Geometry and Trigonometric Representations of Bicomplex Numbers

Luna-Elizarrarás, M. Elena; Shapiro, Michael; Struppa, Daniele C.; Vajiac, Adrian

doi:10.1007/978-3-319-24868-4_4

Cited by 19 publications

(53 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Luna-Elizarrarás et al, 19 the bicomplex spheres were introduced. Let Z 0 = 𝛽 1,0 e+𝛽 2,0 e † ∈ BC and R = r 1 e+r 2 e † ∈ D…”

Section: Möbius Transformation and Propertiesmentioning

confidence: 99%

See 1 more Smart Citation

The Schwarz lemma in bicomplex analysis

Dai

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

We investigate Schwarz lemma in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. The bicomplex is a generalization of complex which has close relations with Fractal geometry, Minkowski Space-Time, Maxwell's equations, Schrödinger equation, and Gaussian pulse wave. In this paper, we first construct a type of bicomplex Möbius transformation and obtain some results: the mapping properties on bicomplex sphere and bicomplex ball, preserving the inverse points with respect to the bicomplex sphere B(0, 1). Then we obtain the Poisson integral formula in bicomplex setting, and by using the Poisson integral formula, we give the Schwarz lemma for bicomplex holomorphic functions in bicomplex setting.Finally, we shall give the Schwarz lemma and the Schwarz-Pick type lemma for holomorphic functions in bicomplex analysis. These results may give new energy for the development of quantum mechanics.

show abstract

“…In Luna-Elizarrarás et al, 19 the bicomplex spheres were introduced. Let Z 0 = 𝛽 1,0 e+𝛽 2,0 e † ∈ BC and R = r 1 e+r 2 e † ∈ D…”

Section: Möbius Transformation and Propertiesmentioning

confidence: 99%

“…The following partial order on hyperbolic numbers, introduced by Luna-Elizarrarás ME in earlier studies. 19,21,31 Given hyperbolic numbers 𝔷 and 𝔴, we say that 𝔷 ≼ 𝔴 if 𝔴 − 𝔷 ∈ D + . When 𝔴 − 𝔷 ∈ D + ∖{0}, we write z ≺ 𝔴.…”

Section: Preliminariesmentioning

confidence: 99%

The Schwarz lemma in bicomplex analysis

Dai

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…This section summarizes the main properties of Bicomplex numbers and Bicomplex valued analytic functions of one complex variables, which can be found, for example, in [8,22,24,32,34].…”

Section: Bicomplex Analytic Functionsmentioning

confidence: 99%

“…The set of the zero divisors of B is denoted by σ(B). The following proposition summarizes some properties of the Bicomplex numbers shown in [8,32].…”

Section: Basic Propertiesmentioning

confidence: 99%

Transmutation operators and complete systems of solutions for the radial Schrödinger equation

Kravchenko

Vicente‐Benítez

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

A transmutation operator boldT transmuting harmonic functions into solutions of the radial Schrödinger equation boldSu:=()△d−qfalse(rfalse)u=0 is studied. The potential q is assumed to be continuously differentiable, and the Schrödinger equation is considered in a star‐shaped domain normalΩ⊂double-struckRd (with d⩾2). Several new properties of the transmutation operator are established including the operator relation r2△d−q(r)T=Tr2△d, valid on C2 functions and its boundedness on the Bergman space. A Fourier‐Jacobi series expansion of the integral transmutation kernel is derived, and with its aid, an infinite system of solutions of the radial Schrödinger equation is obtained, which is shown to be complete with respect to the uniform norm. Explicit construction of the system is derived. In the case of normalΩ being an open ball centered in the origin, the system of solutions represents an orthogonal basis of the corresponding Bergman space.

show abstract

“…For more details about the bicomplex numbers, the readers can refer to [17,18,21,22]. In 2015, the bicomplex Fibonacci and Lucas numbers defined by Nurkan and Güven [20], respectively, as follows:…”

Section: Introductionmentioning

confidence: 99%