1550251515-Classical_Complex_Analysis__Gonzalez_

716 Chapter^9

The recursive formula (9.11-47) can be applied in particular to rational
functions f(x) = P(x)/Q(x), with P and Q prime to each other and the
degree of Q at least two units greater than the degree of P.

Examples 1. Toevaluatef 000 lnxdx/(x^2 +a^2 ),wherea>O. Heref(z)=

1/(z^2 + a^2 ), which satisfies conditions (2), (3) and (4). Since

Res logz = logia = lna + %i7r

z=ia z^2 + a^2 2ia 2ia

(9.11-48) yields

Hence

2 roo _lnxdx +i'lf' roo dx = !'.:.(lna+%i7r)

lo x2 + a2 lo x2 + a2 a

{^00 lnx dx

lo x2 + a2

{^00 dx

lo x2 + a2

71'

= -lna

2a

71'

2a

(9.11-50)

(9.11-51)

where the last integral can be evaluated by elementary means. For a = 1

the first integral gives

[

00

lnxdx = 0

Jo x^2 +1

2. To evaluate f 0
   00

(lnx)^2 dx/(x^2 + a^2 ), a> 0. Since

Res (log z )^2 = (logia )^2 = (ln a )^2 + i'lf' ln a - %Tr2

=~~+~ ~ ~

we get, from (9.11-49),

2

1

00

(lnx)2 dx. 1

00

lnx dx 21

00

~-'--+ 2rn - 7r dx

o x2 + a2 o x2 + a2 o x2 + a2

= !'.:. ((ln a)^2 + i'lf' ln a - ~ 71'^2 ] (9.11-52)

a 4

Hence,

2100 (lnx)2dx -11·2 roo dx = !'.:.[(lna)2 - !71'2]

o x2 + a2 lo x2 + a2 a 4

or,