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Publication Details

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


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