Conformal mappings of plane domains are realized by holomorphic functions with non vanishing derivative.
Therefore complex differentiability plays an important role in all questions related to fundamental properties of such mapping. In contrast to the planar case, in higher dimensions the set of conformal mappings consists only of M¨obius transformations. But unfortunately M¨obius transformations are not monogenic functions and therefore
also not hypercomplex differentiable. However the equivalence between both concepts - hypercomplex differentiability in the sense of [9], [11] and monogenicity - suggests the question whether monogenic functions can play or not a special role for other types of 3D-mappings, for instance, for quasi-conformal ones. Our goal is to present a case study of an approach to 3D-mappings by using particularly easy to handle monogenic homogeneous polynomials as basis for approximating the mapping function. Thereby we extend significantly the results obtained in [3]. From the numerical point of view we apply ideas from complex numerical analysis realizing the approximation via polynomials of a small real parameter

Falcão, M. I.

Malonek, H. R.

Universidade do Minho: RepositoriUM

ICNAAM – 2006 Extended Abstracts, 3 – 7
3D-Mappings Using Monogenic Functions
H. R. Malonek ∗ and M. I. Falcão ∗∗
Departamento de Matematica, Universidade de Aveiro, 3810-193 Aveiro, Portugal
Received 09 July 2006
Key words Clifford Analysis, monogenic functions, 3D-mappings
Subject classification 30G35,41A10
Conformal mappings of plane domains are realized by holomorphic functions with non vanishing derivative.
Therefore complex differentiability plays an important role in all questions related to fundamental properties of
such mapping. In contrast to the planar case, in higher dimensions the set of conformal mappings consists only
of Möbius transformations. But unfortunately Möbius transformations are not monogenic functions and there-
fore also not hypercomplex differentiable. However the equivalence between both concepts - hypercomplex
differentiability in the sense of [9], [11] and monogenicity - suggests the question whether monogenic func-
tions can play or not a special role for other types of 3D-mappings, for instance, for quasi-conformal ones. Our
goal is to present a case study of an approach to 3D-mappings by using particularly easy to handle monogenic
homogeneous polynomials as basis for approximating the mapping function. Thereby we extend significantly
the results obtained in [3]. From the numerical point of view we apply ideas from complex numerical analysis
realizing the approximation via polynomials of a small real parameter.
c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Preliminaries
Many questions concerning the general extension of theoretical and practical conformal mapping methods in
C to the higher dimensional case in the setting of Clifford Analysis have not been answered until now (see
[11, 3]). But, contrary to the case of several complex variables, in Clifford analysis are no restrictions on the real
dimension of being even or odd. This implies that the real 3-dimensional Euclidean space, the most important
space for concrete applications, can in principle be subject to a treatment similar to the complex one. For this
purpose, let {1, e1, e2, e3} be an orthonormal basis of the Euclidean vector space R4 with the (quaternionic)
product given according to the multiplication rules e21 = e
2
2 = e
2
3 = −1, e1e2 = −e2e1 = e3. (see [2]).
As usual, we identify each element x = (x0, x1, x2) ∈ R3 with the paravector (sometimes also called reduced
quaternion) z = x0 + x1e1 + x2e2.
For C1(Ω, R3) define the (reduced) quaternionic Cauchy-Riemann operator D = ∂
∂x0
+ e1
∂
∂x1
+ e2
∂
∂x2
.
Solutions of the differential equations Df = 0 (resp. fD = 0) are called left-monogenic (resp. right-monogenic)
functions in the domain Ω. Let us remind that the differential operator D is not only a formal linear combination
of the real partial derivatives ∂
∂xk
but, when applied to a given function f : Ω → H, is nothing else than an
areolar derivative in the sense of Pompeiu (cf. [13] and [12]). The same is true for the conjugate quaternionic
Cauchy-Riemann operator D = ∂
∂x0
− e1
∂
∂x1
− e2
∂
∂x2
.
But if f is a function monogenic in Ω, its areolar derivative Df is vanishing and this is equivalent with the fact
that the areolar derivative 12Df can be considered as the hypercomplex derivative of the function f . In C for a
complex differentiable function f we have f ′ = df
dz
= 12 (
∂f
∂x
− i∂f
∂y
) = ∂f
∂x
. The same is true in our case, i.e.
1
2Df =
∂f
∂x0
. Obviously, this formula guarantees that the (hypercomplex) derivative of a monogenic function is
again a monogenic function.
In general we have to assume that a monogenic function f has values in H, i.e., it is of the form f(x) =
f0(x) + f1(x)e1 + f2(x)e2 + f3(x)e3, where fk, k = 0, 1, 2, 3 are real valued functions in Ω. But if we are
dealing with mappings from one 3-dimensional domain to another 3-dimensional domain we have to restrict f
∗ e-mail: hrmalon@mat.ua.pt
∗∗ e-mail: mif@math.uminho.pt
c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
4 Malonek and Falcão: 3D Mappings
to be a quaternion-valued function with one identically zero component. This can be done by different choices.
Here we consider f also as a paravector, i.e., as being of the form f(x) = f0(x) + f1(x)e1 + f2(x)e2.
In this case a monogenic function is monogenic from both sides and its components satisfy the Riesz system
∂f0
∂x0
−
∂f1
∂x1
−
∂f2
∂x2
= 0
∂f0
∂x1
+
∂f1
∂x0
= 0
∂f0
∂x2
+
∂f2
∂x0
= 0
∂f1
∂x2
−
∂f2
∂x1
= 0. (1)
2 Polynomial approximation of monogenic 3D-mappings
For our purpose we use monogenic polynomials in terms of two hypercomplex monogenic variables zk = xk −
x0ek = −
zek+ekz
2 , k = 1, 2, following the approach in [10], [12]; for other approaches and notations see eg.
[2, 5, 14]. This leads to generalized powers of degree n that are by convention symbolically written as zn−k1 × z
k
2
and defined as an n-nary symmetric product by
zn−k1 × z
k
2 = z1 × z1 × · · · × z1︸ ︷︷ ︸
n−k times
×z2 × z2 × · · · × z2
︸ ︷︷ ︸
k times
=
1
n!
∑
π(i1,...,in)
zi1 · · · zin ,
where the sum is taken over all permutations of (i1, . . . , in). Generalized powers form a paravector valued basis
for the Taylor series of a monogenic function. Given a paravector-valued function f it is possible to prove (see
[3]) a particular form for its general power series development ([10]):
Theorem 2.1 Let f = f(z) = f0 +f1e1 +f2e2 be a paravector-valued monogenic function of the paravector
z = x0 + x1e1 + x2e2. The Taylor series of f(z) in terms of zk in a neighborhood of the origin is given by
f(z) =
∞∑
n=0
n∑
k=0
(
n
k
)
z1
n−k × z2
k α(n−k, k)
with
α(n−k, k) =
1
n!
∂nf(0)
∂xn−k1 ∂x
k
2
and [α(n−k, k)]2 = [α(n−k−1, k+1)]1 , k = 0, . . . , n, (2)
where
α(n−k, k) = [α(n−k, k)]0 + [α(n−k, k)]1 e1 + [α(n−k, k)]2 e2, n = 1, 2, . . . , k = 0, . . . , n.
Our main goal is to realize approximations of 3D-mappings by partial sums of this type of series. Inspired by
the corresponding complex approach (see [8]) we normalize the series by the conditions given below, restricting
ourselves only to a situation where the interior of a domain should be transformed into the interior of a sphere.
Hence, the considered domain should contain the origin and the origin should be an invariant point under the
considered mapping f . Furthermore,
Df(0) = (
∂
∂x0
−e1
∂
∂x1
−e2
∂
∂x2
)f(0) = 2 and D̃f(0) := (
∂
∂x0
+e1
∂
∂x1
−e2
∂
∂x2
)f(0) = 1. (3)
It is worth noticing that the first condition in (3) means that the hypercomplex derivative at the origin should be
equal to one. All together implies that the general form of the series to be studied is
f(z1, z2) =
1
2
(z1e1 + z2e2) +
∞∑
n=2
n∑
k=0
(
n
k
)
z1
n−k × z2
kα(n−k, k) (4)
and that the coefficients satisfy the compatibility conditions (2).
c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ICNAAM – 2006 Extended Abstracts 5
3 Numerical experiments with monogenic 3D-mappings
As a concrete example, our case study is concerned with the mapping of the interior of the oblate ellipsoid
Eλ, (0 ≤ λ < 1), defined by z(s, t, λ) = x0(s, t, λ)+x1(s, t, λ)e1+x2(s, t, λ)e2 with x0 = (1+λ) cos s, x1 =
2(1 − λ) sin s cos t, x2 = 2(1 − λ) sin s sin t, where 0 ≤ s ≤ π and 0 ≤ t < 2π, into the interior of a ball B. In
fact, Eλ is a small perturbation of the canonical oblate spheroid O := {(x0, x1, x2) : x02 + 14x1
2 + 14x2
2 = 1}
which is mapped into the unit sphere (ww = w20 + w
2
1 + w
2
2 = 1 ) by the linear monogenic function w =
1
2 (z1e1 + z2e2) = x0 +
1
2 (x1 e1 + x2 e2) that appears as the linear part of the normalized series. Notice that Eλ
is described by the hypercomplex equation (1 + λ2)w w − λ(w w + w w) = (1 − λ2)2.
By taking into account several symmetry properties of Eλ, which imply certain invariance properties of the
mapping function f, we could show in [3] that the number of coefficients of the mapping function can be substan-
tially reduced. Indeed, the Taylor series contains only generalized powers of odd order with real coefficients and
therefore the adequate polynomial approximation of f up to a certain degree m can be realized by a polynomial
of the form
ϕm(z1, z2) =
1
2
(z1e1 + z2e2) +
+a(3,0) z
3
1e1 + a(2,1) z
2
1 × z
1
2e2 + a(1,2) z
1
1 × z
2
2e1 + a(0,3) z
3
2e2 +
+a(5,0) z
5
1 e1 + a(4,1) z
4
1 × z2 e2 + a(3,2) z
3
1 × z
2
2 e1 + a(2,3) z
2
1 × z
3
2 e2 +
+a(1,4) z1 × z
4
2 e1 + a(0,5) z
5
2 e2 +
+ · · · . (5)
Due to the fact that on the sphere (i.e. on the boundary of B) the polynomial (5) is a function of the parameter
λ and the value of |ϕ(z1, z2, λ))|2 should be constant and equal to some %2, the corresponding development
of |ϕ(z(s, t, λ))|2 as a polynomial with respect to λ results in a nonlinear system of algebraic equations, with
a(λ)(n−k,k) as unknowns. The corresponding numerical procedures have been executed by using the powerful
Maple-Quatpackage from [7].
Of course, the relatively high number of indeterminate coefficients together with the non-linearity of the
system causes problems for executing the numerical procedures in a reasonable time and with high accuracy.
It is not our aim here to discuss these problems in detail. Instead of this we shall pay attention to an essential
simplification of the mentioned general approach. This simplification consists in the use of a special set of
monogenic basis functions defined and studied to some extend with respect to its algebraic properties in [4],
namely functions of the form
Pk(x) =
k∑
s=0
T ks z
k−s z̄s, with T ks =
1
k + 1
( 32 )(k−s)(
1
2 )(s)
(k − s)!s!
, (6)
where a(r) denotes the Pochhammer symbol, i.e. a(r) :=
Γ(a+r)
Γ(a) , for any integer r > 1, and a(0) := 1. In terms
of generalized powers these polynomials are of the form
Pk(x) = Pk(z1, z2) = ck
n∑
k=0
zn−k1 × z
k
2
(
n
k
)
en−k1 × e
k
2 with ck =



(k−1)!!
k!! , if k is even,
k!!
(k+1)!! , if k is odd.
. (7)
The first polynomials are given by
P0 = 1
P1(z) =
3
4
z +
1
4
z̄; P1(z1, z2) =
1
2
(z1e1 + z2e2)
P2(z) =
5
8
z2 +
1
4
zz̄ +
1
8
z̄2; P2(z1, z2) = −
1
2
(z21 + z
2
2)
P3(z) =
35
64
z3 +
15
64
z2z̄ +
9
64
zz̄2 +
5
64
z̄3
P3(z1, z2) = −
3
8
(z31e1 + z
2
1 × z2e2 + z1 × z
2
2e1 + z
3
2e2)
c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
6 Malonek and Falcão: 3D Mappings
It is easy to prove, in the general case, that a linear combination with real coefficients of these homogeneous poly-
nomials up to degree m has exactly the deduced structure (5) and hence can be used as a monogenic polynomial
for the mapping problem discussed in the beginning of this section, although only one and the same coefficient
appears in every homogeneous degree. Nevertheless, the corresponding numerical experiments have shown, not
only an enormous reduction of numerical costs but also an increasing rate of convergence. Several conjectures
about the efficiency of such special polynomials in mapping problems will be presented in our talk.
In the following figures we present approximations of degree 3 for the image of the ellipsoid E0.01, obtained
by considering the described general approach and the simplified approach corresponding to the use of (6).
−1.0
−1.0
1.0
−0.5
−0.5
0.0
0.5
x1
0.0
x0
0.5
0.0
1.0
0.5
x2−0.5
1.0 −1.0
Figure 1: Image of E0.01: general case
−1.0
1.0
−0.5−1.0
0.5
−0.5
x1
0.0
0.0
0.0
x2
x0
0.5
−0.5
0.5
1.0
−1.0
1.0
Figure 2: Image of E0.01: simplified case
Acknowledgments: The research of the first author was partially supported by the R&D unit Matemática e
Aplicações (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology
(FCT). The research of the second author was partially supported by the Portuguese Foundation for Science and
Technology (FCT) through the research program POCTI.
References
[1] S. Bock, M. I. Falcão, K. Gürlebeck and H. Malonek, A 3-Dimensional Bergman Kernel Method with Applications to
Rectangular Domains, Journal of Computational and Applied Mathematics Vol. 189, 1-2 , 67–79, (2006).
[2] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman 76, Boston-London-Melbourne, (1982).
[3] J. Cruz, M. I. Falcão and H. R. Malonek, 3D-Mappings and their approximations by series of powers of a small
parameter to appear in: Proceedings of IKM2006, Weimar, Germany, July 12-14, edited by K. Gürlebeck and C. Könke.
[4] M. I. Falcão, J. Cruz and H. R. Malonek, Remarks on the generation of monogenic functions, to appear in: Proceedings
of IKM2006, Weimar, Germany, July 12-14, edited by K. Gürlebeck and C. Könke.
[5] R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comment. Math. Helv., 4, 9–20, (1932).
c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ICNAAM – 2006 Extended Abstracts 7
[6] K. Gürlebeck and H. R. Malonek, A Hypercomplex Derivative of Monogenic Functions in Rm+1 and its Applications,
Complex Variables, Theory Appl. 39, 199–228, (1999).
[7] K. Gürlebeck, K. Habetha and W. Sprössig, Funktionentheorie in der Ebene und im Raum, Reihe: Grundstudium
Mathematik, Birkhäuser, (2006).
[8] L. V. Kantorovich and V. I. Krylov, Approximate Methods in Higher Analysis, Interscience Publishers, (1958).
[9] H. Malonek, A new hypercomplex structure of the euclidean space Rm+1 and the concept of hypercomplex differentia-
bility, Complex Variables, 14, 25–33, (1990).
[10] H. Malonek, Power series representation for monogenic functions in Rm+1 based on a permutational product, Complex
Variables, Theory Appl., 15, 181–191, (1990).
[11] H. Malonek. Contributions to a geometric function theory in higher dimensions by Clifford analysis methods: Mono-
genic functions and M-conformal mappings. in: Clifford Analysis and its Applications, edited by F. Brackx et al.,
Kluwer, NATO Sci. Ser. II, Math. Phys. Chem. 25, 213-222, (2001).
[12] H. Malonek, Selected topics in hypercomplex function theory, in: Clifford algebras and potential theory, edited by S.-L.
Eriksson, University of Joensuu, Report Series 7, 111-150, (2004).
[13] M. Pompeiu, Sur une classe de fonctions dúne variable complexe, Rend. Circ. Mat. Palermo, V. 33, 108 – 113, (1912).
[14] A. Sudbery. Quaternionic analysis. Math. Proc. Camb. Phil. Soc., 85:199–225, (1979).
c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


English

3D-Mappings using monogenic functions

Conformal mappings of plane domains are realized by holomorphic functions with non vanishing derivative. Therefore complex differentiability plays an important role in all questions related to fundamental properties of such mapping. In contrast to the planar case, in higher dimensions the set of conformal mappings consists only of M¤obius transformations. But unfortunately M¤obius transformations are not monogenic functions and there-fore also not hypercomplex differentiable. However the equivalence between both concepts- hypercomplex differentiability in the sense of [9], [11] and monogenicity- suggests the question whether monogenic func-tions can play or not a special role for other types of 3D-mappings, for instance, for quasi-conformal ones. Our goal is to present a case study of an approach to 3D-mappings by using particularly easy to handle monogenic homogeneous polynomials as basis for approximating the mapping function. Thereby we extend signicantly the results obtained in [3]. From the numerical point of view we apply ideas from complex numerical analysis realizing the approximation via polynomials of a small real parameter

H. R. Malonek

M. I. Falcao

CiteSeerX

3D-Mappings Using Monogenic Functions

ICNAAM  2006 Extended Abstracts, 3  73D-Mappings Using Monogenic FunctionsH. R. Malonek ∗ and M. I. Falca˜o ∗∗Departamento de Matematica, Universidade de Aveiro, 3810-193 Aveiro, PortugalReceived 09 July 2006Key words Clifford Analysis, monogenic functions, 3D-mappingsSubject classification 30G35,41A10Conformal mappings of plane domains are realized by holomorphic functions with non vanishing derivative.Therefore complex differentiability plays an important role in all questions related to fundamental properties ofsuch mapping. In contrast to the planar case, in higher dimensions the set of conformal mappings consists onlyof M¤obius transformations. But unfortunately M¤obius transformations are not monogenic functions and there-fore also not hypercomplex differentiable. However the equivalence between both concepts - hypercomplexdifferentiability in the sense of [9], [11] and monogenicity - suggests the question whether monogenic func-tions can play or not a special role for other types of 3D-mappings, for instance, for quasi-conformal ones. Ourgoal is to present a case study of an approach to 3D-mappings by using particularly easy to handle monogenichomogeneous polynomials as basis for approximating the mapping function. Thereby we extend signicantlythe results obtained in [3]. From the numerical point of view we apply ideas from complex numerical analysisrealizing the approximation via polynomials of a small real parameter.c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim1 PreliminariesMany questions concerning the general extension of theoretical and practical conformal mapping methods inC to the higher dimensional case in the setting of Clifford Analysis have not been answered until now (see[11, 3]). But, contrary to the case of several complex variables, in Clifford analysis are no restrictions on the realdimension of being even or odd. This implies that the real 3-dimensional Euclidean space, the most importantspace for concrete applications, can in principle be subject to a treatment similar to the complex one. For thispurpose, let {1, e1, e2, e3} be an orthonormal basis of the Euclidean vector space R4 with the (quaternionic)product given according to the multiplication rules e21 = e22 = e23 = −1, e1e2 = −e2e1 = e3. (see [2]).As usual, we identify each element x = (x0, x1, x2) ∈ R3 with the paravector (sometimes also called reducedquaternion) z = x0 + x1e1 + x2e2.For C1(Ω, R3) define the (reduced) quaternionic Cauchy-Riemann operator D = ∂∂x0+ e1∂∂x1+ e2∂∂x2.Solutions of the differential equations Df = 0 (resp. fD = 0) are called left-monogenic (resp. right-monogenic)functions in the domain Ω. Let us remind that the differential operator D is not only a formal linear combinationof the real partial derivatives ∂∂xkbut, when applied to a given function f : Ω → H, is nothing else than anareolar derivative in the sense of Pompeiu (cf. [13] and [12]). The same is true for the conjugate quaternionicCauchy-Riemann operator D = ∂∂x0− e1∂∂x1− e2∂∂x2.But if f is a function monogenic in Ω, its areolar derivative Df is vanishing and this is equivalent with the factthat the areolar derivative 12Df can be considered as the hypercomplex derivative of the function f . In C for acomplex differentiable function f we have f ′ = dfdz= 12 (∂f∂x− i∂f∂y) = ∂f∂x. The same is true in our case, i.e.12Df =∂f∂x0. Obviously, this formula guarantees that the (hypercomplex) derivative of a monogenic function isagain a monogenic function.In general we have to assume that a monogenic function f has values in H, i.e., it is of the form f(x) =f0(x) + f1(x)e1 + f2(x)e2 + f3(x)e3, where fk, k = 0, 1, 2, 3 are real valued functions in Ω. But if we aredealing with mappings from one 3-dimensional domain to another 3-dimensional domain we have to restrict f∗ e-mail: hrmalon@mat.ua.pt∗∗ e-mail: mif@math.uminho.ptc© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim4 Malonek and Falcao: 3D Mappingsto be a quaternion-valued function with one identically zero component. This can be done by different choices.Here we consider f also as a paravector, i.e., as being of the form f(x) = f0(x) + f1(x)e1 + f2(x)e2.In this case a monogenic function is monogenic from both sides and its components satisfy the Riesz system∂f0∂x0−∂f1∂x1−∂f2∂x2= 0∂f0∂x1+∂f1∂x0= 0∂f0∂x2+∂f2∂x0= 0∂f1∂x2−∂f2∂x1= 0. (1)2 Polynomial approximation of monogenic 3D-mappingsFor our purpose we use monogenic polynomials in terms of two hypercomplex monogenic variables zk = xk −x0ek = −zek+ekz2 , k = 1, 2, following the approach in [10], [12]; for other approaches and notations see eg.[2, 5, 14]. This leads to generalized powers of degree n that are by convention symbolically written as zn−k1 × zk2and defined as an n-nary symmetric product byzn−k1 × zk2 = z1 × z1 × · · · × z1︸ ︷︷ ︸n−k times×z2 × z2 × · · · × z2︸ ︷︷ ︸k times=1n!∑pi(i1,...,in)zi1 · · · zin ,where the sum is taken over all permutations of (i1, . . . , in). Generalized powers form a paravector valued basisfor the Taylor series of a monogenic function. Given a paravector-valued function f it is possible to prove (see[3]) a particular form for its general power series development ([10]):Theorem 2.1 Let f = f(z) = f0 +f1e1 +f2e2 be a paravector-valued monogenic function of the paravectorz = x0 + x1e1 + x2e2. The Taylor series of f(z) in terms of zk in a neighborhood of the origin is given byf(z) =∞∑n=0n∑k=0(nk)z1n−k × z2k α(n−k, k)withα(n−k, k) =1n!∂nf(0)∂xn−k1 ∂xk2and [α(n−k, k)]2 = [α(n−k−1, k+1)]1 , k = 0, . . . , n, (2)whereα(n−k, k) = [α(n−k, k)]0 + [α(n−k, k)]1 e1 + [α(n−k, k)]2 e2, n = 1, 2, . . . , k = 0, . . . , n.Our main goal is to realize approximations of 3D-mappings by partial sums of this type of series. Inspired bythe corresponding complex approach (see [8]) we normalize the series by the conditions given below, restrictingourselves only to a situation where the interior of a domain should be transformed into the interior of a sphere.Hence, the considered domain should contain the origin and the origin should be an invariant point under theconsidered mapping f . Furthermore,Df(0) = (∂∂x0−e1∂∂x1−e2∂∂x2)f(0) = 2 and D˜f(0) := (∂∂x0+e1∂∂x1−e2∂∂x2)f(0) = 1. (3)It is worth noticing that the first condition in (3) means that the hypercomplex derivative at the origin should beequal to one. All together implies that the general form of the series to be studied isf(z1, z2) =12(z1e1 + z2e2) +∞∑n=2n∑k=0(nk)z1n−k × z2kα(n−k, k) (4)and that the coefficients satisfy the compatibility conditions (2).c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimICNAAM  2006 Extended Abstracts 53 Numerical experiments with monogenic 3D-mappingsAs a concrete example, our case study is concerned with the mapping of the interior of the oblate ellipsoidEλ, (0 ≤ λ < 1), defined by z(s, t, λ) = x0(s, t, λ)+x1(s, t, λ)e1+x2(s, t, λ)e2 with x0 = (1+λ) cos s, x1 =2(1− λ) sin s cos t, x2 = 2(1− λ) sin s sin t, where 0 ≤ s ≤ pi and 0 ≤ t < 2pi, into the interior of a ball B. Infact, Eλ is a small perturbation of the canonical oblate spheroid O := {(x0, x1, x2) : x02 + 14x12 + 14x22 = 1}which is mapped into the unit sphere (ww = w20 + w21 + w22 = 1 ) by the linear monogenic function w =12 (z1e1 + z2e2) = x0 +12 (x1 e1 + x2 e2) that appears as the linear part of the normalized series. Notice that Eλis described by the hypercomplex equation (1 + λ2)w w − λ(w w + w w) = (1− λ2)2.By taking into account several symmetry properties of Eλ, which imply certain invariance properties of themapping function f, we could show in [3] that the number of coefficients of the mapping function can be substan-tially reduced. Indeed, the Taylor series contains only generalized powers of odd order with real coefficients andtherefore the adequate polynomial approximation of f up to a certain degree m can be realized by a polynomialof the formϕm(z1, z2) =12(z1e1 + z2e2) ++a(3,0) z31e1 + a(2,1) z21 × z12e2 + a(1,2) z11 × z22e1 + a(0,3) z32e2 ++a(5,0) z51 e1 + a(4,1) z41 × z2 e2 + a(3,2) z31 × z22 e1 + a(2,3) z21 × z32 e2 ++a(1,4) z1 × z42 e1 + a(0,5) z52 e2 ++ · · · . (5)Due to the fact that on the sphere (i.e. on the boundary of B) the polynomial (5) is a function of the parameterλ and the value of |ϕ(z1, z2, λ))|2 should be constant and equal to some %2, the corresponding developmentof |ϕ(z(s, t, λ))|2 as a polynomial with respect to λ results in a nonlinear system of algebraic equations, witha(λ)(n−k,k) as unknowns. The corresponding numerical procedures have been executed by using the powerfulMaple-Quatpackage from [7].Of course, the relatively high number of indeterminate coefficients together with the non-linearity of thesystem causes problems for executing the numerical procedures in a reasonable time and with high accuracy.It is not our aim here to discuss these problems in detail. Instead of this we shall pay attention to an essentialsimplification of the mentioned general approach. This simplification consists in the use of a special set ofmonogenic basis functions defined and studied to some extend with respect to its algebraic properties in [4],namely functions of the formPk(x) =k∑s=0T ks zk−s z¯s, with T ks =1k + 1( 32 )(k−s)(12 )(s)(k − s)!s!, (6)where a(r) denotes the Pochhammer symbol, i.e. a(r) :=Γ(a+r)Γ(a) , for any integer r > 1, and a(0) := 1. In termsof generalized powers these polynomials are of the formPk(x) = Pk(z1, z2) = ckn∑k=0zn−k1 × zk2(nk)en−k1 × ek2 with ck =(k−1)!!k!! , if k is even,k!!(k+1)!! , if k is odd.. (7)The first polynomials are given byP0 = 1P1(z) =34z +14z¯; P1(z1, z2) =12(z1e1 + z2e2)P2(z) =58z2 +14zz¯ +18z¯2; P2(z1, z2) = −12(z21 + z22)P3(z) =3564z3 +1564z2z¯ +964zz¯2 +564z¯3P3(z1, z2) = −38(z31e1 + z21 × z2e2 + z1 × z22e1 + z32e2)c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim6 Malonek and Falcao: 3D MappingsIt is easy to prove, in the general case, that a linear combination with real coefficients of these homogeneous poly-nomials up to degree m has exactly the deduced structure (5) and hence can be used as a monogenic polynomialfor the mapping problem discussed in the beginning of this section, although only one and the same coefficientappears in every homogeneous degree. Nevertheless, the corresponding numerical experiments have shown, notonly an enormous reduction of numerical costs but also an increasing rate of convergence. Several conjecturesabout the efficiency of such special polynomials in mapping problems will be presented in our talk.In the following figures we present approximations of degree 3 for the image of the ellipsoid E0.01, obtainedby considering the described general approach and the simplified approach corresponding to the use of (6).−1.0−1.0 1.0−0.5−0.50.00.5x10.0x00.50.01.00.5x2−0.51.0 −1.0Figure 1: Image of E0.01: general case−1.01.0−0.5−1.00.5−0.5x10.00.00.0x2x00.5−0.50.51.0−1.01.0Figure 2: Image of E0.01: simplied caseAcknowledgments: The research of the first author was partially supported by the R&D unit Matem ·atica eAplicac‚oes (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology(FCT). The research of the second author was partially supported by the Portuguese Foundation for Science andTechnology (FCT) through the research program POCTI.References[1] S. Bock, M. I. Falcao, K. G¤urlebeck and H. Malonek, A 3-Dimensional Bergman Kernel Method with Applications toRectangular Domains, Journal of Computational and Applied Mathematics Vol. 189, 1-2 , 6779, (2006).[2] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman 76, Boston-London-Melbourne, (1982).[3] J. Cruz, M. I. Falcao and H. R. Malonek, 3D-Mappings and their approximations by series of powers of a smallparameter to appear in: Proceedings of IKM2006, Weimar, Germany, July 12-14, edited by K. G¤urlebeck and C. K¤onke.[4] M. I. Falcao, J. Cruz and H. R. Malonek, Remarks on the generation of monogenic functions, to appear in: Proceedingsof IKM2006, Weimar, Germany, July 12-14, edited by K. G¤urlebeck and C. K¤onke.[5] R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comment. Math. Helv., 4, 920, (1932).c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimICNAAM  2006 Extended Abstracts 7[6] K. G¤urlebeck and H. R. Malonek, A Hypercomplex Derivative of Monogenic Functions in Rm+1 and its Applications,Complex Variables, Theory Appl. 39, 199228, (1999).[7] K. G¤urlebeck, K. Habetha and W. Spr¤ossig, Funktionentheorie in der Ebene und im Raum, Reihe: GrundstudiumMathematik, Birkh¤auser, (2006).[8] L. V. Kantorovich and V. I. Krylov, Approximate Methods in Higher Analysis, Interscience Publishers, (1958).[9] H. Malonek, A new hypercomplex structure of the euclidean space Rm+1 and the concept of hypercomplex differentia-bility, Complex Variables, 14, 2533, (1990).[10] H. Malonek, Power series representation for monogenic functions in Rm+1 based on a permutational product, ComplexVariables, Theory Appl., 15, 181191, (1990).[11] H. Malonek. Contributions to a geometric function theory in higher dimensions by Clifford analysis methods: Mono-genic functions and M-conformal mappings. in: Clifford Analysis and its Applications, edited by F. Brackx et al.,Kluwer, NATO Sci. Ser. II, Math. Phys. Chem. 25, 213-222, (2001).[12] H. Malonek, Selected topics in hypercomplex function theory, in: Clifford algebras and potential theory, edited by S.-L.Eriksson, University of Joensuu, Report Series 7, 111-150, (2004).[13] M. Pompeiu, Sur une classe de fonctions d·une variable complexe, Rend. Circ. Mat. Palermo, V. 33, 108  113, (1912).[14] A. Sudbery. Quaternionic analysis. Math. Proc. Camb. Phil. Soc., 85:199225, (1979).c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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3D-Mappings using monogenic functions

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