1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications 709

-R x

Fig. 9.21

Under the stated conditions the integrand za f(z) is a single-valued ana-

lytic function on and within c+' except at those poles bk ( k = 1, 2, ... ' m)

that may lie in the upper half-plane. Hence by the residue theorem, we have

or

J zaf(z)dz= i: xaf(x)dx+ J zaf(z)dz+ ip-

8

xaf(x)dx

a+ 71

+ J za f(z) dz+ {R xa f(x) dx + J za f(z) dz= 27ri f ~es za f(z)

_ }p+8 r+ k=l z-bk

72

m

= 27ri'"""' Res za J(z)

L.J z=bk

k=l

(9.11-36)

By letting x = -x'(x' > 0) in the first integral of (9.11-36), we get

1: ( -x't f (-x')(-dx') = eiTra 1R x'a f( -x') dx' (9.11-37)

Also,

lim [ r-

8

zaf(x)dx+ {R xaf(x)dx] =(PV)1R xaf(x)dx

8__,.Q Jr }p+8 r