Page 66 - Calculus with Complex Numbers

for y k: 0. lt follows that

.P
e iz dz JZ'
/ * 0
. :2 + 1 R1 - 1
as R --> co. For R > 1 the integrand has a singularity at z = i inside y where the
residue is
z
e iz ei . j
.
Rej = = .
z=, .:2 + 1 2 .z zzi lie
Therefore
eiz dz 1 zr
= z.n'i x = - .
p' . ::2 + 1 lie e
Hence we have

as R --> co, from which it follows that
* cos .x dx zr
converges = - .
. x2 + 1 c
W e therefore deduce that

though this is of cotlrse immediate from the fact that the integrand is odd in this
Case.

6 .5 Roots of un ity
Suppose we want to evaluate the integral

dz
. :4 + 1 '
;'