Res f À1
ð Þ ¼ z
aÀ1
z¼À1
¼ À1
ð Þ
aÀ1 ¼ e
iπ
À Á aÀ1 ¼ Àe
aiπ
:
Then, we have
I C ¼ 2πi Res f À1
ð Þ ¼ À2πi e
aiπ
:
ð6:249Þ
When R ! 1 and r ! 0, we have
Z
Γ R
z
aÀ1
1 þ z
dz
<
R
aÀ1
R À 1
Á 2πR ! 0 and
Z
Γ 0
z
aÀ1
1 þ z
dz
<
r
aÀ1
1 À r
Á 2πr ! 0: ð6:250Þ
Thus, the second and fourth terms of (6.248) vanish.
Meanwhile, we take the principal value of ln z of (6.244). Since the lines PQ and
Q
0
P
0 are very close to the real axis, using (6.245) with n ¼ 0 we can put θ ¼ 0 on PQ
and θ ¼ 2π on Q
0 P
0
. Then, we have
Z
PQ
z
aÀ1
1 þ z
dz ¼
Z
PQ
e
aÀ1
ð
Þln z
1 þ z
dz ¼
Z
PQ
e
aÀ1
ð
Þln jzj
1 þ z
dz:
ð6:251Þ
Moreover, we have
Z
Q
0 P
0
z
aÀ1
1 þ z
dz ¼
Z
Q
0 P
0
e
aÀ1
ð
Þln z
1 þ z
dz ¼
Z
Q
0 P
0
e
aÀ1
ð
Þ ln z
j jþ2πi
ð
Þ
1 þ z
dz
¼ e
aÀ1
ð
Þ2πi
Z
Q
0 P
0
e
aÀ1
ð
Þln z
j j
1 þ z
dz:
ð6:252Þ
Notice that the argument underneath the branch cut (i.e., the line Q
0 P
0
) is
increased by 2π relative to that over the branch cut.
Considering (6.249) through (6.252) and taking the limit of R ! 1 and r ! 0,
(6.248) is rewritten by
À2πi e
aπi
¼
Z 1
0
e
aÀ1
ð
Þln z
j j
1 þ z
dz þ e
aÀ1
ð
Þ2πi
Z 0
1
e
aÀ1
ð
Þln z
j j
1 þ z
dz
¼ 1 À e
aÀ1
ð
Þ2πi
h
i Z 1
0
e
aÀ1
ð
Þln z
j j
1 þ z
dz
¼ 1 À e
aÀ1
ð
Þ2πi
h
i Z 1
0
z
aÀ1
1 þ z
dz:
ð6:253Þ
In (6.253) we have assumed z to be real and positive (see Fig. 6.29), and so we have
e
(a À 1) ln jz|
¼ e
(a À 1) ln z
¼ z
a À 1 . Therefore, from (6.253) we get
258
6 Theory of Analytic Functions