Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0101200

Bicomplex algebra and function theory

Stefan Rönn
View original source ↗Read source document ↗Export metadata ↓

Abstract

This treatise investigates holomorphic functions defined on the space of bicomplex numbers introduced by Segre. The theory of these functions is associated with Fueter's theory of regular, quaternionic functions. The algebras of quaternions and bicomplex numbers are developed by making use of so-called complex pairs. Special attention is paid to singular bicomplex numbers that lack an inverse. The elementary bicomplex functions are defined and their properties studied. The derivative of a bicomplex function is defined as the limit of a fraction with nonsingular denominator. The existence of the derivative amounts to the validity of the complexified Cauchy-Riemann equations, which characterize the holomorphic bicomplex functions. It is proved that such a function has derivatives of all orders. The bicomplex integral is defined as a line integral. The condition for path independence and the bicomplex generalizations of Cauchy's theorem and integral formula are given. Finally, the relationship between the bicomplex functions and different forms of the Laplace equation is considered. In particular, the four-dimensional Laplace equation is factorized using quaternionic differential operators. The outcome is new classes of bicomplex functions including Fueter's regular functions. It is shown that each class contains differentiable functions.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stefan Rönn. 2001-01-24. Bicomplex algebra and function theory. https://arxiv.org/abs/math/0101200

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions

The celebrated Ohsawa--Takegoshi extension theorem for $L^2$ holomorphic functions on bounded pseudoconvex domains in $\mathbb C^n$ is a fundamental result in the fields of complex analysis and algebraic geometry. In 1995, Ohsawa conjectured that the theorem holds more generally on bounded complete Kähler domains in $\mathbb C^n$. Recently, Chen, Wu and Wang confirmed this conjecture in a special case. In this paper, we extend their result to the case of holomorphic sections of twisted canonical bundles over relatively compact complete Kähler domains in Stein manifolds. As an application, we establish a Hartogs-type extension theorem for plurisubharmonic functions across compact complete pluripolar sets. This result complements a classical theorem of Shiffman and may be regarded as a plurisubharmonic analogue of the Skoda--El Mir extension theorem, thereby filling a gap that appears to have remained open in the literature since at least 1985.

math.CV↗
arXiv2026

The real analytic structure of the Teichmüller space of circle diffeomorphisms with Zygmund continuous derivatives

We apply the methods of simultaneous uniformization and composition operators on Besov spaces to the Teichmüller space $T^Z$ of circle diffeomorphisms with Zygmund continuous derivatives. As consequences, we obtain the following: (1) a new proof of the correspondence between quasiconformal self-homeomorphisms of the unit disk with complex dilatations of linear decay order and their quasisymmetric extensions to the unit circle with regularity in the Zygmund continuously differentiable class; (2) a real-analytic equivalence of $T^Z$ with the real Banach space of Zygmund continuous functions on the unit circle.

math.CV↗
arXiv2026

The Oka principle for holomorphic fibre bundles of Holder-Zygmund classes on strongly pseudoconvex domains

Let \(\overline Ω\) be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let \(h:Z\to \overline Ω\) be a fibre bundle of Hölder-Zygmund class \(Λ^r\), \(r>0\), which is holomorphic over \(Ω\). Assuming that the fibre is an Oka manifold, we prove that every continuous section \(f_0:\overline Ω\to Z\) is homotopic to a section \(f_1:\overline Ω\to Z\) of class \(Λ^r(\overline Ω)\) which is holomorphic on \(Ω\). We also establish the parametric h-principle in this context. As an application, we obtain the Oka principle for the classification of vector bundles and principal bundles of Hölder-Zygmund classes on such domains.

math.CV↗