Since 1992 H. Leutwiler, S. L. Eriksson and others developed in a number of papers
a modified Clifford analysis and, particularly, a modified quaternionic analysis. The modification mainly consists in considering generalized Cauchy-Riemann equations with respect to a hyperbolic metric in a half space. The aim of this paper is to show how through a change of the basic combinatorial relations used in the modified quaternionic analysis a special Maplesoftware that has been developed for the case of the Euclidean metric can directly be used for numerical calculations in the modified theory.Fundação para a Ciência e a Tecnologia (FCT

Falcão, M. I.

Malonek, H. R.

Silva, A. M.

Abstract. Since 1992 H. Leutwiler, S. L. Eriksson and others developed in a number of papers a modified Clifford analysis and, particularly, a modified quaternionic analysis. The modifi-cation mainly consists in considering generalized Cauchy-Riemann equations with respect to a hyperbolic metric in a half space. The aim of this paper is to show how through a change of the basic combinatorial relations used in the modified quaternionic analysis a special Maple-software that has been developed for the case of the Euclidean metric can directly be used for numerical calculations in the modified theory. 1 

H. R. Malonek

M. I. Falcão

A. M. Silva

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MAPLE TOOLS FOR MODIFIED QUATERNIONIC ANALYSIS

At the 16th IKM Bock, Falcão and Gürlebeck presented examples of the application of some specially developed Maple-Software in hypercomplex analysis. Other papers of those authors continued this work and showed the efficiency of such tools for concrete numerical calculations as well as for numerical experiments, supporting the detection of new relationships and even theorems in a highly technical theoretical work. The mentioned software has been developed mainly for the use on mapping problems in the Euclidean spaces of dimension 3 and 4 by means of Bergman kernel methods (BKM), which are related to monogenic functions as solutions of generalized Cauchy-Riemann equations with respect to the Euclidean metric (Riesz system). The developed procedures concerning generalized powers of totally regular variables and the corresponding homogeneous polynomials basically rely on results and conventions introduced in the paper "Power series representation for monogenic functions in Rm+1 based on a permutational product", Complex Variables, 15, No.3, 181-191 (1990) by H. Malonek. Since 1992 H. Leutwiler, S. L. Eriksson and others developed in a number of papers a modified Clifford analysis and, particularly, a modified quaternionic analysis. The modification mainly consists in considering generalized Cauchy-Riemann equations with respect to a hyperbolic metric in a half space. The aim of this contribution is to show how through a change of the basic combinatorial relations used in the modified quaternionic analysis the aforementioned Maple-software (that has been recently published on CD-Rom as integrated part of the text book "Funktionentheorie in der Ebene und im Raum" by K. Gürlebeck, K. Habetha, and W. Sprössig, in the series "Grundstudium Mathematik" of Birkhäuser Verlag, 2006) can directly be used for numerical calculations in the modified theory

Malonek, Helmuth R.

Falcão, M. Irene

Silva, António

Digitale Bibliothek Thüringen

17th International Conference on the Application of ComputerScience and Mathematics in Architecture and Civil EngineeringK. Gu¨rlebeck and C. Ko¨nke (eds.)Weimar, Germany, 12–14 July 2006MAPLE TOOLS FOR MODIFIED QUATERNIONIC ANALYSISH.R. Malonek∗, M.I. Falca˜o, and A.M. Silva∗Departamento de Matema´tica, Universidade de AveiroPortugalE-mail: hrmalon@mat.ua.ptKeywords: modified quaternionic analysis, Maple softwareAbstract. Since 1992 H. Leutwiler, S. L. Eriksson and others developed in a number of papersa modified Clifford analysis and, particularly, a modified quaternionic analysis. The modifi-cation mainly consists in considering generalized Cauchy-Riemann equations with respect to ahyperbolic metric in a half space. The aim of this paper is to show how through a change ofthe basic combinatorial relations used in the modified quaternionic analysis a special Maple-software that has been developed for the case of the Euclidean metric can directly be used fornumerical calculations in the modified theory.11 INTRODUCTIONAt the 16th IKM in 2003 Bock, Falca˜o and Gu¨rlebeck presented in [1] examples for the ap-plication of a special Maple-Software to approximation problems in IR3 based on methods ofClifford analysis. For this purpose they used a special class of generalized holomorphic func-tions which are obtained as solutions of a generalized Cauchy-Riemann system in the Euclideanspace IR3 and by tradition are called monogenic functions (cf. [3]). The paper [2] continued thiswork and shows the efficiency of such tools for concrete numerical calculations as well as fornumerical experiments, supporting the detection of new relationships in highly technical theo-retical work. Recently (2006) this ”QuatPackage” (developed for the practical work in IR3 andalso IR4 ) has been published on a CD-ROM accompanying the text book Funktionentheorie inder Ebene und im Raum by Gu¨rlebeck, Habetha, and W. Spro¨ssig ([6]. The restriction to lowerdimension problems had only to do with the nature of the considered practical problems. In-deed, the algebraic procedures realized in this Maple software can easily be extended to higherdimensions by changing from the use of the quaternion algebra IH to general Clifford algebrasCl0,n.Almost fifteen years ago H. Leutwiler and his collaborators developed in a number of papersa modified Clifford analysis and, particularly, a modified quaternionic analysis. From the veryextensive list we mention here only [7], [4])[8], and [9]. The modification is based on a changefrom the Euclidean metric to generalized Cauchy-Riemann equations with respect to the hyper-bolic metric in a half space. In [9] Leutwiler compares both types as extensions of complexanalysis - the ”mathematical” as related to ”hyperbolic” (non-Euclidean) geometry, the ”physi-cal” as connected with the Euclidean geometry. For this purposes he uses an embedding of IR3into the algebra of quaternions IH in the following way: The points (x, y, t) ∈ IR3 are identifiedwith the so called reduced quaternions z = x + iy + jt, (with i, j as the usual quaternionicnon-commutative units).Let now f = u+iv+jw be a function with values in the set of reduced quaternions. Withoutgoing in the moment further into the details, we remark that the Euclidean metric in dimensionthree gives rise to generalized Cauchy-Riemann equations with respect to (u, v, w) in form ofthe Riesz system(R)ux − vy − wt = 0uy + vx = 0ut + wx = 0vt − wy = 0.Often and in comparison with the hyperbolic case its solutions f = u+ iv + jw are called (R)-solutions, but following [3] the functions f are the already mentioned monogenic functions. Tosee The Riesz system is a generalized Cauchy-Riemann system since we have the followingsimple relationship to the classical C.-R. system. Using Wirtinger’s-derivative the classicalCauchy-Riemann system can be written in one line. Indeed, f = f(z) = f(x + iy) is aholomorphic function in a domain Ω ⊂ C, if∂f∂z¯=12(∂f∂x+ i∂∂y) = 0.In the quaternionic case one uses the differential operatorD =∂∂x+ i∂∂y+ j∂∂t.2and it is easy to check that the Riesz system is equivalent to Df = 0 or fD = 0 (cf. [3]).Notice that the Riesz-system describes the velocity field of a stationary flow of a non-compressible liquid without sources and sinks and therefore really has physical relevance.On the other hand, the hyperbolic version of the generalized Cauchy-Riemann system, i.e.of the modified quaternionic analysis, is given by the so called Hodge system (cf. [9])(H)t(ux − vy − wt) + w = 0uy + vx = 0ut + wx = 0vt − wy = 0.We will call its solution (H)-solutions, following Leutwiler and others. Remarkable in thiscontext is the fact that all positive and negative powers zn of z = x + iy + jt are also reducedquaternions and at the same time (H)-solutions. Since Dz = zD = −1 this is not the same inthe Euclidean case, i.e. z = x + iy + jt is not monogenic. In Leutwilers own words (see e.g.[8] or [9]) this observation was one of the fundamental reasons for asking for a modification ofthe Euclidean case (with its origin in the work of Fueter [5], for example). Of course, the factthat the basic variable z = x + iy + jt as well as all its powers belong to the considered classof functions is a big advantage in the hyperbolic case and the study of (H)-solutions.Due to the aim of this paper, namely to show how through a change of the basic combinatorialrelations used in the modified quaternionic analysis we are able to use approaches that have beendeveloped for the case of monogenic functions we recall facts from [11] and [12]. Most of them- and particularly for German readers - can also be found in [6]. Thereby we restrict ourselvesonly to the technical tools.2 REMARKS ON MONOGENIC GENERALIZED POWERSThe main object of our study are procedures related to the construction of so called mono-genic generalized powers since we saw that the powers of the underlying variable are not mono-genic. We mention here only its appearance as basis of the Taylor series of monogenic functions.Let us start withDefinition 1 Let V+,· be some commutative or non-commutative ring, ak ∈ V (k = 1, . . . , n),then the symmetric “×”-product is defined bya1 × a2 × · · · × an = 1n!∑pi(i1,...,in)ai1ai2 · · · ain (1)where the sum runs over all permutations of all (i1, . . . , in).and use the followingConvention:If the factor aj occurs µj-times in (1), we briefly writea1 × · · · × a1︸ ︷︷ ︸µ1× · · · × an × · · · × an︸ ︷︷ ︸µn(2)= a1µ1 × a2µ2 × · · · × anµn = ~a µ3where µ = (µ1, . . . , µn, ) and set parentheses if the powers are understood in the ordinary way(see [11]).Since the symmetric products of µ1 factors z1 = y − xi and µ2 factors z2 = t − xj aremonogenic functions of homogeneous degree µ1 + µ2 which form a basis for the Taylor seriesof a monogenic function in R3 (see [12]) they are called generalized powers and written as ~zµwith the multi-index µ = (µ1, µ2).In practice it is usual to consider series ordered by powers of the same homogeneous degree.Therefore we set |µ| = n and µ1 = (n− k), µ2 = k; k = 0, 1, . . . n, it holdsTheorem 1 Every convergent L-power series generates in the interior of its domain of conver-gence a monogenic function f(~z) and coincides there with the Taylor series of f(~z), i. e. in aneighborhood of ~z = ~a we havef(z1, z2) =∞∑n=01n!n∑k=0(nk)(z1 − a1)n−k × (z2 − a2)k ∂nf(a1, a2)∂yn−k∂tk.(3)Notice that in our case of bi-monogenic functions the coefficients could also all be written onthe left side of the powers.Obviously, in the neighborhood of the origin the series reduces tof(z1, z2) =∞∑n=01n!n∑k=0(nk)z1n−k × z2k ∂nf(0, 0)∂yn−k∂tk3 BASIS OF (H)-SOLUTIONSStarting from the powers zn Leutwiler deduces two types of elementary polynomials whichcan serve as basis for homogeneous polynomial (H)-solutions of degree n.Definition 2 For n, k ∈ N0 and z = x+ yi+ jt ∈ R3 the so called L-polynomials are definedbyLkn(z) :=1k!∂kzn+k∂yk. (4)4Examples:L00(z) = 1L0n(z) = zn = zL0n−1Lk0(z) = ik = iLk−10 (k ∈ N)L11(z) = iz + ziL11(x+ iy + jt) = −2y + 2xiL12(z) = iz2 + ziz + z2iL12(x+ iy + jt) = −6xy + (3x2 − 3y2 − t2)i− 2ytjL21(z) = −2z + iziL21(x+ iy + jt) = −3x− 3yi− tjL22(z) = −3z2 + zizi+ iziz + iz2iL22(x+ iy + jt) = −6x2 − 6y2 + 2t2 − 12xyi− 4xtjL32(z) = −3iz2 − 3ziz − 3z2i+ iziziL32(x+ iy + jt) = 20xy + (−10x2 + 10y2 + 2t2)i+ 4ytjThe examples show the regular structure of the Lkn(z). In fact Leutwiler proved the followingformula:Lkn(z) =∑µ0+µ1+···+µn=kµν∈{0,1,...k}iµ0ziµ1ziµ2z · · · iµn−1ziµn .Another type of polynomials have been defined in a similar way:Definition 3 For n, k ∈ N0 and z = x+ yi+ jt ∈ R3 the so called E-polynomials are definedbyEkn(z) :=∑µ0+µ1+···+µn=kµν∈{0,1}iµ0ziµ1ziµ2z · · · iµn−1ziµn , (5)for k = 0, 1, . . . , n+ 1,They are related to the L-polynomials byLemma 1 For z ∈ R3 and all n, k ∈ N0, the polynomials Ekn(z) can be obtained as a linearcombination of the polynomials Lkn(z) in the following wayEkn(z) =[ k2]∑p=0(n+ 1p)Lk−2pn (z).4 POLYNOMIAL (H)-SOLUTIONS AND GENERALIZED POWERSIn [9] Leutwiler did not pose the question about the connections between the polynomialbasis functionszn−k1 × zk25of the Riesz system (R) on one side and the (H)-basisEkn(z) or Lkn(z)on the other side. Also other combinatorial expressions for Ekn(z) or Lkn(z) have not beenstudied.Our experience with polynomial solutions of (R)-systems and our interest in the applicationand development of Maple tools for problems in different areas of Clifford analysis led us tothe answer of that question. The following theorem shows the very simple relationship betweenboth combinatorial structures of polynomials, i.e. how the polynomial (H)-solutions can beconstructed with the tools developed for monogenic generalized powers.Theorem 2 For k, n ∈ N0, 0 ≤ k ≤ n+ 1 and z = x+ iy + jt in R3, it holdsLkn(z) =(n+ kk)zn × ikin the sense of the aforementioned convention.Proof:By definitionzn × ik = z × . . .× z︸ ︷︷ ︸n vezes× i× . . .× i︸ ︷︷ ︸k vezes=n!k!(n+ k)!∑Π(i1,...,in+k)ii1zi2 · · · zin+k−1iin+k=1(n+kk) ∑i0+i1+···+in=kiν∈{0,1,...,k}ii0zii1z · · · iin−1ziin=1(n+kk)Lkn(z).This leads toLkn(z) =(n+ kk)L˜kn(z).whereL˜kn(z) = zn × ik.is the so called normalized L− polynomial.We deduced the form of the L− polynomials in terms of the algebraic conventions and thepermutative ”×” product introduced in [11]. The same can be done also for Ekn(z) and thesefact imply that the Maple software developed for (R)-system problems is directly applicable toproblems on (H) - solutions. At the same time new formulas and relationships for problems inthe modified quaternionic analysis setting could be deduced. More technical details, includinga number of new Maple procedures, will be given in the oral presentation.6REFERENCES[1] S. Bock, M. I. Falca˜o and K. Gu¨rlebeck, Bergman kernel functions. K. Gu¨rlebeck,L. Hempel and C. Knke, eds. IKM, 16, Weimar, 2003 : proceedings http://e-pub.uni-weimar.de/volltexte/2005/311/[2] S. Bock, M. I. Falca˜o, K. Gu¨rlebeck and H. Malonek, A 3-Dimensional Bergman Ker-nel Method with Applications to Rectangular Domains, Journal of Computational andApplied Mathematics, Vol. 189, 1-2 , 67–79, 2006.[3] F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, Pitman 76, Boston-London-Melbourne, 1982.[4] S. L. Eriksson-Bique, On modified Clifford analysis, Complex Variables, Theory Appl.45 11–32, 2001.[5] R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comment. Math. Helv.4, 9–20, 1932.[6] K. Gu¨rlebeck, K. Habetha and W. Spro¨ssig. Funktionentheorie in der Ebene und imRaum Reihe: Grundstudium Mathematik, Birkha¨user, 2006.[7] H. Leutwiler, Modified Clifford analysis, Complex Variables, Theory Appl. 17 Nr. 3/4,153–171, 1992.[8] H. Leutwiler, Modified quaternionic analysis in IR3, Complex Variables, Theory Appl.20 Nr. 1-4, 19–51, 1992.[9] H. Leutwiler, Rudiments of a function theory in IR3, Expositiones Math. 14 , 97–123,1996.[10] H. Leutwiler, Quaternionic analysis in IR3 versus its hyperbolic modification . F. Brackset al. eds. Clifford analysis and its applications. Kluwer Acad. Publ., 193–211, 2001.[11] H. Malonek, Power series representation for monogenic functions in IRm+1 based on apermutational product. Complex Variables, Theory Appl. 15, Nr. 3, 181–191, 1990.[12] H. Malonek, Selected topics in hypercomplex function theory, in: Clifford algebras andpotential theory, Eriksson, S.-L.(ed.), University of Joensuu, Report Series 7, 111-150,2004.7

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