1. Introduction
The properties of analytic functions have been given in references [1,2]. The theory of analytic functions was extended to vector valued function in reference [3].
In this paper, we extended the theory of vector valued function to locally convex space.
Let
be a complete Hausdorff locally convex space on the real or complex domain
, and
be the sufficient directed set of semi norms which generates the topology of
. We denote the ad joint space of
by
, i.e.
is the set of linear bounded functions on
.
Definition 1 Let
be a vector function defined on a domain
with values in
. If there is an element
such that the difference quotient
tends weakly(strongly) to
as
, we call
the weakly (strongly) derivative of
at
. We also say that
is weakly (strongly) derivative at
in
. We call
weakly (strongly) derivative in
.
Definition 2 A vector function
is
1) weakly continuous at
if
for each
.
2) strongly continuous at
if
for each
.
Definition 3 A vector function
is said to be regular in
if
is regular for every
, where range of
is in
. If a vector valued function
is regular in
, then
is called an entire function or said to be entire.
Theorem 1 [4] (Cauchy) If
is a regular vector-valued function on the domain
with values in the locally convex space
. Let
be a closed path in
, and assume that
is homologous to zero in
, then
![](https://www.scirp.org/html/12-7400907\0d285dff-d2ce-4cdd-9799-5f1b15f667b1.jpg)
where c is a circle.
Proof For any linear bounded functional
, we have
Hence
![](https://www.scirp.org/html/12-7400907\0c4b66db-6087-4aae-9536-59393f767285.jpg)
Theorem 2 [5] (Cauchy integral formula) Let
be a regular vector-valued function on the domain
with values in the locally convex space
. Let
be a closed path in
, and assume that
is homologous to zero in
, and let
be in
and not on
. Then
(1)
where
is the index of the point
with respect to the curve
.
Proof For any linear bounded functional
, we have
.
Then
![](https://www.scirp.org/html/12-7400907\891d1f5c-38f8-4d25-b418-edd3e3a68742.jpg)
2. The Main Conclusions
Theorem 3 Given the power series
. (2)
Set
. Then the power series (2)
is absolutely convergent for
and divergent for
. The power series (2) convergence to a regular function on
with values in
, the convergence being uniform in every circle of radius less than
.
Proof First, we will prove the power series (2) is absolutely convergent for
and divergent for
.
By Theorem 1, for any
, we have
where
.
Let
, then
![](https://www.scirp.org/html/12-7400907\3e29e934-0a6c-4e04-8601-480ad0949d1f.jpg)
where
. Thus the power series (2) is absolutely convergence. But for
, if we suppose the power series (2) is convergence, it is contradict with the radius is
. So the power series (2) is absolutely convergent for
and divergent for
.
Secondly, for any linear bounded functional
, we have
.
The right side series convergence to a regular function on
with values in
. So
is regular in the circle and the convergence being uniform.
Definition 4 Let
have an isolated singularity at
and let
(3)
where
(4)
be its Laurent Expansions about
. The residue of
at
is the coefficient
. Denote this by
.
Theorem 4 Let
be a regular vector-valued function except for a finite number of points
in the domain
. Let
be a closed path in
, and assume that
is homologous to zero in
, and let
be in
and not on
. Then
(5)
Proof For any linear bounded functional
, we have
.
Then
![](https://www.scirp.org/html/12-7400907\3f86ffa7-02bf-4a63-a6cf-6076a8d41229.jpg)
Theorem 5
1) If
has a pole of order one at a point ![](https://www.scirp.org/html/12-7400907\f78f2428-d449-44cc-a6b9-80b11704a309.jpg)
then
(6)
2) If
has a pole of order
at a point
then
(7)
Proof Because
has a pole of order
at a point
, then
can be written in the form
![](https://www.scirp.org/html/12-7400907\7ae27285-06e7-4182-87bf-d1c3dbed78d6.jpg)
where
is regular and nonzero at
.
So
has a power series representation
![](https://www.scirp.org/html/12-7400907\5c1598ba-baf6-48bb-aa24-179d72a9ecd1.jpg)
in some neighborhood of
. It follows that
![](https://www.scirp.org/html/12-7400907\0b122c0b-f8de-4727-b0b3-1f1ab602de72.jpg)
in some neighborhood of
. Then we have formula (7)
![](https://www.scirp.org/html/12-7400907\55eb9cec-0512-480c-b470-314d3be63d55.jpg)
Obviously, when
, the formula (7) is formula (6).
Theorem 6 If
![](https://www.scirp.org/html/12-7400907\1029b61c-ed3a-4e7e-9626-1605f375b2bc.jpg)
where
for
and if
exists, then
exist and has a pole with order
at
.
Proof Since
![](https://www.scirp.org/html/12-7400907\86242ab4-aeb2-4713-aae7-61c8d767b339.jpg)
For any linear bounded functional
, we have
as
where
is sufficiently small. Thus
.
It follows that
![](https://www.scirp.org/html/12-7400907\9ffcbf3e-b73f-449a-84fb-77b45fadd059.jpg)
Therefore
![](https://www.scirp.org/html/12-7400907\27c32e53-936b-4e1b-8a62-2c37fe7a36af.jpg)
where ![](https://www.scirp.org/html/12-7400907\40ccc0b9-1b06-427b-8e8e-6801347c1f6b.jpg)
Remark:
exist, this condition is important.
For example, in
, we define
, where
and For any linear bounded functional ![](https://www.scirp.org/html/12-7400907\0420ab84-3946-44c4-8469-37981de26237.jpg)
.
Thus
is a B-algebra, and
. We set
where
and
. It follows that
is zero with order one, but
![](https://www.scirp.org/html/12-7400907\f4ce4f53-ad08-4dd1-9d10-e4a302864ad9.jpg)
With order three.
Theorem 7 If
and
are regular in
with values in
and if
,
, the points
having a limit point in
, then
in
.
Proof For any linear bounded functional
, we have
![](https://www.scirp.org/html/12-7400907\5fa788fb-c4fe-4eb3-8823-71d822cbbfdb.jpg)
So
.
Theorem 8 Let
be defined in a domain
of the extended plane and on its boundary
, regular in
and strongly continuous in
. If
then either
or
in
.
Proof For any linear bounded functional
, we have
.
But except
is constant,
. So either
or
in
.
Remark: Unlike the classical case,
may have a minimum other zero in
as the following example shows.
For example, Let
be a Banach space of complex pairs,
, where
.
Set
Then
,
for ![](https://www.scirp.org/html/12-7400907\d31b9369-5942-4bf0-85ad-3e35361f7451.jpg)
and
for
.
Theorem 9 If
is regular in
, and if
is bounded in
, then
constant element.
Proof For any linear bounded functional
, we have
.
So
is bounded in
, then
is constant.
Suppose
is not constant, then exist two point
such that
.
Thus exist
satisfy
.
This is contradict with
is constant. So
constant element.
Theorem 10 If
is regular in the unit circle, satisfy the condition
and
. Then
.
Proof For any linear bounded functional
, we have
.
Since every point
, their exist a bounded function
such that
.
So
.
Then
.