Page 64 - Calculus with Complex Numbers

dz J:A 1 -* 0
+
2
l z2 1 Y 1
/

-
R

as R --> co.
Putting all this information together we have
R dx dz dz dz dz
= = - = Jr- -> Jr
R .x2 + 1 . :2 + 1 . z2 + 1 . z2 + 1 . z2 + 1
- /1 / /2 /2
as R --> co. Hence we deduce that
agreeing with what we found in Section 6. 1.

/
2
7
6.3 Failure of j --> 0
lt is essential to the success of the method outlined in Section 6.2 that we can prove
J --> 0. For example, corlsider the integral
oo x dx
. x2 + 1'

ln this case we have
z z 1
Res o = - = -.
,
C
z= 'z'f + 1 2.: i l

=
Therefore
z dz 1
= 2.zr i x - = n'i
r . :2 + 1 2
for R > 1. lf we could prove that

zdz
-- > 0
/ . z2 + 1
2
as R --> co, then we could deduce that
R
.x dx c dz c dz c dz
= = - --> n.i
s .r2 + 1 . :2 + 1 . z2 + 1 . z2 + 1
- /1 / /2