Here let us pause for a second to consider the differentiation of a function.
Suppose that a mathematical function is defined on a real domain (i.e., a real number
line). Consider whether that function is differentiable at a certain point x 0 of the real
number line. On this occasion, we can approach x 0 only from two directions, i.e.,
from the side of x < x 0 (from the left) or from the side of x 0 < x (from the right); see
Fig. 6.10a. Meanwhile, suppose that a mathematical function is defined on a
complex domain (i.e., a complex plane). Also consider whether the function is
differentiable at a certain point z 0 of the complex plane. In this case, we can approach
z 0 from continuously varying directions; see Fig. 6.10b where only four directions
are depicted.
In this context let us think of a simple example.
Example 6.1 Let f (x, y) be a function described by
f x, y
ð Þ ¼ 2x þ iy ¼ z þ x:
ð6:45Þ
As remarked above, substituting (6.43) for (6.45), we could have
h z, z
Ã
ð
Þ ¼ 2 Á
1
2
z þ z
Ã
ð
Þþi À
i
2
z À z
Ã
ð
Þ
h
i
¼ z þ z
Ã
þ
1
2
z À z
Ã
ð
Þ¼
1
2
3z þ z
Ã
ð
Þ,
where f (x, y) and h(z, z
à ) denote different functional forms. The derivative (6.45)
varies depending on a way z 0 is approached. For instance, think of
df
dz
0
¼ lim
z!0
f 0 þ z
ð
ÞÀf 0
ð Þ
z
¼ lim
x!0, y!0
2x þ iy
x þ iy
¼ lim
x!0, y!0
2x
2
þ y
2
À ixy
x 2 þ y 2
:
Suppose that the differentiation is taken along a straight line in a complex plane
represented by iy ¼ (ik)x (k, x, y : real). Then, we have
df
dz
0
¼ lim
x!0, y!0
2x
2
þ k
2 x
2
À ikx
2
x 2 þ k
2 x 2
¼ lim
x!0, y!0
2 þ k
2
À ik
1 þ k
2
¼
2 þ k
2
À ik
1 þ k
2
: ð6:46Þ
However, this means that
df
dz
0
takes varying values depending upon k. Namely,
df
dz
0
cannot uniquely be defined but depends on different ways to approach the origin
of the complex plane. Thus, we find that the derivative takes different values
depending on straight lines along which the differentiation is taken. This means
that f (x, y) is not differentiable or analytic at z ¼ 0.
Meanwhile, think of g(z) expressed as
g z
ð Þ ¼ x þ iy ¼ z:
ð6:47Þ
In this case, we get
6.2 Analytic Functions of a Complex Variable
201
Suppose that a mathematical function is defined on a real domain (i.e., a real number
line). Consider whether that function is differentiable at a certain point x 0 of the real
number line. On this occasion, we can approach x 0 only from two directions, i.e.,
from the side of x < x 0 (from the left) or from the side of x 0 < x (from the right); see
Fig. 6.10a. Meanwhile, suppose that a mathematical function is defined on a
complex domain (i.e., a complex plane). Also consider whether the function is
differentiable at a certain point z 0 of the complex plane. In this case, we can approach
z 0 from continuously varying directions; see Fig. 6.10b where only four directions
are depicted.
In this context let us think of a simple example.
Example 6.1 Let f (x, y) be a function described by
f x, y
ð Þ ¼ 2x þ iy ¼ z þ x:
ð6:45Þ
As remarked above, substituting (6.43) for (6.45), we could have
h z, z
Ã
ð
Þ ¼ 2 Á
1
2
z þ z
Ã
ð
Þþi À
i
2
z À z
Ã
ð
Þ
h
i
¼ z þ z
Ã
þ
1
2
z À z
Ã
ð
Þ¼
1
2
3z þ z
Ã
ð
Þ,
where f (x, y) and h(z, z
à ) denote different functional forms. The derivative (6.45)
varies depending on a way z 0 is approached. For instance, think of
df
dz
0
¼ lim
z!0
f 0 þ z
ð
ÞÀf 0
ð Þ
z
¼ lim
x!0, y!0
2x þ iy
x þ iy
¼ lim
x!0, y!0
2x
2
þ y
2
À ixy
x 2 þ y 2
:
Suppose that the differentiation is taken along a straight line in a complex plane
represented by iy ¼ (ik)x (k, x, y : real). Then, we have
df
dz
0
¼ lim
x!0, y!0
2x
2
þ k
2 x
2
À ikx
2
x 2 þ k
2 x 2
¼ lim
x!0, y!0
2 þ k
2
À ik
1 þ k
2
¼
2 þ k
2
À ik
1 þ k
2
: ð6:46Þ
However, this means that
df
dz
0
takes varying values depending upon k. Namely,
df
dz
0
cannot uniquely be defined but depends on different ways to approach the origin
of the complex plane. Thus, we find that the derivative takes different values
depending on straight lines along which the differentiation is taken. This means
that f (x, y) is not differentiable or analytic at z ¼ 0.
Meanwhile, think of g(z) expressed as
g z
ð Þ ¼ x þ iy ¼ z:
ð6:47Þ
In this case, we get
6.2 Analytic Functions of a Complex Variable
201