1
x À a
ð
Þ x À b
ð
Þ
¼
1
a À b
1
x À a
À
1
x À b
:
ð6:214Þ
Then, (6.213) is described by
I ¼
1
a À b
Z 1
À1
cos z
z À a
À
cos z
z À b
dz
¼
1
2 a À b
ð
Þ
Z 1
À1
e
iz
þ e
Àiz
z À a
À
e
iz
þ e
Àiz
z À b
dz:
ð6:215Þ
We wish to consider the integration by dividing the integrand and calculate each
term individually. First, we estimate
Z 1
À1
e
iz
z À a
dz:
ð6:216Þ
To apply the Jordan’s lemma to the integration of (6.216), we use the upper
contour C that consists of the real axis, Γ a , and Γ R ; see Fig. 6.23a. As usual, we
consider a following integral:
lim
R!1
I C ¼ P
Z 1
À1
e
iz
z À a
dz þ
Z
Γ a
e
iz
z À a
dz þ
Z
Γ 1
e
iz
z À a
dz,
ð6:217Þ
where Γ a represents an infinitesimally small semicircle around z ¼ a. Notice that
(6.217) is obtained when we are considering R ! 1 in Fig. 6.23a. The third term of
(6.217) vanishes due to the Jordan’s lemma. With the second term of (6.217), we
change variable z À a ! z and rewrite it as
0
−
Γ
Γ
(a)
0
−
Γ
Γ
(b)
Fig. 6.23 Contour for the integration of
cos z
zÀa
ð
ÞzÀb ð
Þ that appears in Example 6.10. (a) Along the upper
contour C. (b) Along the lower contour C
0
246
6 Theory of Analytic Functions