Page 98 - Calculus with Complex Numbers

= j ei (x-biy) j= j eix-y j= j eixe-y j = e-y :j( 1 on yz.

as R --> co.

The residue at z = - 1 + i is c-1 -i (li . Therefore,
co eix ty .x - 1 - i
= 7(e .
x2 + 2
. .x + 2

The residue at c = û? is
2 + 1 û)7 + 0)5 .//'-/ i
2
*
2
4*3 4 4 2.$//--
/
The residue at c = 0)3 is
* 6 + 1 û)6 + 1 a)5 + .)7 yxi i
4*9 4* 4 4 2.$ ,7-1'
,
.:2 + 1 n'RLRI + 1)
dz s 4 (R > 1) --> O as R --> co.
/ . :4 + 1 R - 1
2
Use a pizza slice contour with angle 2>/5.
8. Observe that for .x real
sin.x 3 3 sin x - sin 3.x 3c01 - e3ix
.
= = Im ,
. x 4.x3 4x3
and that for z complex

3 eiz - e3iz 3 (1 + iz - z2/ 21 + ...) - (1 + 3ïc - 9:2/ 21 + ... )
4c3 4c3