1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

318 CHAPTER 8 • RESIDUE THEORY

has simple poles at the points t 1 = 2 on the x-axis and z 1 = -1 + iJ3 in the

upper half-plane. By Theorem 8.5,

1

(^00) x dx
P.V. ~
8
= 27riRes [/, z1] + 7riRes (/,ti]
-oo x
2

. -1 - i v'3 .1 7rJ3
= 7rt + 7rt- = --.
12 6 6

• EXAMPLE 8. 21 Evaluate P.V. J::' 00 ;.~~ by using a computer a lgebra sys-
  tem.

S olution Computer algebra systems such as Mathematica or MAPLE give the

indefinite integral

J

~ Arctan!J:f Log (t - 2) Log (t^2 + 2t + 4) ()

t 3 - 8 - 2J3 + 6 + 12 - g t.

Lo f (t-2)2]

However, for real numbers, we should write the second term as g 12 and

use the equivalent formula:

Arctan7J Log [(t-2)

2

) Log (t^2 + 2t + 4)

g(t)= 2J3 + 12 + 12.

This antiderivative has the property that lim g (t) = -oo, as shown in Figure

t - 2

8.5. We also compute

. 7rv'3

ltm g(t) = -

12

and

t~oo

$

0.5

--0.5

• 1rJ3

lim g (t) = -

2

• ,
  i--oo 1

s = g(t)

Figure 8.5

t dt

Graphofs= g(t)= f t 3 _

8

.