1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications

Exercises 9.5

Show that:

1

(^00) dx 11"J3

1. (PV) _ a a = -3 2 (a > 0)

00 x +a a

1

(^00) dx 271"

2. (PV) 3 3 = ---r;;- (a > 0)

o x +a 3v3a2

1

(^00) x +5

3. (PV) -oo x4 - 1Qx2 + 9 dx = 0


4. (PV) loo x4 dx = ~
   lo x^6 -1 2J3


5. (PV) loo (x2 + 2) dx 771"
   _ 00 x^6 + 4x^4 - x^2 - 4 = - 30


6. (PV) loo sin ax dx = ~ (1-e-ab) (a> 0, b > 0)

lo x(x^2 + b^2 ) 2b^2

1

00 • 2 + 2

7. ~ x 2 ~ 2 dx =
   2

7l"b 2 [a^2 + e-b(b^2 - a^2 )] (a> 0, b > 0)

0 x x +

1

(^00) sin 7l"X cos 11"X d
S. 00 x(2x - 1) x - -71"

9. (PV) loo sin 7l"X dx = ~ 11"( e-11' - 3)
   _ 00 x^5 - x 2


10. loo sin3 x dx = 37!"
    lo x^3 8
    [Hint: Use sin^3 x =^3 / 4 sin x -^1 / 4 sin 3x = Im(^3 / 4 eix -1/ 4 e^3 ix).]

ll.100 si::x dx = ~71"

100 sin ax dx 71"

12. (PV) lo x(x 2 _ b 2 ) = 2 b 2 (cos ab - 1) (a > O, b > O)

1

00

sin ax dx 71" [ 1 -ab ]

13. lo x(x2+b2)2 = 2b4 1-2e _(ab+2) (a>O,b>O)


14. loo cos2ax ~ cos2bx dx = 11"(b-a) (a~ O,b ~ 0)
    10 x

l la +ioo. e=t {1 fort>O

15. (PV)-. - dz = H(t) = % fort= 0
    27l"z a-ioo Z

0 fort<O

705

where a> 0. (Hint: Use as a contour the vertical segment z =a+ ir,

-R ::::; r ::::; R, followed by the semicircle rt: z = a+ Rei^9 , R > a,

(^1) /271" ::::; () ::::; %71", for the case t > 0, and followed by the semicircle r;-: