684 Chapter9ButHence
lim [p 'f/d() = 0
R-+0 la
and we obtainlim J f(z) dz= iL((J - a)
R-+0Remark If the arc 'Y has its center at the point a -/:-0, so that "(: z - a =
Rei^8 , a~()~ (J, it suffices to replace assumption 2 in Lemma 9.4 by21 • (z --a)f(z) ~ L as R--> 0 for all z E 'YNote that if f is analytic in NHa) and has a simple pole at a, then
lim(z - a)f(z) = L = Resf(z)
z--+a z=aLemma 9.5 If f is analytic in some open set G containing the real interval
[a, b], except at a simple pole x = c (a < c < b), then the principal value
of 1: f ( x) dx exists.
Proof We recall that if f ( x) has an infinite discontinuity at some interior
point c of [a, b], the principal value of 1: f(x) dx, denoted (PV) 1: f(x) dx,
is defined to belim [lc-e f(x)dx + [b f(x)dx]
e-+O+ a lc+e(9.10-7)provided that the limit exists.
By the Laurent expansion theorem, in some deleted neighborhood of c
we have
A
f(z) = - + g(z)
z-c
(9.10-8)where A is a constant and g(z) is analytic in that neighborhood. However,
since both f ( z) and A/ ( z -c) are analytic in G - { c}, the analyticity of g
extends to the whole component of G containing [a, b] (Fig. 9.11).
Let 0 < e < min(c - a, b - c). We have, using (9.10-8),
(PV) lb f(x)dx