A note on a generalized Joukowski transformation

Malonek, Helmuth R.; Almeida, R. De

doi:10.1016/j.aml.2010.05.006

Cited by 25 publications

(23 citation statements)

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“…Remark 9 One fact that should be stressed here is that for a suitable choice of inputs to the Jacobian (2) the function (3) coincides (up to a real constant) with the 3D Joukowski monogenic transformation introduced in [31] (see Property (iii) above and Example 5 below).…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In particular, it is proved that a certain change of the solid angle is necessary and sufficient for the property of the function to be monogenic and orthogonal to the (infinite dimensional subspace) of paramonogenic constants (i.e monogenic functions with identically vanishing hypercomplex derivative). First global results were considered by H. Malonek et al in [30,31]; see also [3,8]. The authors studied the global behavior of higher dimensional analogues of the exponential function and the classical Joukowski transformation in the quaternionic and Clifford analyses contexts.…”

Section: Introductionmentioning

confidence: 99%

“…Notice that this function coincides (up to a real constant) exactly with the aforementioned Joukowski transformation. For more details the interested reader is suggested to check some of the existing pioneering works, see [3,31].…”

Section: Numerical Examplesmentioning

confidence: 99%

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Local distortion of M-conformal mappings

Morais

Nolder

2014

Applied Mathematics and Computation

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A conformal mapping in a plane domain locally maps circles to circles. More generally, quasiconformal mappings locally map circles to ellipses of bounded distortion. In this work, we study the corresponding situation for solutions to Stein-Weiss systems in the (n + 1)D Euclidean space. This class of solutions coincides with the subset of monogenic quasiconformal mappings with nonvanishing hypercomplex derivatives (named M-conformal mappings). In the theoretical part of this work, we prove that an M-conformal mapping locally maps the unit sphere onto explicitly characterized ellipsoids and vice versa. Together with the geometric interpretation of the hypercomplex derivative, dilatations and distortions of these mappings are estimated. This includes a description of the interplay between the Jacobian determinant and the (hypercomplex) derivative of a monogenic function. Also, we look at this in the context of functions valued in non-Euclidean Clifford algebras, in particular the split complex numbers. Then we discuss quasiconformal radial mappings and their relations with the Cauchy kernel and p-monogenic mappings. This is followed by the consideration of quadratic M-conformal mappings. In the applications part of this work, we provide the reader with some numerical examples that demonstrate the effectiveness of our approach.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Local distortion of M-conformal mappings

Morais

Nolder

2014

Applied Mathematics and Computation

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“…Their ideas, based on Appell's concept of power-like polynomials [1], have allowed recently to develop a systematic study of Appell sequences as tools for other applied problems, like quasi-conformal mappings [21] or construction of classes of generalized classical polynomials [4,5]. Since the construction of power-like monogenic functions was of general interest in Clifford analysis, the study of sets of Appell polynomials has developed meanwhile in several directions and has been realized with different methods and by different authors.…”

Section: Historical Remarksmentioning

confidence: 99%

On Paravector Valued Homogeneous Monogenic Polynomials with Binomial Expansion

Malonek

Falcão

2012

Adv. Appl. Clifford Algebras

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“…[2]), allowed to develop a systematic study of Appell sequences as tools for other applied problems, like quasi-conformal mappings (cf. [22]) or construction of classes of generalized classical polynomials (cf. [5], [6]).…”

Section: Sir Arthur Eddington (In: Relativity Theory Of Protons and Ementioning

confidence: 99%

Matrix Representations of a Special Polynomial Sequence in Arbitrary Dimension

Cação

Falcão

Malonek

2012

Comput. Methods Funct. Theory

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This paper provides an insight into different structures of a special polynomial sequence of binomial type in higher dimensions with values in a Clifford algebra. The elements of the special polynomial sequence are homogeneous hypercomplex differentiable (monogenic) functions of different degrees and their matrix representation allows to prove their recursive construction in analogy to the complex power functions. This property can somehow be considered as a compensation for the loss of multiplicativity caused by the non-commutativity of the underlying algebra.

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A note on a generalized Joukowski transformation

Cited by 25 publications

References 1 publication

Local distortion of M-conformal mappings

Local distortion of M-conformal mappings

On Paravector Valued Homogeneous Monogenic Polynomials with Binomial Expansion

Matrix Representations of a Special Polynomial Sequence in Arbitrary Dimension

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