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338 K. Gürlebeck and J. Morais

we can consider

f(x)

∼

f(0) +linf(x),

where f(0) is to be understood as a translation of the ellipsoid generated by the linear

part. Since, by assumption, linf maps balls to our special ellipsoids, we can apply

Theorem 3, and for an arbitrary rotation R, its associated symmetric matrix has the

form

where

A is a diagonal matrix, and its elements satisfy the following relation:

∂

∂x

[linf]

∂

∂x

[linf]

∂

∂x

[linf]

=0. (11)

On the other hand, having in mind identiﬁcation (4), we write

linf(x) = x

∂

∂x

[f]

(0) +x

∂

∂x

[f]

(0) +x

∂

∂x

[f]

(0)

−e



∂

∂x

[f]

(0) +x

∂

∂x

[f]

(0) +x

∂

∂x

[f]

(0)



−e



∂

∂x

[f]

(0) +x

∂

∂x

[f]

(0) +x

∂

∂x

[f]

(0)



Then, it comes directly from the structure of the matrix associated to linf together

with the relation (11) that the conditions of f at the origin satisfy the following

system:

⎧

⎪

⎨

⎪

⎩

∂

∂x

[f]

(0) +

∂

∂x

[f]

(0) +

∂

∂x

[f]

(0) =0,

∂

∂x

[f]

(0) +

∂

∂x

[f]

(0) =0,

∂

∂x

[f]

(0) +

∂

∂x

[f]

(0) =0,

∂

∂x

[f]

(0) +

∂

∂x

[f]

(0) =0,

which means, in a compact form, that Df(0) =0.

Extending this argument to an arbitrary point a ∈Ω by translation of our Taylor

series, we obtain that the function f satisﬁes at each point of the domain the Cauchy–

Riemann equation (i.e., Df(a) =0). Hence, f is monogenic, and this completes our

proof. 

According to the fact that the function f has rank 3, it is natural to ask for the

properties of the inverse mapping. We ﬁrst again restrict ourselves to linear functions

and work with ellipsoids having the property that the reciprocal of the length of

one semiaxis is equal to the sum of the reciprocals of the lengths of the other two

semiaxes.

Geometric Characterization of M-Conformal Mappings 339

By the next corollary, we state that a linear map between special regions in R

has an inverse function that is monogenic.

Corollary 1 Let linf be an A-valued linear function deﬁned in a domain Ω of

with nonvanishing Jacobian determinant. Then, linf maps an ellipsoid (with the

properties described above) onto a ball if and only if its inverse function linf

−1

monogenic.

After this direct consequence of Theorem 3 we show now that the previous corol-

lary can be generalized to arbitrary real-analytic functions which have the described

local mapping properties. A function with nonvanishing linear part that maps (in

the small) the special class of ellipsoids onto balls has an inverse function that is

monogenic.

Corollary 2 Let f be an A-valued real-analytic function deﬁned in a domain Ω of

with nonvanishing Jacobian determinant. Then, f maps locally an ellipsoid (with

the properties described above) onto a ball if and only if its inverse function f

−1

monogenic.

The noncommutative nature of the quaternionic algebra raises new obstacles in

the study of inverses of M-conformal mappings. In general, the inverse of a mono-

genic function is not monogenic. In the following we will study at least some prop-

erties of the linear part of the inverse of an A-valued monogenic function.

Theorem 6 Let f be an A-valued real-analytic function deﬁned in a domain Ω

of R

with nonvanishing Jacobian determinant and such that it maps locally an

ellipsoid (with the properties described above) onto a ball. Then, its linear part

satisﬁes the following properties:

1. curl(linf) =0;

2. |div(linf(x))|=



−α +β +γ, α minimum,

α −β +γ, β minimum,

α +β −γ, γ minimum;

3. |div(linf(x))|∈



[α, β +γ ],αminimum,

[β,α +γ ],βminimum,

[γ,α +β],γminimum;

D linf =0 if

a. γ =

(3 ±

√

5)β or β =

(3 ±

√

5)γ , α minimum;

b. γ =

(

√

5 ±1)α, β minimum;

c. β =

(

√

5 ±1)α, γ minimum.

Proof We have seen in Corollary 1 that the inverse of linf, which maps a ball onto

the special ellipsoid, is monogenic. Also, it is easily seen that the matrix associated

to linf

−1

, here denoted by A, is symmetric. From linear algebra it is well known

340 K. Gürlebeck and J. Morais

that the matrix that represents the original function linf, A

−1

, is also symmetric.

This means clearly that curl(linf) =0.

Denoting by μ

,μ

the eigenvalues of A

−1

, it is clear that

div



linf(x)



=tr



−1



=μ

+μ

where λ

,λ

stand for the eigenvalues of A.

Now, assume without loss of generality that div(linf(x)) is positive and the pa-

rameter α assumes the minimum value of the semiaxes, i.e.,

Then, Conditions 2 and 3 are immediate since

=−α +β +γ =

+γ

+βγ

β +γ

. (12)

For Property 4, we assume again without loss of generality that α is the minimum

value. Then, from the previous equality it follows

D linf =

+λ

+3λ

(λ

+λ

)λ



Remark 6 Taking into account the expression on the right-hand side of (12), it is

clear that div(linf(x)) cannot be zero, and consequently linf cannot be monogenic.

Remark 7 In Property 4 it is shown that for some special choices of the parameters

α, β, and γ , the linear part of the inverse of an A-valued monogenic function is

antimonogenic.

5 Observations and Perspectives

With the obtained results, we are able to characterize monogenic mappings com-

pletely by their local geometric properties. Having in mind the desired analogy to

conformal mappings in the plane, one can ask for more special properties. In the

here presented approach, we relate the shapes of 3D-objects in the domain of def-

inition and in the image of a mapping. In the paper [17] a certain ratio between

surfaces and volumes was the background for the deﬁnition of M-conformal map-

pings. Nothing has been said about the transformation of angles. We want to study

by examples here only the transformation of right angles under monogenic transfor-

mations. The idea behind is to map orthogonal grids from the sphere to the surface

of more general domains. Looking for the moment again only at local properties, we

study the problem for linear monogenic mappings. We have seen before that each

Geometric Characterization of M-Conformal Mappings 341

Fig. 2 Images of curves on

the unit ball under the linear

monogenic transformation

f(x) =2x

Fig. 3 Images of curves on

the unit ball under the linear

monogenic transformation

f(x) =4x

+5x

−x

monogenic linear function can be transformed by a rotation such that the represent-

ing matrix has diagonal form. All angles are invariant under this rotation. Studying

now the question under which conditions the images of meridians and latitude cir-

cles intersect orthogonally, we ﬁnd the additional condition that b

−c

= 0. The

ﬁrst solution b

= c

deﬁnes a special nondegenerate linear and monogenic func-

tion. For b

=1 and c

=1, Fig. 2 visualizes the images of a meridian and a circle

with latitude π/4. Indeed, we can see that the right angles are preserved.

To get an impression that this preservation is not automatically satisﬁed, we con-

sider a second example. Here we see that the angles are changing. The exact angle

in Fig. 3 is about 122

◦

Looking now at the second solution of the extra equation, we obtain b

=−c

This implies immediately that a

= 0 and our mapping degenerates because it

does not depend on x

. Such a mapping is a so-called hyperholomorphic constant

(see [7]). A hyperholomorphic constant is a monogenic function with identically

vanishing hypercomplex derivative. Moreover, taking into account that the set of

complex numbers is isomorphic to the (commutative) even subalgebra C

0,2

,itis

easy to prove that such a hyperholomorphic constant is isomorphic to an antiholo-

morphic function in the (x

)-plane (see [19], pp. 10–11). This means that such

a mapping will preserve all angles in the (x

)-plane (not necessarily the orienta-

tion).

Analyzing the condition b

again, we ﬁnd that this is exactly the condition

for a linear monogenic function to be orthogonal to the subspace of hyperholo-

morphic constants in L

(B). Moreover, in [9] it was shown that each monogenic

342 K. Gürlebeck and J. Morais

A-valued function f permits an orthogonal decomposition

f(x) :=f(0) +g(x) +h(x), (13)

where h is a hyperholomorphic constant and g is the so-called principal part of f.

Collecting all these properties, we understand from the simple example that lo-

cally each monogenic function can be decomposed as an orthogonal sum of a prin-

cipal monogenic mapping, preserving right angles as described, and a hyperholo-

morphic constant, preserving all angles in a plane.

At the moment it is still an open problem how the described local mapping prop-

erties can be connected with the global mapping properties of M-conformal map-

pings. There is already one paper [2], where the authors study the global behavior

of a generalized Joukowski mapping in the three-dimensional space.

Acknowledgement The second author is sponsored by Fundação para a Ciência e a Tec-

nologia—FCT via the Ph.D./grant SFRH/BD/19174/2004.

References

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2. De Almeida, R., Malonek, H.R.: On a higher dimensional analogue of the Joukowski trans-

formation. Num. Anal. Appl. Math., AIP Conf. Proc. 1048, 630–633 (2008)

3. Bock, S., Falcão, M.I., Gürlebeck, K., Malonek, H.R.: A 3-dimensional Bergman kernel

method with applications to rectangular domains. J. Comput. Appl. Math. 189, 6–79 (2006)

4. Boone, B.: Bergmankern en conforme albeelding: een excursie in drie dimensies. Proefschrift,

Universiteit Gent Faculteit van de Wetenshappen (1991–1992)

5. Cerejeiras, P., Gürlebeck, K., Kähler, U., Malonek, H.R.: A quaternionic Beltrami-type equa-

tion and the existence of local homeomorphic solutions. ZAA 20, 17–34 (2001)

6. Delanghe, R., Kraußhar, R.S., Malonek, H.R.: Differentiability of functions with values in

some real associative algebras: approaches to an old problem. Bull. Soc. R. Sci. Liège 70(4–6),

231–249 (2001)

7. Gürlebeck, K., Malonek, H.R.: A hypercomplex derivative of monogenic functions in R

n+1

Complex Var. 39, 199–228 (1999)

8. Gürlebeck, K., Morais, J.: On mapping properties of monogenic functions. CUBO Math. J.

11(01), 73–100 (2009)

9. Gürlebeck, K., Morais, J.: Bohr type theorems for monogenic power series. Comput. Methods

Funct. Theory 9(2), 633–651 (2009)

10. Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems.

Math. Research, vol. 56. Akademieverlag, Berlin (1989)

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York (1997)

12. Kraußhar, R.S., Malonek, H.R.: A characterization of conformal mappings in R

by a formal

differentiability condition. Bull. Soc. R. Sci. Liège 70, 35–49 (2001)

13. Kravchenko, V.V.: Applied Quaternionic Analysis. Research and Exposition in Mathematics,

p. 28. Heldermann, Berlin (2003)

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Physics. Research Notes in Mathematics. Pitman, London (1996)

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Application de l’analyse à la géometrie, G. Monge, Paris, pp. 609–616 (1850).

Geometric Characterization of M-Conformal Mappings 343

16. Malonek, H.R.: Power series representation for monogenic functions in R

m+1

basedona

permutational product. Complex Var. Theory Appl. 15, 181–191 (1990)

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Clifford Analysis and its Applications. Kluwer, Dordrecht (2001)

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Fluid Flow Problems with Quaternionic

Analysis—An Alternative Conception

K. Gürlebeck and W. Sprößig

Abstract This article deals with some classes of ﬂuid ﬂow problems under given

initial-value and boundary-value conditions. Using a quaternionic operator calculus,

representations of solutions are constructed. For the case of a bounded velocity,

a numerically stable semi-discretization procedure for the solution of the problem

is presented.

1 Introduction

The study of hydrodynamic problems leads J. d’Alembert (1752) and L. Euler

(1777) to the notion of a

holomorphic function in the plane, long before A. Cauchy

(1814) started with his comprehensive work. B. Riemann used in his papers the

connection between holomorphic functions and planar ﬂows of ﬂuids and heat. On

the other hand, complex analysis initiated new solution methods for boundary value

problems in several research areas of mathematical physics, in particular, in pla-

nar ﬂuid- and aerodynamics (N.E. Joukowski 1911) and elasticity theory (G.W.

Kolossov 1909).

Although real quaternions were available since Sir W.R. Hamilton’s invention

in 1843, further 60 years should lapse until the ﬁrst paper in quaternionic analysis

could appear. This paper by A.C. Dixon from the University of Belfast was enti-

tled On the Newtonian Potential. In the thirties of the last century, G.N. Moisil and

N. Teodorescu from the University of Cluj, on the one hand, and the group around

the Swiss mathematician R. Fueter, on the other hand, published a series of fun-

damental papers on quaternionic analysis. The ﬁrst closed representation of real

Clifford analysis (= generalization of complex analysis in R

), which can also be

W. Sprößig (



)

Fakultaet fuer Mathematik und Informatik, TU Bergakademie Freiberg, Prueferstraße 9,

09596 Freiberg, Germany

e-mail: sproessig@math.tu-freiberg.de

E. Bayro-Corrochano, G. Scheuermann (eds.), Geometric Algebra Computing,

345

346 K. Gürlebeck and W. Sprößig

seen as a generalization of quaternionic analysis, was written by the Belgian group

around R. Delanghe in 1982 [2].

The solution of boundary-value problems of partial differential equations by

function theoretic methods needs a corresponding knowledge on the behavior of

Cauchy’s parameter integral near to the boundary. The Russian Y.V. Sokhotzki

(1873) and the Slovenian mathematician J. Plemelj (1908) found independently of

each other jump formulae for Cauchy’s integral. In 1912 D. Pompeiu proved the

formula

u(z) =

2πi



∂

u(ζ )

ζ −z

dζ −



∂

f(ζ)

ζ −z

dξ dη, (1.1)

where G ⊂R

is a bounded domain with piecewise smooth boundary ∂G, and ζ =

ξ +iη. This formula bears today the name

formula of Borel–Pompeiu.G.C.Moisil

proved in 1930 a corresponding analogue in R

. The weakly singular integral oper-

ator in the second term of Borel–Pompeiu’s formula is called

Teodorescu transform.

The Russian mathematician A.W. Bitzadse found in 1953 a quaternionic analogue

to Cauchy’s integral in the plane for compact Lyapunov surfaces and quaternionic

Hölder continuous functions. T.G. Gegelia proved in 1968 its L

-boundedness. This

result was generalized in 1982 by R. Coifman, A. McIntosh, and Y. Meyer on Lips-

chitz surfaces in R

Based on three operators: a Cauchy–Riemann-type operator, a Teodorescu trans-

form, and a generalized Cauchy-type integral operator in the books [7, 8], an opera-

tor calculus was developed, acting on quaternion-valued or Clifford algebra-valued

functions. A new alternative approach for the treatment of ﬂuid ﬂow problems in

the space could be established using Plemelj–Sokhotzki-type formulae and Borel–

Pompeiu’s formula by the help of the mentioned operators. Roughly speaking, a

calculus for the spatial ﬂuid ﬂow problems was developed which is analogous to

the known complex analytic approach in the plane, advantageously used for a long

time.

In this paper we will give a survey on the state of the art. For this reason we will

omit the most proofs and refer for that to special research papers. We describe the

basics of the applied operator calculus, construct ﬁrst representation formulas for the

steady-state case, and generalize them to the time-dependent case. Finally, we dis-

cuss a semi-discretization method for the numerical solution of the initial-boundary-

value problems for the Navier–Stokes equations and certain coupled problems in-

volving the Navier–Stokes equations. Our discussion is based on the assumption that

a well-adapted numerical method for the remaining elliptic boundary-value prob-

lems is available. For a discussion of this problem, we refer to [6]. If the velocity is

bounded, then the presented semi-discretization is numerically stable.

2 Operator Calculus

In this section we introduce a general operator conception. By suitable examples we

intend to demonstrate the general ideas of this approach.

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 347

2.1 Operator Triple

Let X, Y, Z be Banach spaces, and let T,Tr, P be bounded linear operators such

that:

(i) T :X →imT ⊂Y is injective.

(ii) Tr :Y →Z is a generalized trace operator.

(iii) The operator P :im Tr →Y satisﬁes P Tr Pu=Pu.

We assume:

(iv) im TrT ⊂ker P .

(v) im T ∩ker Tr ={0}.

It is easy to see that im T ∩im P ={0}. The reader can ﬁnd the proofs in [12].

Theorem 2.1 (Mean value formula [12]) Set imT ⊕ im P =: Y

⊂ Y. There is a

unique linear operator L with D(L) =Y

and L :D(L) →X such that

u =P Tr u +TLu. (2.1)

Corollary 2.2 [12] The following relations between the operators L, P , and T are

valid:

(i) The operator L is the left inverse to the operator T , i.e., LT =I .

(ii) Set R := TL. Then R is a projection onto Y

with im R =im T .

(iii) kerL =imP Tr.

Deﬁnition 2.3 (L-Holomorphy) Elements u ∈ker L ∩Y are called L

-holomorphic.

L is called

algebraic derivative. P Tr is called the initial projection, and T is denoted

general Teodorescu transform.

Remark 2.4 From the point of view of a general operator theory T is also called

algebraic integral.

2.2 Plemelj-Type Projections

Set P

:= Tr P : im Tr → Z and Q

:= I − P

. Then the following properties are

valid:

(i) The operators P

are idempotent, i.e., we have P

and Q

and

furthermore Q

=0.

(ii) An element ξ ∈ Z is the generalized trace of an element u from ker L if and

only if P

ξ =ξ .

(iii) We have Q

ξ =Tr TLu.