310 CHAPTER 8 • RESIDUE THEORYthe residues, we obtainRes[/, i] = ~i and Res [f, 2i) =
1
i
2
.Using Theorem 8.3, we conclude that- EXAMPLE 8.16 Evaluate f~ 00 (of: 4 )s ·
Solution The integrand f (z) = (z•! 4 )" has a pole of order 3 at the point
2i, which is the only singularity of f in the upper half-plane. Computing the
residue, we getRes If 2il = ~ lim d
(^2 1)
' 2 z - 2i dz^2 (z + 2i)^3
1 d - 3
= -lim
2 • -2i dz (z + 2i)^4
1 12 - 3i
= - lim = -
2 · -2; (z + 2i)^5 512Th ere1ore, < J"" -oo (x•+4)" dx -^2 1l't · (-s 12 3i) - 256. 3,.
-------~EXERCISES FOR SECTION 8.3Use residues to evaluate1 J"" x2da;. -oo (x2 + 16)2.
(^2) • J-"" 00 ~ x (^2) + 16'
3 J"" x dx
· - oo (x2 + 9 )2 ·
4 J"" x+3 dx
- -oo (x2 + 9 )2 ·
5 J"" 2x 2 + 3 d
. - oo (x2 + 9) 2 x.
(^6). J"" -oox~ 4 +4'