Properties of sums of some elementary functions and modeling of transitional and other processes

Yuri Shestopaloff

Published 2008-11-07, updated 2012-11-24Version 3

The article presents mathematical generalization of results which originated as solutions of practical problems, in particular, the modeling of transitional processes in electrical circuits and problems of resource allocation. However, the presented findings have broader meaning and can be used for approximation of transitional and other processes in different areas of science and technology. We present discovered properties of sums of polynomial, power, and exponential functions of one variable. It is shown that for an equation composed of one type of function there is a corresponding equation composed of functions of the other type. The number of real solutions of such equations and the number of characteristic points of certain appropriate corresponding functions are closely related. In particular, we introduce a method similar to Descartes Rule of Signs that allows finding the maximum number of real solutions for the power equation and equation composed of sums of exponential functions. The discovered properties of these functions allow us to improve the adequacy of mathematical models of real phenomena.