Abstract
It is known that functions involving natural numbers are generalized to the real ones, for instance the gamma function can be viewed as a generalization of the factorial operator. In this paper, we propose to generalize the repetition of an operation over a function (composition, derivatives and integrals) toward the field of reals. It means repeating q times an operation over a function, where q is a real number. As a result, it is explained what functional and analytical dimensional extensions are and it is given a proof to theorems related to the indeterminate terms. The main finding is that every real number is expressible as a bijection of an infinite sum of elements whose coefficients are real numbers and their main values are either an indeterminate value or an infinite value. The concept of series of indeterminate values becomes relevant, as a novelty to operate with infinite, zero and indeterminate terms, which cannot be deductible from the non-standard analysis.
Highlights
The fact in Mathematical Analysis of the supposed impossibility of operating with indeterminate terms is known
We propose to generalize the repetition of an operation over a function toward the field of reals. It means repeating q times an operation over a function, where q is a real number. It is explained what functional and analytical dimensional extensions are and it is given a proof to theorems related to the indeterminate terms
The main finding is that every real number is expressible as a bijection of an infinite sum of elements whose coefficients are real numbers and their main values are either an indeterminate value or an infinite value
Summary
The fact in Mathematical Analysis of the supposed impossibility of operating with indeterminate terms is known. 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. Some questions arise: Why should n be 1, 2, 3? Some questions arise: Why should n be 1, 2, 3? Is
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