Information Technology Reference
In-Depth Information
where j is the imaginary unit defined by the equation
j
2
=−
1. It can be verified that
+
,
×
the definitions for
satisfy all the postulates of the Field.
2.3.1.2 Cauchy-Riemann Equations and Liouville's Theorem
In real domain, the property of differentiability is not a very strong property for
functions of real variables. It is surprisingly true that study of complex function for
differentiability (analyticity) is a different topic from real analysis. The power and
importance of complex numbers cannot be exploited until a full theory of analytic
(holomorphic) function is developed. Interested readers may consult the theory of
complex numbers for details, this section only presents the brief discussion on differ-
entiability necessary for the development of learning algorithm in neurocomputing.
Definition 2.3
A complex valued function
f
:
C
−→
C
is said to be
analytic
(com-
plex differentiable) at
C
if the following limit exists at every point z in the
complex plane. If the function is analytic over the whole finite complex plane, it is
said to be an entire function.
z
∈
f
(
z
+
ʴ
z
)
−
f
(
z
)
Limit
ʴ
z
→
0
(2.1)
ʴ
z
The definition demands that the function be differentiable at every point in some
neighborhood of the point z. The function
f
is said to be differentiable at z when its
derivative at
z
exists. The limit will be called the first derivative of
f
at
z
and denoted
by
f
(
z
)
.
Further, there are a pair of equations that the first-order partial derivatives of the
component functions of a function (
f
) with complex variable, must satisfy at a point
when the derivative of
f
exists there.
Let
f
be a complex valued function. Cauchy-Riemann
equations (
CR
) are pair of equations that first-order partial derivatives of the com-
ponent functions,
U
and
V
, of function f must satisfy at a point when the derivative
of
f
exist there.
Definition 2.4
The first order partial derivatives of component function of
f
must
exist. A complex valued function
f
(
z
)
=
U
(
x
,
y
)
+
jV
(
x
,
y
)
(
z
)
=
U
(
x
,
y
)
+
jV
(
x
,
y
)
is said to satisfy
CR
equations if the following equalities hold:
ʴ
U
ʴ
x
=
ʴ
V
ʴ
U
ʴ
y
=−
ʴ
V
and
(2.2)
ʴ
y
ʴ
x
It can be shown that every analytic function satisfies CR equations. The converse
is true if an additional condition of continuity of the partial derivatives of the CR is
assumed (Ahlfors 1979).
Above equations, not only give the derivative of
f
in terms of partial derivatives
of component function, but also they (
CR
equations) are necessary conditions for the
Search WWH ::
Custom Search