Application of Differentialintegral Functions
The article is devoted to the development and implementation of new mathematical functions, differentialintegral functions that provide differentiation and integration operations not only according to existing algorithms described in textbooks on higher mathematics, but also by substituting a certain parameter k into formulas developed in advance, forming the necessary derivatives and integrals from these formulas. The Purpose of the Research: The expansion of the concept of number, in particular, in classical mechanics, physics, optics and other sciences, including biological and economic, which makes it possible to expand some understanding of the essence of space, time and their derivatives. Materials and Methods: The idea of fractional space, time and its application is given. The usual elementary functions and the Laplace transform were chosen as the object of research. New functions, differentialintegral functions, have been developed for them. A graphical representation of these functions is given, based on the example of the calculation of the sine wave. Examples of calculating these functions for elementary functions are given. Of particular interest is the differentialintegral function, in which the parameter k is a complex number s, s = a + i • b, although in general, the parameter k can be any function of a real or complex argument, as well as the differentialintegral function itself. Research Results: As a result of the research, it is shown how the Laplace transform and Borel’s theorem are used to calculate differentialintegral functions. It is shown how to use these functions to carry out differentiation and integration. It is presented how fractional derivatives and fractional integrals should be obtained. Dependencies for their calculation are obtained. Examples of their application for such functions as cos(x), exp(x) and loudness curves in music, Fletcher-Manson or Robinson-Dadson curves are shown. Conclusions: Studies show the possibility of a wide application of differentialintegration functions in modern scientific research. These functions can be used both in office and in specialized programs where calculations of fractional derivatives and fractional integrals are needed.
