Page 62 - Calculus with Complex Numbers

E v a I u a t i o n o f i n f i n i t e

r e a I i n t e g r a I s

6 . I C onve rgence
For integrals of the form

-
/--
J'lxldx

the problem of convergence arises. W e shall deline the Cauchy principal value
(CPV) of such an integral to be

*
ytwl,2.v - lim jR y(.v)J.v,
.J-. s-xoa -s

and say the integral converges whenever this limit exists. For example consider
the integral

oa dx
. x2 + 1'

W e have

R dx R zr
A
-

2
s .x2 + 1 gtan-l wj = 2tan-1 R --> 2 x - = zr
=
O dx
= % .
. x2 + 1