1. Introduction
The fact in Mathematical Analysis of the supposed impossibility of operating with indeterminate terms is known. The following article introduces a series of concepts that allow formulating and proving theorems that states that under certain conditions, operations with indeterminate terms give an ordinary real value. In the study of calculus, some derivation and integration methods are learned, as well as the great importance that these mathematical tools have in science and engineering. This is what we know as ordinary or integer integral and differential calculus. Here some questions arise: Why should n be 1, 2, 3? Is there a possibility that n is a real number? [1]. Mathematicians who have proposed different definitions for the fractional calculation (keeping in mind n as a fraction) have been, among others: Lacroix, Euler, Riemann, Caputo etc., each with their own reasoning [1] [2]. In this work, we focus on generalizing the derivative of a power function to the field of reals and taking as reasoning that there is a set of mathematical functions expressible as an infinite sum of power functions, by means of the distributive property of the derivative with respect to the sum, we generalize the concept of continuous derivative to developable functions as power series. Concepts like dimensional analytical extension arise naturally. Then we use the sine function, because it is developable as a series of powers and at the same time, due to its properties of periodicity and complementarity with the cosine function, it has an extra generalization for its development as a continuous derivative. This double definition for the sine function is a bridge that allows us to arrive at the quasi-paradoxical result of the ind-series (series of zero, indeterminate and infinite terms).
2. The Continuous Derivative Operator
2.1. The Exponential Function
Let
where
and
. Then
We can generalize the repetition of derivative operator to the field of reals
where
.
2.2. The Power Function
Let
where
and
. Then
We can generalize the repetition of derivative operator to the field of reals
where
So
3. Functional and Analytical Dimensional Extensions
3.1. Functional Extension
Let
such that
then:
such that
is a functional extension of
if: there exists at least one value of y, whereby it is true that:
3.2. Analytical Dimensional Extension
As a consequence of the expressed above, let
be continuously operable by an integral or derivative operator along
.
In case of the derived function, we have
.
So
is a special case of functional extension, that we can call analytical dimensional extension of
.
It is true that
.
4. Application of the Concept to Another Functions
4.1. Development in Power Series
Note that
So, it is followed that all function expressible by a power series have real derivative, and consequently a-d extension.
4.2. The Sine Function
Using the Taylor development formula, we can express the sine function as follows:
On the other hand, we use another generalization, trough the following basic property:
So
Therefore
At the right side of the equality we observe that appears
, that in some cases could be equal to
, but by intuition we try to avoid a situation where the series term were indeterminate or infinite.
Concepts such as infinite or infinitesimal are based on the hyper-real numbers. The infinitesimals would be numbers smaller than any conventional real number, and their respective inverses would correspond to “infinite” or “unbounded” numbers.
Therefore:
If
then
and consequently
.
But by its own definition the
function never equals zero.
Consider for one instance that
is a number. Then we have a series with terms whose values are zeroes, infinite and minus infinite and the series equals a real ordinary number.
Let’s call them the ind-series or
. As sin function takes values in the
interval, and the same is bijectable to the
numbers (through a reciprocal function, for example). Then we are in conditions to prove some theorems:
5. Formal Proof
Definition 1 (Continuous Derivative of power function).
Be
where
. Is the Continuous Derivative of
of q order.
Lemma 1 (Continuous Derivative of a function developable by power series).
Be
developable by power series as
.
where
and
. And it is called the Continuous Derivative of
of q order.
Proof of Lemma 1.
Using the distributive property of the derivative operator respect to the sum of functions and the Definition 1, the lemma is proved. □
Lemma 2 (Symmetry of values of sine function).
Be
Proof of Lemma 2.
Using the properties of the sine function is proved.
The same way,
□
Definition 2 (Functional extension).
Be
is a functional extension of
if:
Property 1 (Derivative or zero order).
Proof of property 1.
Using Definition 1 and
, the property is proved. □
Definition 3 (Analytical-dimensional extension or a-d extension).
is called a-d extension of
. And it is a special case of functional extension because:
(by Property 1)
i.e.:
Definition 4 (Alternate a-d extension of sine function).
Using the ordinary derivative definition and basic properties of sine function:
easily
where
. So using a process of generalization to
where
.
Principle 1 (Equivalence of
and
).
We take as true that:
Given particular values
and
. You can compute both a-d extensions and it is verifiable the coincidence of both outcomes.
Definition 5 (Ind-series).
It is called ind-series.
Theorem 1 (the ind-series Theorem).
Every real number y is expressible as a reciprocal value of ind-series
and
where
, whose core domain is
and
and
are symmetric respect to the ordinates
if
and
are in the
interval and
if both of them are in the
interval.
Proof of Theorem 1.
(by Property 1)
(by Definition 3 and Principle 1)
(By Definition 5)
(By lemma 2)
so
.
Be
.
Then
.
The steps above prove the Theorem. □
5.1. Analysis of Terms in Series of Indeterminate Values
If
then the general term is
If
then the general term is
If
then the general term is
where
.
So at first sight you probably observe that, for instance
,
where
.
It seems to break the axioms of arithmetic, but you have to take into account that you are operating with infinite amount of terms.
Besides, you have to make another analysis on the set of real numbers where the
function is not defined.
Remember that the indeterminate are:
5.2. Components of the Series
Based on Section 7, we can express the ind-series to its eventual and deeper analysis as follows:
where:
is the partial sum where
(
component)
is the partial sum where
(00 component)
is the partial sum where
(0 component)
Therefore we can offer a notation for real numbers, called trienial.
Be
then:
could be expressible for the following possibilities:
where
represents the
components,
the 00 component, and
the 0 component. And
belongs to
.
5.3. A Last Point to Analyze
Perhaps an issue pending analysis is to investigate the correspondence between a polynomial series
and a polynomial series
.
In this case we can carry out a similar analysis as Section 6.
5.4. Example of the Method
5.5. Conclusion
The facts that this class of series is not directly computational and any convergence criterion cannot be applied to them, stand out and give relevance to those above mentioned theorems.
The approaches offered by the non-standard analysis and the way it is showed to operate with the “unbounded” and “infinitesimal” numbers do not lead to some outcome [3].