1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

8.3 • IMPROPER INTEGRALS OF RATIONAL FUNCTIONS 307

provided the limit exists. If f is defined for all real x, then the integral of f over
(-oo, oo) is defined by

J

oo f (x) dx = lim r

0

f (x) dx + lim rb f (x) dx,

-oo a--^00 Ja t>-oo}o

(8-7)

provided both limits exist. If the integral in Equation (8-7) exists, we can obtain
its value by taking a single limit:

(8-8)

For some functions the limit on the right side of Equation (8-8) exists, but the
limit on the right side of Equation (8-7) doesn't exist.

•EXAMPLE 8.13 Jim J!Rxdx = lim [~

2

• <-:l

2

] = 0, but Equation (8-

R-oo R-oo
7) tells us that the improper integral of j(x) = x over (-00,00) doesn't exist.
Therefore, we can use Equation (8-8) to extend the notion of the value of an
improper integral, as Definition 8.2 indicates.

Definition 8.2: Cauchy principal value

Let f (x) be a continuous real-valued function for all x. The Cauchy principal
value (P.V.) of the integral J~ 00 f (x) dx is defined by

P.v. J

00

f(x)dx= Jim JR f(x)dx,

• oo R-oo - R

provided the limit exists.

Example 8.13 shows that P.V. J~ 00 x dx = 0.

• EXAMPLE 8.14 The Cauchy principal value of J~ 00 ,,,2~ 1 dx is

P.V.

J

oo l dx = lim JR l dx

_ 00 x^2 + 1 R~oo - R x^2 + 1

= lim [Arc tan R -Arc tan (-R)]

R-oo

'Tr -'Tr

= - - - ='Tr.

2 2