American Review of Mathematics and Statistics

Logarithmic Mean and the Difference Calculus
Shu Tsuchida

Abstract
A logarithmic mean (log-mean) has very useful properties and has been applied over a broad range of fields. Nevertheless, there are more interesting and important areas to solve using this log-mean. The main discussions revolve around two areas. One is the decomposition of a difference or ratio of a function between two periods into two forms: an additive decomposition (AD) and a multiplicative decomposition (MD).To derive an AD and/or an MD for the function; our method (difference calculus) employs a log-mean. Using this method, we derive the ADs and MDs for many functions and compare some results with those derived by the finite-difference calculus (conventional calculus) and the differential calculus. The other area of focus is to show the close correspondences between a difference quotient derived by the difference calculus and a differential quotient, generally called a derivative. From these discussions, we demonstrate three points: our difference calculus has many advantages over the conventional one, some of the results obtained by our calculus can/cannot be used as discrete approximations to those obtained by the differential calculus, and some expressions produced by the difference quotients can/cannot be used as discrete approximations to the differential equations.

Full Text: PDF     DOI: 10.15640/arms.v6n1a2