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
as
where 
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
.