A Generalization of Calculus for Use with Continuous or Discrete Variables

TL;DR

This paper introduces a unified calculus framework for both continuous and discrete variables, developing analogues of calculus functions, integrals, and control system analysis tools, with practical computational methods included.

Contribution

It presents a novel generalized calculus that unifies discrete and continuous analysis, including new functions, integrals, and a control system transform, distinct from existing time scales and quantum calculus approaches.

Findings

Discrete exponential function maintains key properties of continuous exponential.

Unified integral evaluation involving special functions like zeta and digamma.

Development of the $K_{\Delta x}$ transform for control system analysis.

Abstract

This document introduces a generalization of calculus that treats both continuous and discrete variables on an equal footing. This generalization of calculus was developed independently of the "Calculus on Time Scales" literature but may be seen to have interesting overlap with it as well as with the "h-Calculus" of the book Quantum Calculus by V. Kac and P. Cheung. As in the time scales literature, we first derive discrete analogues of all the common continuous calculus functions with an eye to maintaining as much similarity as possible between these discrete analogues and their continuous forebears. For example, in order to maintain the crucial property that the derivative of an exponential is a constant times itself, we replace the continuous exponential, $e^{ax}$, with a discrete function, $e_{\Delta x}(a,x) = [1+a \Delta x]^{x / \Delta x}$. Next, we develop a unified method of evaluating integrals of discrete variables. We discover that summations such as the Riemann Zeta Function, the Hurwitz Zeta Function, and the Digamma Function frequently appear in evaluating such integrals. Thus, we subsume these functions into a generalization of the natural logarithm, which we name "$lnd(n,\Delta x,x)$", and evaluate many types of discrete variable integrals in terms of it. We provide a computer program, LNDX, to evaluate the $lnd(n,\Delta x,x)$ function. Then, we develop a theory of control system analysis based on what we name a "$K_{\Delta x}$ Transform," which is related to the well-known Z Transform but has advantages beyond it. In closing, we highlight the fact that this document is structured somewhat like a textbook with many sample problems and solutions in the hope that it will be readily understood and found useful by those with even an undergraduate understanding of calculus and control systems.