C 2 , C 3 , and C 4 (i.e., convergence circles of radius 1). The said region R is colored
pale green in Fig. 6.17. There are four petals as shown.
We can view Fig. 6.17 as follows: For instance, f 1 (z) and f i (z) are defined in C 1
and its inside and C 2 and its inside, respectively, excluding z ¼ 0. We have
f 1 (z) ¼ f i (z) within the petal P 1 (overlapped part as shown). Consequently, from
the statement of Theorem 6.16, f 1 (z) f i (z) throughout the region encircled by C 1
and C 2 (excluding z ¼ 0). Repeating this procedure, we get
f 1 z
ð Þ f i z
ð Þ f À1 z
ð Þ f Ài z
ð Þ:
Thus, we obtain the same functional entity throughout R except for the origin
(z ¼ 0). Further continuing similar procedures, we finally reach the entire complex
plane except for the origin, i.e., ℂ À {0} regarding the domain of analyticity of 1/z.
We further compare the above results with those for analytic function defined in
the real domain. The tailor’s expansion of f (x) ¼ 1/x (x: real with x 6 ¼ 0) around
a (a 6 ¼ 0) reads as
f x
ð Þ ¼
1
x
¼ f a
ð Þ þ f
0 a
ð Þ þ Á Á Á þ
f
n
ð Þ a
ð Þ
n!
þ Á Á Á ¼
X 1
n¼0
À1
ð Þ
n
a nþ1 x À a
ð
Þ
n : ð6:142Þ
This is exactly the same form as that of (6.119) aside from notation of the
argument.
The aforementioned procedure that determines the behavior of an analytic function outside the region where that function was originally defined is called analytic
continuation or analytic prolongation. Comparing (6.119) and (6.142), we can see
that (6.119) is a consequence of the analytic continuation of 1/x from the real domain
to the complex domain.
6.7 Calculus of Residues
The theorem of residues and its application to calculus of various integrals are one of
central themes of the theory of analytic functions.
The Cauchy’s integral theorem (Theorem 6.10) tells us that if f (z) is analytic in a
simply connected region R , we have
I
C
f z
ð Þdz ¼ 0,
ð6:84Þ
where C is a closed curve within R . As we have already seen, this is not necessarily
the case if f (z) has singular points in R . Meanwhile, if we look at (6.130), we are
aware of a simple but important fact. That is, replacing n with 1, we have
6.7 Calculus of Residues
227
pale green in Fig. 6.17. There are four petals as shown.
We can view Fig. 6.17 as follows: For instance, f 1 (z) and f i (z) are defined in C 1
and its inside and C 2 and its inside, respectively, excluding z ¼ 0. We have
f 1 (z) ¼ f i (z) within the petal P 1 (overlapped part as shown). Consequently, from
the statement of Theorem 6.16, f 1 (z) f i (z) throughout the region encircled by C 1
and C 2 (excluding z ¼ 0). Repeating this procedure, we get
f 1 z
ð Þ f i z
ð Þ f À1 z
ð Þ f Ài z
ð Þ:
Thus, we obtain the same functional entity throughout R except for the origin
(z ¼ 0). Further continuing similar procedures, we finally reach the entire complex
plane except for the origin, i.e., ℂ À {0} regarding the domain of analyticity of 1/z.
We further compare the above results with those for analytic function defined in
the real domain. The tailor’s expansion of f (x) ¼ 1/x (x: real with x 6 ¼ 0) around
a (a 6 ¼ 0) reads as
f x
ð Þ ¼
1
x
¼ f a
ð Þ þ f
0 a
ð Þ þ Á Á Á þ
f
n
ð Þ a
ð Þ
n!
þ Á Á Á ¼
X 1
n¼0
À1
ð Þ
n
a nþ1 x À a
ð
Þ
n : ð6:142Þ
This is exactly the same form as that of (6.119) aside from notation of the
argument.
The aforementioned procedure that determines the behavior of an analytic function outside the region where that function was originally defined is called analytic
continuation or analytic prolongation. Comparing (6.119) and (6.142), we can see
that (6.119) is a consequence of the analytic continuation of 1/x from the real domain
to the complex domain.
6.7 Calculus of Residues
The theorem of residues and its application to calculus of various integrals are one of
central themes of the theory of analytic functions.
The Cauchy’s integral theorem (Theorem 6.10) tells us that if f (z) is analytic in a
simply connected region R , we have
I
C
f z
ð Þdz ¼ 0,
ð6:84Þ
where C is a closed curve within R . As we have already seen, this is not necessarily
the case if f (z) has singular points in R . Meanwhile, if we look at (6.130), we are
aware of a simple but important fact. That is, replacing n with 1, we have
6.7 Calculus of Residues
227