De Moivre and Euler Formulas for Hyper-Dual Numbers
Abstract
The purpose of this study is to extend classical De Moivre and Euler formulas to the algebra of hyper-dual numbers and to investigate their implications for powers and roots. Hyper-dual numbers form a commutative ring with nilpotent elements that enable exact propagation of first- and second-order differentials. In addition to the fundamental operations, including conjugation, inversion, and Taylor expansion, it presents that every nonzero hyper-dual number admits a multiplicative normal form of the type $a(1+\theta_{1}\varepsilon)(1+\widehat{\theta}\,\varepsilon^{\ast})$. Based on this representation, Euler- and logarithm-type identities are derived, together with a general power formula valid for all integers. Using this framework, the existence and structure of $n$th roots are characterized: when the scalar part is positive and $n$ is even, two distinct roots occur; when $n$ is odd, a unique root exists; and when the scalar part vanishes, nilpotent root families appear in the quadratic case. Illustrative examples are provided to demonstrate the computation of roots and to verify consistency with the hyper-dual Taylor calculus. The findings extend known quaternionic and split-quaternionic results to the hyper-dual setting, contributing tools that combine symbolic manipulation of powers and roots with exact first- and second-order derivative propagation. These tools have potential applications in geometry, kinematics, and automatic differentiation.
Keywords
Dual numbers, hyper-dual numbers, De Moivre's formula, roots of hyper-dual numbers
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