preprints.org > engineering > electrical and electronic engineering > doi: 10.20944/preprints202410.0628.v1 (registering DOI)

Preprint Article Version 1 This version is not peer-reviewed

Version 1 : Received: 8 October 2024 / Approved: 8 October 2024 / Online: 9 October 2024 (14:40:48 CEST)

Some mathematical problems reach such complexity that their solution and engineering interpretation is no longer possible, or at least extremely difficult for researchers without the use of artificial intelligence tools. Mathematical relations for such problems are very difficult to interpret. In connection with the development of means for mathematical calculations, such problems have partially lost their relevance. However, new problems can be posed in mathematics, for which the existing means of mathematical calculations may still be insufficient. Presumably, such problems include the problem of differentiation and integration to a complex degree. Derivation of various functions is widely used in many branches of mathematics, technology, and science. Historically, derivation was known for cases where the exponent index of the degree of derivation was a positive integer value, which meant the multiplicity of taking the derivation operation. Later, this operation was extended with the notion that the exponent index can also be negative, which means multiple integration. Derivation to a negative power is defined as integration, and integration to a negative power is defined as derivation. Later, the concept of the possibility of fractional (non-integer) derivation and, accordingly, fractional integration was developed and widely used. This extension of the mathematical tool proved to be of considerable practical value, since it permits the design and implementation of more efficient controllers for systems having negative feedback, for instance. Publications about taking the derivative to a purely imaginary degree have already appeared, but, apparently, the question of differentiation was also discussed in the literature, in which the degree of taking the derivative would be expressed by a complex number. The article suggests an approach to solving this problem, which may not have been discussed yet. If this complex number, denoting the degree of differentiation, has a positive real part, the operation is better called a special form of differentiation, but if the real part of the degree of differentiation is negative, then the operation is more consistent with the concept of integration. Formally, inverting the exponent of the degree of differentiation turns the operation into integration and vice versa. Throughout history, it has been repeatedly confirmed that mathematics occasionally solves problems that at the time of their discovery have no obvious applied value; however, the development of the theory is valuable in and of itself, even if there is currently no obvious applied value for such development. In addition, experience has shown that every new mathematical tool will ultimately be applied to a significant practical issue.

automation; fractional derivation; fractional integration; Laplace transform; complex numbers

Engineering, Electrical and Electronic Engineering

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