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
where c is a circle.
Proof For any linear bounded functional, we have
Hence
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
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
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
Theorem 5
1) If has a pole of order one at a point
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
where is regular and nonzero at.
So has a power series representation
in some neighborhood of. It follows that
in some neighborhood of. Then we have formula (7)
Obviously, when, the formula (7) is formula (6).
Theorem 6 If
where for and if exists, then exist and has a pole with order at.
Proof Since
For any linear bounded functional, we have
aswhere is sufficiently small. Thus
.
It follows that
Therefore
where
Remark: exist, this condition is important.
For example, in, we define, where and For any linear bounded functional
.
Thus is a B-algebra, and. We set
where and. It follows that is zero with order one, but
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
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
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
.