z ¼ À1 þ re
iθ ,
ð6:176Þ
with the first term of RHS we have
A 0
Z
Γ À1
ire
iθ dθ
re iθ ¼ A 0
Z
Γ À1
idθ ¼ A 0
Z 0
π
idθ ¼ ÀA 0
Z π
0
idθ ¼ ÀA 0 iπ:
ð6:177Þ
Note that in (6.177) the contour integration along Γ À1 was performed in the
decreasing direction of the argument θ from π to 0. Thus, (6.175) is rewritten as
Z
Γ À1
dz
1 þ z
ð
Þ z 2 À z þ 1
ð
Þ
¼ ÀA 0 iπ þ A n i
X 1
n¼1
r
n
Z 0
π
e
inθ dθ,
ð6:178Þ
where with the last equality we exchanged the order of the summation and integration. The second term of RHS of (6.178) vanishes when r ! 0. Thus, (6.171) is
further rewritten as
I ¼ lim
r!0
I ¼ 2πi Res f e
iπ=3
þ A 0 iπ:
ð6:179Þ
We have only one simple pole for f (z) at z ¼ e
iπ/3 within the semicircle contour C.
Then, using (6.148), we have
Res f e
iπ=3
¼ f z
ð Þ z À e
iπ=3
z¼e iπ=3
¼
1
1 þ e iπ=3
ð
Þe iπ=3 À e Àiπ=3
ð
Þ
¼ À
1
6
1 þ
ffiffi ffi
3
p
i
:
ð6:180Þ
The above calculus is straightforward, but (6.37) can be conveniently used. Thus,
finally we get
I ¼ À
πi
3
1 þ
ffiffi ffi
3
p
i
þ
πi
3
¼
ffiffi ffi
3
p π
3
¼
π
ffiffi ffi
3
p :
ð6:181Þ
That is, I % 1:8138:
ð6:182Þ
To avoid any ambiguity of the principal value, we properly define it as [8]
P
Z 1
À1
f x
ð Þdx lim
R!1
Z R
ÀR
f x
ð Þdx:
238
6 Theory of Analytic Functions