53.3. EXERCISES 1697

(c)∫

∞

−∞

dx(x2+a2)(x2+b2)

,a,b > 0

10. Evaluate the following improper integrals.

(a)∫

∞

0cosax

(x2+b2)2 dx

(b)∫

∞

0xsinx

(x2+a2)2 dx

11. Find the Cauchy principle value of the integral∫∞

−∞

sinx(x2 +1)(x−1)

dx

defined as

limε→0+

(∫ 1−ε

−∞

sinx(x2 +1)(x−1)

dx+∫

∞

1+ε

sinx(x2 +1)(x−1)

dx).

12. Find a formula for the integral∫

∞

−∞

dx

(1+x2)n+1 where n is a nonnegative integer.

13. Find∫

∞

−∞

sin2 xx2 dx.

14. If m < n for m and n integers, show∫∞

0

x2m

1+ x2n dx =π

2n1

sin( 2m+1

2n π) .

15. Find∫

∞

−∞

1

(1+x4)2 dx.

16. Find∫

∞

0ln(x)1+x2 dx = 0.

17. Suppose f has an isolated singularity at α. Show the singularity is essential if andonly if the principal part of the Laurent series of f has infinitely many terms. Thatis, show f (z) = ∑

∞k=0 ak (z−α)k +∑

∞k=1

bk(z−α)k where infinitely many of the bk are

nonzero.

18. Suppose Ω is a bounded open set and fn is analytic on Ω and continuous on Ω.Suppose also that fn→ f uniformly on Ω and that f ̸= 0 on ∂Ω. Show that for all nlarge enough, fn and f have the same number of zeros on Ω provided the zeros arecounted according to multiplicity.