Document Type

Article

Rights

This item is available under a Creative Commons License for non-commercial use only

Disciplines

2.2 ELECTRICAL, ELECTRONIC, INFORMATION ENGINEERING

Publication Details

Journal of Advances in Applied Mathematics, Vol. 6, No. 2, April 2021

Abstract

We consider the consequence of breaking with a fundamental result in complex analysisby lettingi2=±1wherei=√−1is the basic unit of all imaginary numbers. An analysis of theMandelbrot set for this case shows that a demarcation between a Fractal and a Euclidean object ispossible based oni2=−1andi2= +1, respectively. Further, we consider the transient behaviourassociated with the two cases to produce a range of non-standard sets in which a Fractal geometricstructure is transformed into a Euclidean object. In the case of the Mandelbrot set, the Euclideanobject is a square whose properties are investigate. Coupled with the associated Julia sets and othercomplex plane mappings, this approach provides the potential to generate a wide range of newsemi-fractal structures which are visually interesting and may be of artistic merit. In this context,we present a mathematical paradox which explores the idea thati2=±1. This is based on couplinga well known result of the Riemann zeta function (i.e.ζ(0) =−1/2) with the Grandi’s series, bothbeing examples of Ramanujan sums. We then explore the significance of this result in regard to aninterpretation of the fundamental field equations of Quantum Mechanics when a Higgs field is takento be produced by an imaginary massimsuch that(±im)2= +m2. A set of new field equationsare derived and studied. This includes an evaluation of the propagators (the free space Green’sfunctions) which exhibit decay characteristics over very short (sub-atomic) distances

DOI

https://dx.doi.org/10.22606/jaam.2021.62001


Share

COinS