Page 97 - Calculus with Complex Numbers
P. 97
6 Eval uation of i nfi nite real i ntegrals
1 h
as singularities at z = +2/ .
. z2 + 4
The residue at z = li is 1/4ï (differentiate the denominator).
dz zr R
:é - (R > 2) --> 0 as R --> co.
/ . :2 + 4 R'L - 4
2
vlz ,,.p
/
./' (c2 + l)(c2 + 4.) ' (.p2 - 1)(.p2 - 4.) CR > 2) -' '' as R --> co.
2
To get the residue we have to compute the Laurent expansion. Put t = z - i .
Then we have
(using the binomial theorem with exponent -2). Therefore the residue is 1/4ï .
as R --> co.
as R --> co.
C<' cosw dx C<' eix dx
= Re .
. x2 + 2 . x + 2 -oo .x2 + 2 .x + 2
