Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
then
and so
Z 1
1
dx
1 ‡ x 4 ˆ 2i
4 …ei=4 ‡ e i=4 †ˆ sin 4 ˆ p
2
Z 1
0
dx
1 ‡ x 4 ˆ 1
2
Z 1
1
dx
1 ‡ x 4 ˆ p
2 :
2
Example 6.29
Show that
Z 1
1
x 2 dx
…x 2 ‡ 1† 2 …x 2 ‡ 2x ‡ 2† ˆ 7
50 :
Solution:
The poles of
z 2
f …z† ˆ
…z 2 ‡ 1† 2 …z 2 ‡ 2z ‡ 2†
enclosed by the contour of Fig. 6.17 are z ˆ i of order 2 and z ˆ1 ‡ i of order 1.
The residue at z ˆ i is
" #
d
lim
z!i dz …z z 2
9i 12
i†2
…z ‡ i† 21 …z i† 2 ˆ
…z 2 ‡ 2z ‡ 2† 100 :
The residue at z ˆ1 ‡ i is
Therefore
lim …z ‡ 1 i† z 2
z!1‡i …z 2 ‡ 1† 2 …z ‡ 1 i†…z ‡ 1 ‡ i† ˆ 3 4i
25 :
Z 1
1
x 2
dx
…x 2 ‡ 1† 2 …x 2 ‡ 2x ‡ 2† ˆ 2i 9i 12
100 ‡ 3 4i
25
ˆ 7
50 :
Integrals of the rational functions of sin and cos
Z 2
0
G…sin ; cos †d
G…sin ; cos † is a real rational function of sin and cos ®nite on the interval
0 2. Let z ˆ e i , then
dz ˆ ie i d; or d ˆ dz=iz; sin ˆ…z z 1 †=2i; cos ˆ…z ‡ z 1 †=2
286