fo(z)dz (9.11-63)
'Yt
ButJ f^0 (z) dz= - 27l"i ~/o(z) = 27l"i ~ z~ 2 fo( :, )
r+= 27l"i Res _!__ J A+ 2Bz^1 + Cz^12
z^1 =0 z^12= 27l"i Res --;.. h(z')
z^1 =0 Z^1where h(z') = J A+ 2Bz^1 + Cz'^2 • Since h(z') is analytic in the vicinity
of the origin, we haveh(z') = h(O) + h'(O)z' + · · ·
for lz'I < 1/b. Hence
Res -2.._h(z')=h'(O)= !!_ = _!!__ =-i_!!_
z^1 =0 z^12 JA iJ=A J=A