3D deformations by means of monogenic functions

3D Deformations by means of Monogenic Functions

†

J. Morais

‡

Technical University of Mining, Freiberg, Germany

Email: [email protected]

M. Ferreira

School of Technology and Management, Polytechnic Institute of Leiria, Portugal

2411-901 Leiria, Portugal. Email: [email protected]

and

CIDMA - Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, Portugal.

Email: [email protected]

Abstract

In this paper the authors compute the coecient of quasiconformality for monogenic functions in an arbitrary ball of the Euclidean space R3_{. This quantication may be needed in applications}

but also appear to be of intrinsic interest. The main tool used is a 3D Fourier series development of monogenic functions in terms of a special set of solid spherical monogenics. Ultimately, we present some examples showing the applicability of our approach.

MSC 2000: 30G35; 30C65

Keywords: Quaternion analysis; Riesz system; monogenic functions; quasiconformal mappings.

1 Introduction

By a classical theorem of Liouville [28], the usual denition of conformal mappings in the n-dimensional

Euclidean space R

n

_{when n ≥ 3 applies only to the restricted set of Möbius transformations. For}

this reason, the theory of conformal mappings is essentially restricted to 2D settings. Actually, for

quasiconformal mappings the situation is quite dierent; there exist several possible ways to give an

innitesimal notion of quasiconformality. In order to explain our standing position, let us consider a

homeomorphism f of a domain D ⊂ R

n

_{onto a domain D}

′

_{⊂ R}

n

_{. For x ∈ D, r > 0 and a 3D closed}

ball B(x, r) ⊂ D of center x and radius r, we set

L

f

(x, r) := max

|x−y|=r

|f(y) − f(x)|,

l

f

(x, r) := min

|x−y|=r

|f(y) − f(x)|,

k

f

(x, r) =

L

f

(x, r)

l

f

(x, r)

.

The coecient of quasiconformality (or linear dilatation) of f at x ∈ D is dened as k

f

(x) = lim sup

r→0

k

f

(x, r)

.

For our purposes here, the orientation preserving homeomorphism f : D → D

′

is called k-quasiconformal

†_{Accepted author's manuscript (AAM) published in [Math. Meth. Appl. Sci., 36(7) (2013),}

780793] [DOI: 10.1002/mma.2625]. The nal publication is available at link.springer.com via http://onlinelibrary.wiley.com/doi/10.1002/mma.2625/abstract

in D

′

if k

f

(x)

is bounded in D and k

f

(x)

≤ k for almost all points x ∈ D. Then a mapping is called

qua-siconformal if it is k-quaqua-siconformal for some k (1 ≤ k < ∞). The distortion theory of quaqua-siconformal

mappings is usually connected to problems of obtaining global distortion bounds under quasiconformal

deformations from the local bounds of k

f

. The major sources of the theory of quasiconformal

map-pings in the plane are found in the works of Grötzsch [14], Lavrent'ev [25], Ahlfors [1], and Teichmüller

[41]. In the meantime, quasiconformal mappings became a classical object of analysis due to its rich

behavior and wide range of applicability in various elds of mathematics such as discrete group theory,

mathematical physics, complex dierential geometry, medical image analysis, and probability theory.

Higher dimensional quasiconformal mappings were rst introduced by Lavrent'ev in 1938 [26], followed

over a period of several years by a series of famous works: Ahlfors [2], Gehring [13], and Väisälä [42, 43].

The reader can consult the book of Rickman [37] for a more recent treatment of the subject.

In general, the study of quasiconformal mappings is important for the construction of analytic

map-pings with specied dynamics, which can be used as coordinate transformations in various problems,

e.g. in the study of partial dierential equations [7]. The dilatation of quasiconformal mappings

con-trols the way conformal invariants, such as moduli of annuli, can be changed. Especially, in the theory

of renormalizations, we can measure how distant a map is from another one in terms of dilatations

of conjugacies. For that it is essential to have an estimate for the dilatation of the quasiconformal

mapping. For such applications in higher dimensions the study of the quasiconformality factor is

fun-damental. Recent statistical studies have shown that quasi-conformal mappings can also be useful

for approximate recovery of the boundary shape of domains in inverse problems, e.g. in scattering,

diraction problems and tomography [20, 21, 22, 23].

The discussion about the extension of theoretical and practical quasiconformal mapping techniques

in the quaternion analysis setting, has originated many questions. Yet a large number of investigations

[12, 15, 24, 29, 40] were carried out in connection with the goal of studying monogenic functions by a

corresponding dierentiability concept or by the existence of a well-dened hypercomplex derivative.

However, up to now there are feeble attempts to characterize monogenic functions via a generalized

conformality concept. It was Malonek who rst introduced the concept of monogenic conformal

map-pings [30]. These mapmap-pings are called in [30] M-conformal mapmap-pings. The relation of this concept

with the geometric interpretation of the hypercomplex derivative

1

2

D

allowed it to complete the theory

of monogenic functions by providing an accounting for the still missing geometric characterization of

those functions. In [32] Malonek et al. studied the existence of local homeomorphims for quaternionic

Beltrami-type equations, and determined a necessary and sucient criterion that relates the

hyper-complex derivative of a quaternion monogenic function and its corresponding Jacobian determinant.

However, according to our current knowledge nothing has been done in the study of the

quasiconfor-mality factor.

Some of the recent interest in this subject was stimulated by the works of Gürlebeck et al. [16, 17,

18, 19] (cf. [35], Ch.4) in which it is proved that the class of monogenic functions with nonvanishing

Jacobian determinant can be dened as a special subclass of quasi-conformal mappings. More precisely,

it is shown that monogenic functions map locally balls onto explicitly characterized ellipsoids and vice

versa. Besides this, methods used in [18] show that these considerations include the description of the

interplay between the Jacobian determinant and the hypercomplex derivative of a monogenic function

also. In continuation of these studies, our goal is to show how do the coecient of quasiconformality

looks like in the case of a monogenic function dened in a ball of R

3

_{with values in the reduced}

quaternions (identied with R

3

_{). This is particularly rewarding since the computation of this coecient}

will give us the information of the ratio of the major to minor axes of the aforementioned ellipsoids.

In our approach, as we shall see, the key tool is the representation of a monogenic 3D Fourier series in

terms of a special set of solid spherical monogenics. From our point of view these results can be seen as

a step towards for a deeper understanding on the local behavior of monogenic mappings. Later, just as

importantly, this can be used as a basis to study the global behavior of such mappings. In general, it

is still open how to provide the description of monogenic functions via their global geometric mapping

properties. The interest in questions of this type has increased in connection with constructing a

theory of monogenic mappings. A rst global result was considered recently by Almeida and Malonek

in [3, 31] (cf. [11]). The authors have studied the global behavior of the higher dimensional analogue

of the classical Joukowski transform in the context of Cliord analysis.

2 Notation and denitions

The present section collects some denitions and basic properties of quaternion analysis that will be

needed throughout the text. Let H := {z = z

0

+ z

1

i + z

2

j + z

3

k, z

i

∈ R, i = 0, 1, 2, 3} be the real

quaternion algebra, where the imaginary units i, j, and k are subject to the multiplication rules:

i

2

₌

_j

2

₌

_k

2

₌

_{−1; ij = k = −ji, jk = i = −kj, ki = j = −ik.}

As usual, the real vector space R

4

_{may be embedded in H by identifying the element z := (z}

_{0, z1, z2, z3}

₎

_∈

R

4

_{with z := z}

0

+ z

1

i + z

2

j + z

3

k ∈ H. Let us make a notation convention. Consider the subset

A := span

_R

{1, i, j} of H, then the real vector space R

3

_{may be embedded in A via the identication of}

x := (x0, x1, x2

)

∈ R

3

with the reduced quaternion x := x

0

+ x

1

i + x

2

j ∈ A. To this end, throughout

the text, we will often use the symbol x to represent a point in R

3

_{and x to represent the corresponding}

reduced quaternion. Also, it may be worthwhile to point out that A is a real vectorial subspace, but

not a subalgebra of H. Like in the complex case, Sc(x) = x

0

and Vec(x) = x

1

i + x

2

j dene the scalar

and vector parts of x. The conjugate of x is the reduced quaternion x = x

0

− x1

i

− x2

j

, and the

norm |x| of x is dened by |x|

2

_{= xx = xx = x}

2

0

+ x

21

+ x

22

, and it coincides with its corresponding

Euclidean norm as a vector in R

3

_{. Let Ω be an open subset of R}

3

_{with a piecewise smooth boundary.}

We say that f : Ω −→ A, f(x) = [f(x)]

0

+ [f (x)]

1

i + [f(x)]

2

j is a reduced quaternion-valued function

or, in other words, an A-valued function, where [f]

i

(i = 0, 1, 2)

are real-valued functions dened in Ω.

Properties (like integrability, continuity or dierentiability) that are ascribed to f have to be fullled

by all components [f]

i

. We further introduce the real-linear Hilbert space of square integrable A-valued

functions dened in B(x, r), that we denote by L

2

(B(x, r);

A; R). The scalar inner product is dened

by

< f , g >

_L 2(B(x,r);A;R)

=

∫

B(x,r)

Sc(f g) dV ,

(1)

where dV denotes the volume of B(x, r). For continuously real-dierentiable A-valued functions f,

the reader may be familiar with the (reduced) quaternionic operators D = ∂

x0

+

i∂

x1

+

j∂

x2

and

D = ∂

x0

− i∂

x1

− j∂

x2

, which are called generalized Cauchy-Riemann (resp. conjugate generalized

Cauchy-Riemann) operators on R

3

_{. Namely, a continuously real-dierentiable A-valued function f is}

said to be monogenic in B(x, r) if Df = 0 in B(x, r), which is equivalent to the system

(R)



















∂[f ]

0

∂x

0

−

∂[f ]

1

∂x

1

−

∂[f ]

2

∂x

2

= 0

∂[f ]0

∂x

1

+

∂[f ]1

∂x

0

= 0,

∂[f ]0

∂x

2

+

∂[f ]2

∂x

0

= 0,

∂[f ]1

∂x

2

−

∂[f ]2

∂x

1

= 0

.

We may point out that the previous system is called the Riesz system [38], and it generalizes the classical

Cauchy-Riemann system in the plane. Following [27], the solutions of the system (R) are called

(R)-solutions. The subspace of polynomial (R)-solutions of degree n will be denoted by R

+

_{(B(x, r);}

_{A; n).}

In [27], it is shown that the space R

+

_{(B(x, r);}

_{A; n) has dimension 2n + 3. We also denote by}

R

+

_{(B(x, r);}

_{A) := L2}

_{(B(x, r);}

_{A; R) ∩ ker D the space of square integrable A-valued monogenic}

func-tions dened in B(x, r).

For brevity, assume in the sequel that f is an A-valued monogenic function. Furthermore, let

J

f

=

(

∂

xj

[f ]

i

)

2

i,j=0

be the Jacobian of f and suppose that the mapping f preserves the orientation.

There is a great dierence in the properties of holomorphic conformal 2D-mappings and monogenic

quasiconformal 3D-mappings. Methods used in [16, 17, 18, 19] and [35] Ch.4, provide us with an idea

of the interaction between quasiconformality and monogenic functions. To be precise, every function

f

realizes locally in the neighborhood of a xed point x = x

∗

a M-conformal mapping if and only if

det J

f

(x)

|

x=x∗

̸= 0.

Theorem 1 (see [19]) Let f be an A-valued real-analytic function dened in Ω with non-vanishing

Jacobian determinant. Then, the function f is monogenic if and only if it maps locally a ball onto an

ellipsoid with the property that the reciprocal of the length of one semi-axis is equal to the sum of the

reciprocals of the lengths of the other two semi-axes.

Ultimately, we conclude this section by recalling a suitable set of special monogenic polynomials in

the space R

+

_{(B(x, r);}

_{A; n), which was introduced in [8] (see also [9]) by applying the hypercomplex}

derivative

1

2

D

(see [15, 34, 40]) to a standard system of spherical harmonics as considered e.g. in

[39]. First, consider x with polar coordinates (r, θ

1, θ2

): x

0

= r cos θ

1

, x

1

= r sin θ

1

cos θ

2

, and x

2

=

r sin θ1

sin θ

2

, where 0 < r < ∞, 0 < θ

1

≤ π, and 0 < θ2

≤ 2π. To this end, we rst select the set of

homogeneous harmonic polynomials,

{U

l,† n+1

, V

m,†

n+1

: l = 0, 1, . . . , n + 1, m = 1, . . . , n + 1

}

n∈N0

(2)

with the notations U

l,†

n+1

:= r

n+1

U

n+1l

and V

m,†

n+1

:= r

n+1

V

n+1m

; it is formed by the extensions in the ball

of the spherical harmonics U

l

n+1

(θ

1

, θ

2

) = P

n+1l

(cos θ

1

)T

l

(cos θ

2

) (l = 0, . . . , n + 1)

, and V

n+1m

(θ

1

, θ

2

) =

P

_n+1m

(cos θ

1

) sin θ

2

U

m−1

(cos θ

2

) (m = 1, . . . , n + 1)

. Here, P

n+1l

stands for the Ferrer's associated

Legendre functions of degree n + 1 and order l of the rst kind, T

l

and U

m−1

are the Chebyshev

polynomials of the rst and second kinds, respectively. We further assume the reader to be familiar

with the fact that whenever l = 0 the corresponding associated Legendre function P

0

n+1

coincides with

the Legendre polynomial P

n+1

, and P

n+1l

are the zero functions for l ≥ n+2. Based on the factorization

of the 3D Laplace operator by ∆

3

= DD = DD

, for each n ∈ N

0

we do apply the operator

1₂

D

to each

basis element of (2). We obtain then the following set of 2n + 3 homogeneous monogenic polynomials

{X

l,†

n

, Y

m,n †

: l = 0, . . . , n + 1, m = 1, . . . , n + 1

},

(3)

where X

l,†

n

:= r

n

X

ln

and Y

m,†

n

:= r

n

Y

mn

. The fundamental references for these polynomials and their

properties are [35] and [36]. The explicit expressions of the mentioned polynomials are given in the

next proposition.

Proposition 1 (see [36]) The homogenous monogenic polynomials (3) can be represented in the

fol-lowing way

X

0,_n†

:= r

n

[

(n + 1)

2

U

0 n

+

1

2

U

1 n

i +

1

2

V

1 n

j

]

X

m,_n †

:= r

n

[

(n + m + 1)

2

U

m n

+

1

4

R

m,− n

i +

1

4

S

m,+ n

j

]

Y

m,_n †

:= r

n

[

(n + m + 1)

2

V

m n

+

1

4

S

m,− n

i

−

1

4

R

m,+ n

j

]

,

with the notations R

m,±

n

(θ

1, θ2

) := U

nm+1

± (n + m + 1)(n + m)U

nm−1

, and S

m,±

n

(θ

1, θ2

) := V

nm+1

± (n +

m + 1)(n + m)V

_nm−1

, where m = 1, . . . , n + 1. For a more unied formulation we remind the reader

that the spherical harmonics U

m

n

and V

nm

are the zero function for m ≥ n + 1.

The next lemma shows that, for each n ∈ N

0

, the set (3) is composed by orthogonal polynomials

with respect to the scalar inner product (1).

Lemma 1 (see [8, 9]) For each n = 0, 1, . . ., the polynomials X

l,†

n

(l = 0, . . . , n + 1)

and Y

m,n †

(m =

1, . . . , n + 1)

form a complete orthogonal system in R

+

(B(x, r);

A), and their norms are explicitly given

by

∥X

0,† n

∥

L2(B(x,r);A;R)

=

√

r

2n+3

2n + 3

π (n + 1),

∥X

m,†n

∥

L2(B(x,r);A;R)

=

∥Y

m,† n

∥

L2(B(x,r);A;R)

=

√

r

2n+3

2n + 3

π

2

(n + 1)

(n + 1 + m)!

(n + 1

− m)!

.

Therefore, the Fourier series of f (centered at the origin) with respect to the referred orthogonal system

in R

+

_{(B(x, r);}

_{A) is dened by}

f =

∞

∑

n=0

√

2n + 3

π(n + 1) r

2n+3

[

X

0,_n†

a

0_n

+

n+1

∑

m=1

√

2

(n + 1

− m)!

(n + 1 + m)!

(

X

m,_n †

a

m_n

+ Y

m,_n †

b

m_n

)]

,

(4)

where for each n ∈ N

0

, a

0n

, a

mn

, b

mn

(m = 1, . . . , n+1)

are the associated (real-valued) Fourier coecients.

3 The distance |f(y) − f(x)| for A-valued monogenic functions

In this section we compute the coecient of quasiconformality for any function dened in R

+

_{(B(x, r);}

_A).

For the reader's convenience and sake of easy reference, we will now briey discuss this subject,

fol-lowing mainly the notations introduced in [33]. It should also be remarked that in some cases the idea

is to work with the usual Taylor series expansion of a monogenic function in symmetric powers. We

prefer here to consider a special monogenic Fourier series expansion in terms of homogenous monogenic

polynomials.

To begin with, we shall nd a general expression for the distance |f(y)−f(x)|. Whence, we consider

down an expansion of the form

f (y)

− f(x) =

∞

∑

n=0

√

2n + 3

π(n + 1) r

2n+3

{ (

X

0,_n†

(y)

− X

0,_n†

(x)

)

a

0_n

+

n+1

∑

m=1

√

2

(n + 1

− m)!

(n + 1 + m)!

[(

X

m,_n †

(y)

− X

m,_n †

(x)

)

a

m_n

+

(

Y

m,_n †

(y)

− Y

m,_n †

(x)

)

b

m_n

] }

.

A rst straightforward computation shows that |f(y)−f(x)| =

√∑

2

i=0

([f (y)

− f(x)]

i

)

2

₌

√

_F

2 0

+ F

12

+ F

22

,

where

F0 := [f (y)− f(x)]0= ∞ ∑ n=0 √ 2n + 3 π(n + 1) r2n+3 { (n + 1) 2 ( U_n0,†(y)− U_n0,†(x))a0_n + n ∑ m=1 √ 2(n + 1− m)! (n + 1 + m)! (n + 1 + m) 2 [( U_nm,†(y)− U_nm,†(x))am_n +(V_nm,†(y)− V_nm,†(x))bm_n] } , (5) F1 := [f (y)− f(x)]1= ∞ ∑ n=0 √ 2n + 3 π(n + 1) r2n+3 { 1 2 ( U_n1,†(y)− U_n1,†(x))a0_n + n+1_∑ m=1 √ 2(n + 1− m)! (n + 1 + m)! 1 4 [(

U_nm+1,†(y)− U_nm+1,†(x) + (n + 1 + m)(n + m)(U_nm−1,†(x)− U_nm−1,†(y)))am_n

+ (V_nm+1,†(y)− V_nm+1,†(x) + (n + 1 + m)(n + m)(V_nm−1,†(x)− V_nm−1,†(y)))bm_n] } (6)

and

F2 := [f (y)− f(x)]₂= ∞ ∑ n=0 √ 2n + 3 π(n + 1) r2n+3 { 1 2 ( V_n1,†(y)− V_n1,†(x))a0_n + n+1 ∑ m=1 √ 2(n + 1− m)! (n + 1 + m)! 1 4 [( V_nm+1,†(y)− V_nm+1,†(x) + (n + 1 + m)(n + m)(V_nm−1,†(y)− V_nm−1,†(x)))am_n

− (U_nm+1,†(y)− U_nm+1,†(x) + (n + 1 + m)(n + m)(U_nm−1,†(y)− U_nm−1,†(y)))bm_n]

}

. (7)

The proof of the main result needs a series of technical preparations. Therefore, let us note the following lemma to our initial calculation:

Lemma 2 [35] The homogeneous harmonic polynomials Ul,†

n (l = 0, . . . , n)and Vnm,† (m = 1, . . . , n) are given in Cartesian coordinates as:

U_nl,†(x) = [∑n−l2 ] k=0 [l 2] ∑ j=0 2βn,l,kxn₀−2k−l|x|2k(−1)j ( l 2j ) xl₁−2jx2j₂ , V_nm,†(x) = [n−m∑2 ] k=0 [m−1∑2 ] j=0 2βn,m,kxn₀−2k−m|x|2k(−1)j ( m 2j + 1 ) xm₁−2jx2j₂ ,

where the upper bound [s] denotes, as usual, the integer part of s ∈ R and the coecients βn,l,k are dened in the following way

βn,l,k= (−1)k 1 2n+1 ( 2n− 2k n− k )( n− k k ) (n− 2k)l.

Let us introduce a few abbreviations. In the sequel, we will make use of the following system of spherical coordinates:

y0 = x0+ r cos θ1= x0+ rc1

y1 = x1+ r cos θ2sin θ1= x1+ rc2s1

y2 = x2+ r sin θ2sin θ1= x2+ rs2s1 (8)

with si= sin θi and ci = cos θi (i = 1, 2). This being so, we have

|y|2_{= y}2

0+ y12+ y22 = (x0+ rc1)2+ (x1+ rc2s1)2+ (x2+ rs2s1)2

= x2₀+ x2₁+ x₂2+ r2+ 2r(x0c1+ x1c2s1+ x2s1s2) = |x|2+ 2rΛ + r2,

with Λ = x0c1+ x1c2s1+ x2s1s2.In cartesian coordinates it reads Λ = 1/r(< x, y > −|x|2). From this we have

that (|y|2)k =(|x|2+ (2rΛ + r2))k= k ∑ h=0 ( k h ) (|x|2)k−h(2rΛ + r2)h.

By dening the binomial function

fh+q:= ( k h ) ( h q ) (|x|2)k−h(2Λ)h−q

satisfying 0 ≤ h + q ≤ 2k, 0 ≤ q ≤ h ≤ k and fh+q= 0if h > k, we can rewrite the factor (|y|2)k as (|y|2)k =

2k

∑ h+q=0

fh+qrh+q. (9)

Using the spherical coordinates (3) and the new relation (9) it is possible to write Ul,†

n and Vnm,† as polynomial functions in r. Next we formulate the result.

Lemma 3 The polynomials Ul,_† n and V

m,_†

n can be written in terms of the spherical coordinates (3) by

U_nl,†(y) = [∑n−l2 ] k=0 [l 2] ∑ j=0 2βn,l,k(−1)j ( l 2j ) _∑n t=0 ( _t ∑ p=0 g_(k,p)1 g2_(j,t−p) ) rt (10) with g1_(k,t 1):= t1 ∑ p=0 ( n− 2k − l p ) xn₀−2k−l−pcp₁fh+q=t1−p, g2_(j,t2):= t2 ∑ p=0 ( l− 2j p )( 2j t2− p ) cp₂st2 1s t2−p 2 x l_−2j−p 1 x 2j−(t2−p) 2

and V_nm,†(y) = [n−m 2 ] ∑ k=0 [m−1 2 ] ∑ j=0 2βn,m,k(−1)j ( m 2j + 1 )∑n t=0 ( _t ∑ p=0 h1_(k,p)h2_(j,t−p) ) rt (11) where h1_(k,t 1)= t1 ∑ p=0 ( n− 2k − m p ) xn₀−2k−m−pcp₁fh+q=t1−p, h2_(j,t 2)= t2 ∑ p=0 ( m− 2j p )( 2j t2− p ) cp₂st2 1s t2−p 2 x m−2j−p 1 x 2j−(t2−p) 2 .

Proof: For sake of simplicity, we just present the proof of the expression (10). Using spherical coordinates (3) and (9), a rst straightforward computation shows that

U_nl,†(y) = [∑n−l2 ] k=0 [l 2] ∑ j=0 2βn,l,k y₀n−2k−l|y|2k(−1)j ( l 2j ) y₁l−2jy₂2j = [n−l 2 ] ∑ k=0 [l 2] ∑ j=0 2βn,l,k (x0+ rc1)n−2k−l   ∑2k h+q=0 fh+qrh+q   ×(−1)j ( l 2j ) (x1+ rc2s1)l−2j(x2+ rs1s2)2j. (12)

First we consider the expansions of the factors (x0+ rc1)n−2k−l, (x1+ rc2s1)l−2j and (x2+ rs1s2)2j :

(x0+ rc1)n−2k−l = n−2k−l∑ i1=0 ( n− 2k − l i1 ) xn−2k−l−i1 0 (rc1)i1, (x1+ rc2s1)l−2j = l_∑−2j i2=0 ( l− 2j i2 ) xl−2j−i2 1 (rc2s1)i2, (x2+ rs1s2)2j = 2j ∑ i3=0 ( 2j i3 ) x2j−i3 2 (rs1s2) i3_.

Now, using the Cauchy product we obtain (x0+ rc1)n−2k−l   ∑2k h+q=0 fh+qrh+q   = n∑−l t1=0 (_t 1 ∑ p=0 ( n− 2k − l p ) xn₀−2k−l−pcp₁fh+q=t1−p ) rt1_. (13) By dening g_(k,t1 1):= t1 ∑ p=0 ( n− 2k − l p ) xn₀−2k−l−pcp₁fh+q=t1−p we can write (x0+ rc1)n−2k−l   ∑2k h+q=0 fh+qrh+q   = ∑n−l t1=0 g_(k,t1 1)r t1_. (14)

In (14) terms vanish whenever p > n − 2k − l or t1− p > 2k. The rst restriction is satised by formula (14).

For the second restriction we know that the binomial function fh+q is dened for 0 ≤ h + q ≤ 2k, being zero if

h + q < 0or h + q > 2k. Therefore, the second restriction is also satised by formula (14). Similarly we obtain:

(x1+ rc2s1)l−2j(x2+ rs1s2)2j = l_∑−2j i2=0 ( l− 2j i2 ) xl−2j−i2 1 (rc2s1) i2 2j ∑ i3=0 ( 2j i3 ) x2j−i3 2 (rs1s2) i3 = l ∑ t2=0 (_t 2 ∑ p=0 ( l− 2j p ) xl₁−2j−p(c2s1)p ( 2j t2− p ) x2j−(t2−p) 2 (s1s2) t2−p ) rt2 = l ∑ t2=0 (_t 2 ∑ p=0 ( l− 2j p )( 2j t2− p ) cp₂st2 1 s t2−p 2 x l−2j−p 1 x 2j_−(t2−p) 2 ) rt2_. By dening g2_(j,t 2):= t2 ∑ p=0 ( l− 2j p )( 2j t2− p ) cp₂st2 1 s t2−p 2 x l−2j−p 1 x 2j_−(t2−p) 2 we can write (x1+ rc2s1)l−2j(x2+ rs1s2)2j= l ∑ t2=0 g2_(j,t 2)r t2_. (15)

In formula (15) terms vanish when p > l − 2j or t2− p > 2j. It is readily seen that these restrictions are encoded

in formula (15). Finally, we compute the product between (14) and (15): (x0+ rc1)n−2k−l   ∑2k h+q=0 fh+qrh+q   (x1+ rc2s1)l−2j(x2+ rs1s2)2j = n_−l ∑ t1=0 g_(k,t1 ₁₎rt1 l ∑ t2=0 g2_(j,t2)rt2 = n ∑ t=0 ( _t ∑ p=0 g_(k,p)1 g_(j,t2 _−p) ) rt. (16)

In formula (16) terms vanish when p > n − l or t − p > l. Replacing (16) in (12) the polynomial Ul,†

n is nally given by U_nl,†(y) = [∑n−l2 ] k=0 [l 2] ∑ j=0 2βn,l,k(−1)j ( l 2j ) _∑n t=0 ( _t ∑ p=0 g1_(k,p)g_(j,t2 _−p) ) rt.

In the same way, it can be proved the expression (11) for the polynomials Vm,† n .

We are now ready to proceed to the aim of the present paper, which consists of obtaining expressions for

F0, F1, and F2 in series of r. By Lemma 3 and straightforward computations we can compute the following

dierences: U_n0,†(y)− U_n0,†(x) = [n 2] ∑ k=0 2βn,0,k n ∑ t=1 ( _t ∑ p=0 p ∑ p1=0 ( n− 2k p1 ) xn−2k−p1 0 c p1 1 fh+q=p_−p1 ) rt = n ∑ t=1 [n 2] ∑ k=0 2βn,0,k ( _t ∑ p=0 p ∑ p1=0 ( n− 2k p1 ) xn−2k−p1 0 c p1 1 fh+q=p−p1 ) rt, (17)

U_nm,†(y)− U_nm,†(x) = = n ∑ t=1 [n−m∑2 ] k=0 [m 2] ∑ j=0 2βn,m,k(−1)j ( m 2j )_∑t p=0 ( _p ∑ p2=0 ( n− 2k − m p2 ) xn−2k−m−p2 0 c p2 1 fh+q=p_−p2 ) (_t−p ∑ p3=0 ( m− 2j p3 )( 2j t− p − p3 ) cp3 2 s t_−p 1 s t_−p−p3 2 x m_−2j−p3 1 x 2j−(t−p−p3) 2 ) rt (18) and, moreover V_nm,†(y)− V_nm,†(x) = = n ∑ t=1 [n−m 2 ] ∑ k=0 [m−1 2 ] ∑ j=0 2βn,m,k(−1)j ( m 2j + 1 )∑t p=0 ( _p ∑ p4=0 ( n− 2k − m p4 ) xn−2k−m−p4 0 c p4 1 fh+q=p−p4 ) (_t−p ∑ p5=0 ( m− 2j p5 )( 2j t− p − p5 ) cp5 2 s t−p 1 s t−p−p5 2 x m−2j−p5 1 x 2j_{−(t−p−p}5) 2 ) rt. (19)

Replacing (17), (18), and (19) in (5), (6), and (7) we obtain

F0= ∞ ∑ n=1 n ∑ t=1 f_0,(k,j,t)(n) rt, F1= ∞ ∑ n=1 n ∑ t=1 f_1,(k,j,t)(n) rt, F2= ∞ ∑ n=1 n ∑ t=1 f_2,(k,j,t)(n) rt, (20) with f_0,(k,j,t)(n) = √ 2n + 3 π(n + 1) { (n + 1) 2 ( e U0,_n†(y)− eU0,_n†(x) ) a0_n + n ∑ m=1 √ 2(n + 1− m)! (n + 1 + m)! (n + 1 + m) 2 [( e Um,n †(y)− eUm,n †(x) ) amn + ( e Vm,n †(y)− eVm,n †(x) ) bmn ] } .

For the remaining factors, we have

f_1,(k,j,t)(n) = √ 2n + 3 π(n + 1) { 1 2 ( e U1,_n†(y)− eU1,_n†(x) ) a0_n + n+1 ∑ m=1 √ 2(n + 1− m)! (n + 1 + m)! 1 4 [( e Um+1,n †(y)− eUm+1,n †(x) + (n + 1 + m)(n + m)( eUmn−1,†(x)− eUmn−1,†(y)) ) amn + ( e Vm+1,n †(y)− eVnm+1,†(x) + (n + 1 + m)(n + m)( eVmn−1,†(x)− eVnm−1,†(y)) ) bmn ] } and, f_2,(k,j,t)(n) = √ 2n + 3 π(n + 1) { 1 2 ( e V_n1,†(y)− eV1,_n†(x) ) a0_n + n+1 ∑ m=1 √ 2(n + 1− m)! (n + 1 + m)! 1 4 [( e Vm+1,_n †(y)− eVm+1,_n †(x) + (n + 1 + m)(n + m)( eVm_n−1,†(x)− eV_nm−1,†(y)) ) am_n −(Uem+1,n †(y)− eU m+1,_† n (x) + (n + 1 + m)(n + m)( eU m_−1,† n (x)− eU m_−1,† n (y)) ) bmn ] }

where for any m = 1, . . . , n + 1 e U0,_n†(y)− eU0,_n†(x) = [n 2] ∑ k=0 2βn,0,k ( _t ∑ p=0 p ∑ p1=0 ( n− 2k p1 ) xn−2k−p1 0 c p1 1 fh+q=p−p1 ) , (21) e Um,_n †(y)− eUm,_n †(x) = = [n−m∑2 ] k=0 [m 2] ∑ j=0 2βn,m,k(−1)j ( m 2j )_∑t p=0 ( _p ∑ p2=0 ( n− 2k − m p2 ) xn−2k−m−p2 0 c p2 1 fh+q=p_−p2 ) (_t−p ∑ p3=0 ( m− 2j p3 )( 2j t− p − p3 ) cp3 2 s t_−p 1 s t_−p−p3 2 x m_−2j−p3 1 x 2j−(t−p−p3) 2 ) (22) and e Vm,_n †(y)− eVm,_n †(x) = = [n−m 2 ] ∑ k=0 [m−1 2 ] ∑ j=0 2βn,m,k(−1)j ( m 2j + 1 )_∑t p=0 ( _p ∑ p4=0 ( n− 2k − m p4 ) xn−2k−m−p4 0 c p4 1 fh+q=p−p4 ) (_t−p ∑ p5=0 ( m− 2j p5 )( 2j t− p − p5 ) cp5 2 s t−p 1 s t−p−p5 2 x m−2j−p5 1 x 2j_{−(t−p−p}5) 2 ) . (23)

Since the series (20) are absolutely convergent we can write the functions F0, F1, and F2as polynomials functions

in r, namely F0= ∞ ∑ t=1 ∞ ∑ n=t f_0,(k,j,t)(n) rt, F1= ∞ ∑ t=1 ∞ ∑ n=t f_1,(k,j,t)(n) rt, F2= ∞ ∑ t=1 ∞ ∑ n=t f_2,(k,j,t)(n) rt, (24)

Now, considering the square of the functions F0, F1and F2 we obtain F₀2= ∞ ∑ p=1 a0,prp+1, F12= ∞ ∑ p=1 a1,prp+1, F22= ∞ ∑ p=1 a2,prp+1, (25) where ai,p:= p ∑ u=1 ∞ ∑ n=u f_i,(k,j,u)(n) ∞ ∑ v=p_−u+1 f_i,(k,j,p(v) _−u+1), i = 0, 1, 2. Thus, it follows F02+ F12+ F22= ∞ ∑ p=1 Aprp+1 (26)

where Ap:= a0,p+ a1,p+ a2,p. In a compact form we nally obtain |f(y) − f(x)| =√F2 0 + F12+ F22= v u u t∑∞ p=1 Aprp+1. (27)

4 Expansion factor and estimation of the quasiconformality factor

In this section we give explicitly the coecient of quasiconformality of any function dened in R+_{(B(x, r);A)}

depending on its Fourier coecients. Next we formulate the result.

Theorem 2 (On the coecient of quasiconformality) Let f be an A-valued function dened in R+_{(B(x, r);A).}

The quasiconformality factor kf(x) is given by

kf(x) := √ max A1 min A1 , where A1= a0,1+ a1,1+ a2,1, ai,1= ∞ ∑ n=1 f_i,(k,j,1)(n) ∞ ∑ v=1 f_i,(k,j,1)(v) (i = 0, 1, 2).

Proof: For a given A-valued monogenic function, we have proved in the previous section that

|f(y) − f(x)| = v u u t∑∞ p=1 Aprp+1= r √ A1+ A2r + . . . + Anrn−1+ . . . . (28) Taking the limit r → 0 we obtain

lim

r→0[A1+ A2r + . . . + Anr

n−1_{+ . . .] = A}

1. (29)

By using the denition of quasiconformality

kf(x) = lim sup r→0 kf(x, r) = lim sup r→0 Lf(x, r) lf(x, r) ,

we can conclude that

kf(x) = lim sup r_→0

max√A1+ A2r + . . . Anrn−1+ . . . min√A1+ A2r + . . . Anrn−1+ . . . that gives an estimation for the factor k of the quasiconformality of f, namely

kf(x) := √ max A1 min A1 (30) where A1= a0,1+ a1,1+ a2,1, ai,1= ∞ ∑ n=1 f_i,(k,j,1)(n) ∞ ∑ v=1 f_i,(k,j,1)(v) (i = 0, 1, 2).

For the expressions f(n)

i,(k,j,1), i = 0, 1, 2,considering t = 1 in (21), (22), and (23) we obtain:

˜ U_n0,†(y)− ˜U_n0,†(x) = [n 2] ∑ k=0 2βn,0,k(xn₀−2k|x|2k+ 2kxn₀−2k(|x|2)k−1Λ + (n− 2k)xn₀−2k−1c1|x|2k) ˜ U_nm,†(y)− ˜U_nm,†(x) = = [n−m 2 ] ∑ k=0 [m 2] ∑ j=0 2βn,m,k(−1)j ( m 2j ) [( xn₀−2k−m|x|2k) (2js1s2xm₁−2jx2j₂−1+ (m− 2j)c2s1xm₁−2j−1x2j₂ ) +(2kxn₀−2k−m(|x|2)k−1Λ + (n− 2k − m)xn₀−2k−m−1c1|x|2k) (2jxm₁−2jx2j₂ )]

and, moreover ˜ V_nm,†(y)− ˜V_nm,†(x) = = [n−m∑2 ] k=0 [m−1∑2 ] j=0 2βn,m,k(−1)j ( m 2j + 1 ) [( xn₀−2k−m|x|2k) (2js1s2xm₁−2jx2j₂−1+ (m− 2j)c2s1xm₁−2j−1x2j₂ ) + +(2kxn₀−2k−m(|x|2)k−1Λ + (n− 2k − m)xn₀−2k−m−1c1|x|2k) (2jxm₁−2jx2j₂ )] , where Λ = x0c1+ x1c2s1+ x2s1s2.

5 Examples

In this section we will compute the quasiconformality constant for some basis polynomials given in (3). For degree 1, there is only one quasiconformal monogenic polynomial, which is X0,_†

1 = x0+ x1 2 i + x2 2 j. For this polynomial we obtain A1 = 1₂ √ 4 cos2_(θ

1) + sin2(θ1), and, therefore, k_X0,†

1 (x) =

√

2, independent of the point

x ∈ R3 chosen. The other monogenic polynomials of degree 1 are not quasiconformal since kf(x) = 0. For

example, for X1,† 1 = 3x1 2 − 3x0 2 iwe obtain A1= 3 2 √ cos2_(θ

2) sin2(θ1) + cos2(θ1). Since in this case the minimum

is zero, we have kX1,₁†(x) =∞. For degree 2 there are some monogenic polynomials that are quasiconformal in

some region of R3_{. For example, for the polynomial} X0,₂†= 6x 2 0− 3x21− 3x22 4 + 3x0x1 2 i + 3x0x2 2 j (31) we obtain A1 = 3 4 (

4x2₂− 4x2₂cos2(θ2) sin2(θ1)− 8x0x2cos(θ1) sin(θ2) sin(θ1) + 4x20sin 2_(θ

1) −8x0x1cos(θ1) cos(θ2) sin(θ1) + 4x21− 4x

2 1sin 2_(θ 2) sin2(θ1) + 16x20cos 2_(θ 1)

+ 8x1x2cos(θ2) sin2(θ1) sin(θ2)

)1 2

. (32)

Using the computer algebra system Maple, we nd that the maximum and the mininum of (32), for each point

xare given by max A1= 3 4 [( 216|x0| ( x40+ ( x21+ x 2 2 ) x20+ 7 27 ( x21+ x 2 2 )2) √ 9 x2 0+ 4 x21+ 4 x22 + 648 ( x40+ 7 9x 2 0 ( x21+ x22 ) +1 9 ( x21+ x22 )2) ( x20+ 4 9 ( x21+ x22 )))/ ( ( 4 x22+ 4 x21+ 9 x20− 3 |x0| √ 9 x2 0+ 4 x21+ 4 x22 ) ( 9 x20− 3 |x0| √ 9 x2 0+ 4 x21+ 4 x22+ 2 x 2 1+ 2 x22 ) )]1₂ (33) and min A1= 3 2|x0|. (34)

Therefore, for x0 ̸= 0 the quasiconformality factor k_X0,†

2

(x) can be computed by (30). From (34) we can see that the quasiconformality constant kX0,₂†(x)behaves as O

( 1

|x0|

)

.For x0= 0the polynomial (31) is no longer

quasiconformal.

In the next table we compute the quasiconformality factor kX0,₂†(x) in dierent points: x = (x0, x1, x2) k_X0,† 2 (x) (₁ 2, 1 2, 1 2 ) 1.600485190 ( −1,1 5, 2 ) 1.734354994 (2,−3, 5) 1.943892809 (₁ 10, 1 5, 3 ) 5.532086985 ( ₁ 100, 1, 0 ) 10.02552988

For degree 3 or higher there are also monogenic polynomials that are quasiconformal in some region of R3_{. We}

proceed with the monogenic polynomial of degree 3

X0,₃† =−3x0x2₂− 3x0x2₁+ 2x3₀+ ( 3x2₀x1−3 4x 3 1− 3 4x1x 2 2 ) i + ( −3 4x 2 1x2− 3 4x 3 2+ 3x 2 0x2 ) j . (35)

For this polynomial we obtain

A1 = [( 81x4₂+ 90(x2₁+ 4x2₀)x2₂+ 9 (x1− 2x0) 2 (x1+ 2x0) 2) sin2(θ2) + 144 cos (θ2) x2x1 ( x2₁+ x2₂+ 6x₀2)sin (θ2) + 9 cos2(θ2) ( x4₂+(10x2₁− 8x₀2)x2₂+ 40x2₀x2₁ + 9x4₁+ 16x4₀)]sin (θ1) 2

+ 144(x2₂+ x2₁− 4x2₀)cos (θ1) x0(sin (θ2) x2+ x1cos (θ2)) sin (θ1)

+ 144 cos2(θ1)

(

4x40+ x42+ 2x22x21+ x41

)

(36) Using the computer algebra system Maple, we nd that the maximum and the mininum of (36), for each point

x,are given by max A1= 9 (( x21+ x 2 2+ 12 7x 2 0 ) (( x21+ x 2 2 )2 −16 9x 4 0 ) √ 49x4 1+ 88x20x21+ 98x21x22+ 144x40+ 88x20x22+ 49x42 +7x82+ ( 128 7 x 2 0+ 28x 2 1 ) x62+ ( 42x41+ 384 7 x 2 0x 2 1+ 3040 63 x 4 0 ) x42 + ( 2048 63 x 6 0+ 384 7 x 2 0x 4 1+ 28x 6 1+ 6080 63 x 4 0x 2 1 ) x22+ 256 7 x 8 0+ 3040 63 x 4 0x 4 1+ 128 7 x 2 0x 6 1+ 2048 63 x 6 0x 2 1+ 7x 8 1 )1 2 / [ 4 (( x21+ x 2 2− 12 7x 2 0 ) √ 49x4 1+ 88x20x21+ 98x21x22+ 144x40+ 88x20x22+ 49x42+ 7x 4 2+ ( −40 7 x 2 0+ 14x 2 1 ) x22 +7x41− 40 7x 2 0x 2 1+ 144 7 x 4 0 )] (37) and min A1= 3 44x 2 0− x 2 1− x 2 2. (38)

From (38) we can see that the minimum of A1 is zero along the cone x20 =

x21+ x22

4 . Since in this cone the maximum of A1 is non-zero we conclude that the quasiconformality factor k_X0,†

3 (x) in this case behaves as

O ( 1 |4x2 0− x21− x22| ) .For x2₀=x 2 1+ x 2 2

4 the polynomial (35) is no longer quasiconformal. In the next table we compute the quasiconformality factor kX0,₃†(x)in dierent points:

x = (x0, x1, x2) k_X0,† 3 (x) (₁ 2, 1 2, 1 2 ) 2.485741007 ( −1,1 5, 2 ) 21.21482516 (2,−3, 5) 2.626259389 (₁ 10, 1 5, 3 ) 1.737541999 ( ₁ 100, 1, 0 ) 1.732545819

Next gures show the deformation of the ball B(x, r), centered at x with radius r, by the polynomials (31) and (35). Figures 1 to 3 correspond to the deformation by the polynomial (31) and Figures 4 to 6 correspond to the deformation by the polynomial (35). In Figures 1, 2, 4, and 5 we can observe that the ball is mapped onto an ellipsoid, but in gures 3 and 6 the ball is not mapped onto an ellipsoid since in the considered ball the mappings are not longer quasiconformal.

–0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5 0.3 0.35 0.4 0.45 0.5 –20.5 –20 –19.5 –19 –18.5 –9.5 –9 –8.5 14.5 15 15.5 16 –1.6 –1.2 –1 –0.8 –0.4 –0.8 –0.4 0 0.4 0.8 0 0.2 Fig. 1: x =(1 2, 1 2, 1 2 ) , r = 1₈ Fig. 2: x = (2, −3, 5) , r = 1₈ Fig. 3: x =(₁₀₀1 , 1, 0), r = 1₂ –0.6 –0.55 –0.5 –0.45 –0.4 0.15 0.2 0.25 0.12 0.16 0.2 0.24 –5 –4 –3 –2 –1 0 –2 –1 –22 –21 –20 –19 –4 –2 0 2 4 –6 –4 –2 0 2 –1 0 1 2 Fig. 4: x =(1 2, 1 2, 1 2 ) , r = ₂₀1 Fig. 5: x =(₁₀1,1₅, 3), r = ₁₀1 Fig. 6: x =(₁₀₀1 , 1, 0), r = 1

Acknowledgements

The rst author acknowledges nancial support from the Foundation for Science and Technology (FCT) via the post-doctoral grant SFRH/BPD/66342/2009. The second author's research is supported by FEDER funds through COMPETE-Operational Programme Factors of Competitiveness (Programa Operacional Factores de Competitividade"��) and by Portuguese founds through the Center for Research and Development in Math-ematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia"��), within project PEst-C/MAT/UI4106/2011 with COM-PETE number FCOMP-01-0124-FEDER-022690.

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