New definitions of exponential, hyperbolic and trigonometric functions on time scales

TL;DR

This paper introduces two new exponential function definitions on time scales, enabling consistent hyperbolic and trigonometric functions that preserve key properties and identities, bridging continuous and discrete analysis.

Contribution

It presents novel exponential definitions on time scales based on Cayley transformation and exact discretization, extending classical functions with preserved identities.

Findings

Functions map imaginary axis into the unit circle

Pythagorean identities hold exactly on any time scale

Cayley-motivated functions satisfy similar dynamic equations as continuous cases

Abstract

We propose two new definitions of the exponential function on time scales. The first definition is based on the Cayley transformation while the second one is a natural extension of exact discretizations. Our eponential functions map the imaginary axis into the unit circle. Therefore, it is possible to define hyperbolic and trigonometric functions on time scales in a standard way. The resulting functions preserve most of the qualitative properties of the corresponding continuous functions. In particular, Pythagorean trigonometric identities hold exactly on any time scale. Dynamic equations satisfied by Cayley-motivated functions have a natural similarity to the corresponding diferential equations. The exact discretization is less convenient as far as dynamic equations and differentiation is concerned.