698 Chapter9
L~
~
. f+
b2 bj
~ r
-R 0 p-r p p+r R xFig. 9.16
By Lem.ma 9.5,
lim {jp-r f(x)dx+ [R f(x)dx} = (PV)jR f(x)dx
r-+O -R }p+r -R
By Lemma 9.4,
lim r->0 f f(z) dz= i(O - 7r) = -7ri Res z=p f(z)
"Y-and by Lemma 9.3,
Hm jf(z)dz=O
R-+oo
r+Hence if we take limits in (9.11-19) first as r-+ O, and then as R-+ oo, we
see that (PV) f ~ 00 f ( x) dx exists, and that
(PV) 1_: f(x) dx = 27ri ~ ~1~ f(z) + 11"i~~:f(z) (9.11-20)
Now, if instead of a simple pole p off in the real axis we haven simple
poles p 1 , P2, ... , Pn, then by applying the same process to each Pr. we
find that
roo m n
(PV) }_
00f(x) dx = 27ri I; ~1~ f(z) + 7ri L Ji~~ f(z)
k=l r=l= 27ri [~ !1~ f(z) + % ~ ~~ f(z)] (9.11-21)
We note that only one-half of the residues of the poles lying on the real
axis are taken into account. This is as if only half of each pole lying on