f z
ð Þ ¼
X 1
k¼0
A k z À a
ð
Þ
k þ
X 1
k¼1
A Àk
1
z À a
ð
Þ
k
ð6:134Þ
so that the constitution of the Laurent’s series can explicitly be recognized. The
second term is called a singular part (or principal part) of f (z) at z ¼ a. The
singularity is classified as follows:
(1) If (6.134) lacks the singular part (i.e., A Àk ¼ 0 for k ¼ 1, 2, Á Á Á), the singular
point is said to be a removable singularity. In this case, we have
f z
ð Þ ¼
X 1
k¼0
A k z À a
ð
Þ
k :
ð6:135Þ
If in (6.135) we define suitably as
f a
ð Þ A 0 ,
f (z) can be regarded as analytic. Examples include the following case where a
sin function is expanded as the Taylor’s series:
sin z ¼ z À
z
3
3!
þ
z
5
5!
À Á Á Á þ
À1
ð Þ
nÀ1 z
2nÀ1
2n À 1
ð
Þ!
þ Á Á Á ¼
X 1
n¼1
À1
ð Þ
nÀ1 z
2nÀ1
2n À 1
ð
Þ!
: ð6:136Þ
Hence, if we define
f z
ð Þ
sin z
z
and f 0
ð Þ lim
z!0
sin z
z
¼ 1,
ð6:137Þ
z ¼ 0 is a removable singularity.
(2) Suppose that in (6.134) we have a certain positive integer such that
A Àn 6 ¼ 0 but A À nþ1
ð
Þ ¼ A À nþ2
ð
Þ ¼ Á Á Á ¼ 0,
ð6:138Þ
the function f (z) is said to have a pole of order n at z ¼ a. If, in particular, n ¼ 1
in (6.138), i.e.,
A À1 6 ¼ 0 but A À2 ¼ A À3 ¼ Á Á Á ¼ 0,
ð6:139Þ
the function f (z) is said to have a simple pole. This special case is important in
the calculation of various integrals (vide infra). In the above cases, (6.134) can
be rewritten as
224
6 Theory of Analytic Functions