Page 65 - Calculus with Complex Numbers

as A -+ co. svhich is nonsense because
R
.x dx
= 0
- R . x2 + 1
for all R since the integrand is odd. ln fact
z dz
A 0
j . z2 + 1
z
v'
in this case. 1ts value is
z dz z dz z dz
= - = n.i
/ . :2 + 1 . :2 + 1 . z2 + 1
2 ;' /1
for all R > 1.

6.4 I ntegrals i nvolvi ng cos x, sin .x
Consider the integral
* cos .x dx

xz .y.1 .
The contotlr integral

cos c dz
;' . :2 + 1 '

I cos zI2 = cos2 .x + sinhz y

(c = .x + iy) is unbounded in the upper half plane. lnstead we use
iz
e try.z
. :2 + 1 '
:
;'
observing first that the integral we require

Co cosx dx C'O eix dx
.
= Re ,
. x2 + 1 . x 2 + 1
and second that eiz is bounded in the upper half plane since