DMA Departamento de Matemática e Aplicações Universidade do Minho

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1 Univeridade do Minho DMA Departamento de Matemática e Aplicaçõe Univeridade do Minho Campu de Gualtar Braga Portugal Univeridade do Minho Ecola de Ciência Departamento de Matemática e Aplicaçõe 3D Mapping by Generalized Joukowki Tranformation Carla Cruz a M.I. Falcão b H.R. Malonek a a Departamento de Matemática, Univeridade de Aveiro, Portugal b Departamento de Matemática e Aplicaçõe, Univeridade do Minho, Portugal Information Keyword: Generalized Joukowki tranformation, quaiconformal mapping, hypercomplex differentiable function. Original publication: Computational Science and it Application, Lecture Note in Computer Science, vol. 6784, pp , DOI:.7/ Abtract The claical Joukowki tranformation play an important role in different application of conformal mapping, in particular in the tudy of flow around the o-called Joukowki airfoil. In the 98 H. Haruki and M. Barran tudied generalized Joukowki tranformation of higher order in the complex plane from the view point of functional equation. The aim of our contribution i to tudy the analogue of thoe generalized Joukowki tranformation in Euclidean pace of arbitrary higher dimenion by method of hypercomplex analyi. They reveal new inight in the ue of generalized holomorphic function a tool for quai-conformal mapping. The computational experience focu on 3D-mapping of order and their propertie and viualization for different geometric configuration, but our approach i not retricted neither with repect to the dimenion nor to the order. Introduction and Notation Firt of all we refer ome baic notation ued in hypercomplex analyi. Let {e, e,..., e m } be an orthonormal bai of the Euclidean vector pace R m with the non-commutative product according to the multiplication rule e k e l +e l e k = δ kl, k, l =,..., m, where δ kl i the Kronecker ymbol. The et {e A : A {,..., m}} with e A = e h e h... e hr, h < < h r m, e = e =, form a bai of the m -dimenional Clifford algebra Cl,m over R. Let R m+ be embedded in Cl,m by identifying x, x,..., x m R m+ with the algebra element x = x +x A := pan R {, e,..., e m } Cl,m. The element of A are called paravector and x = Scx and x = Vecx = e x + +e m x m are the calar rep. vector part of the paravector x. The

2 3D Mapping by Generalized Joukowki Tranformation conjugate of x i given by = x x and the norm x of x i defined by x = x = x = x +x + +x m. Conequently, any non-zero x ha an invere defined by x = x. We conider function of the form fz = A f Aze A, where f A z are real valued, i.e. Cl,m -valued function defined in ome open ubet Ω R m+. Continuity and real differentiability of f in Ω are defined componentwie. The generalized Cauchy-Riemann operator in R m+, m, i defined by where := + x, :=, x := e + + e m. x x x m The higher dimenional analogue of an holomorphic function i uually defined a C Ω-function f atifying the equation f = rep. f = which i the hypercomplex form of a generalized Cauchy-Riemann ytem. By hitorical reaon it i called left monogenic rep. right monogenic [3]. An equivalent definition of monogenic function i that f i hypercomplex differentiable in Ω in the ene of [9], [6], i.e. that for f exit a uniquely defined areolar derivative f in each point of Ω ee alo [8]. Then f i automatically real differentiable and f can be expreed by the real partial derivative a f = / f, where := x i the conjugate Cauchy-Riemann operator. Since a hypercomplex differentiable function belong to the kernel of, it follow that in fact f = f = x f like in the complex cae. Due to the role of the complex derivative in the tudy of conformal tranformation in C, it wa natural to invetigate the role of the hypercomplex derivative from the view point of quai-conformal mapping in R m+ [7, ]. Indeed, conformal mapping in real Euclidean pace of dimenion higher than are retricted to Möbiu tranformation Liouville theorem which are not monogenic function. But obviouly, thi doe not mean that monogenic function cannot play an important role in application to the more general cla of quai-conformal mapping, intenively tudied by real and everal complex variable method o far. The advantage of hypercomplex method applicable to Euclidean pace of arbitrary real dimenion not only of even dimenion like in the cae of C n -method i already evident for one of the mot important cae in practical application, i.e. the lowet odd dimenional cae of R +. Beide other practical reaon, it till allow directly viualization of all geometric mapping propertie. Of coure, quai-conformal 3D-mapping demand more computational capacitie than conformal D-mapping. But ince hypercomplex analyi method are developed in analogy with complex method [, 4, 6,,,, 9], the expectation on their efficiency for olving 3D-mapping problem are in general very high. Neverthele, a ytematical work on thi ubject i till miing, preumably becaue of miing familiarity with hypercomplex method and their ue in practical problem. Therefore we intend to give ome inight in thi ubject alo by comparion with the complex D cae and by D- and 3D-plot produced with Mathematica for their viualization. Worth noticing that in thi paper only baic computational apect are dicued. A deeper function theoretic analyi, for intance, the relationhip of the hypercomplex derivative and the Jacobian matrix, wa not the aim of thi work. However, in [5], the reader can find the correponding reult for the hypercomplex cae of generalized Joukowki tranformation of order k =. Generalized Joukowki Tranformation in the Complex Plane In [3] and later in [] H. Haruki and M. Barran tudied pecific functional equation whoe unique olution i given by w = wz = zk + z k, where k i a poitive integer. The function w = w + iw i aid to be a generalized Joukowki tranformation of order k. It map the unit circle into the interval [, ] of the real axi in the w-plane traced k time. Obviouly there i no eential difference between tranforming the unit circle into the real interval [, ] and tranforming it into the imaginary interval [ i, i]. If we ue modified polar coordinate in the form For k = ee [5] or [6], where thi modified treatment of the Joukowki tranformation wa ued for the firt time. It allow to connect the D cae more directly with the correponding hypercomplex 3D cae, where the unit phere S ha a purely vector-valued image in analogy to the purely imaginary image of S.

3 Carla Cruz, M.I. Falcão and H.R. Malonek 3 z = ρe i π ϕ = ρin ϕ + i co ϕ, for ϕ [, π], then we obtain the interval [ i, i] a the image of the unit circle S under the mapping w = wz = zk z k. Moreover the real and imaginary part of w are obtained in the following form w = ρ k ρ k co k π kϕ, w = ρ k + ρ k in k π kϕ. Circle of radiu ρ are tranformed onto confocal ellipe with emi-axi a = ρk ρ k, b = ρ k + ρ k and foci w = i and w = i. Figure and how the well known image of dik with radii equal or greater than one under the mapping w, for k =. To tre the double covering of the egment [ i, i], in the cae ρ =, we preent the image of both emi-dik eparately. The retriction to black and white figure uggeted to ue alo dotted line. The ame i analogouly done for k = with dahed and dotted line. Figure 3 and 4 are the k = analogue of Fig. and. We underline the fact that, in thi cae, the mapping function i 4-fold when ρ = and -fold for ρ >. Complex Cae - Order k = Figure : The two unit emi-dik and their correponding image.3.3 Figure : The image of dik of radiu ρ =.5 and ρ = 3 3 Generalized Joukowki Tranformation in R m+ In [5] the higher dimenional analogue of the claical Joukowki tranform for m and k = ha been tudied in detail for the firt time. For it generalization to the cae of arbitrary order k, we apply two monogenic paravector-valued function which generalize z k and z k in C. They are defined for m by k k k Pk m x = c m x k x = T k mx k 3 and where c k m = k = T k m and = E m x = T k m = k! m k =, 4 x m+ m+ k k!! m,

4 4 3D Mapping by Generalized Joukowki Tranformation Complex Cae - Order k = Figure 3: The image of the quarter-dik Figure 4: The image of emi-dik of radiu ρ =.5 and ρ = 3 with m k denoting the Pochhammer ymbol. Propertie of the polynomial Pk m x, which form an example of a hypercomplex Appell equence, a well a of the fundamental olution E m x of the generalized Cauchy- Riemann ytem ee Sect. can be found in [7], repectively [8]. Analogouly to [5], the propoed higher dimenional analogue of the Joukowki tranformation i given by Definition. Let x = x +x A = R m+ Cl,m. The generalized hypercomplex Joukowki tranformation of order k i defined a Jk m x = α k Pk m x + k E m k x, 5 m k where α k i a real contant and E k m x denote the hypercomplex derivative of order k, for k. Formula 5 with α = 3 i the generalized hypercomplex Joukowki tranformation conidered in [5]. In fact, 5 can be expreed only in term of thoe monogenic polynomial of type Pk m, if we ue the Kelvin tranform for harmonic function in everal real variable. The connection of monogenic function with harmonic function relie on the fact that monogenic function are alo harmonic function, ince the Laplace operator for m + -real variable i factorized by the generalized Cauchy-Riemann operator and it conjugate operator ee Sect. in the form =. Often thi property, well known from Complex Analyi, i conidered a the eential reaon why hypercomplex analyi or, more general Clifford Analyi ee the title of [3], could be conidered a a refinement of Harmonic Analyi. For our purpoe we adapt the notation of [8] and ue Definition. Given a monogenic, paravector-valued, and homogeneou function f of degree k, then the monogenic homogeneou function of degree k + m defined in R m+ \{} i called the Kelvin tranform of f. I[f]x := E m xfx 6 The Kelvin tranform of an harmonic function generalize the inverion on the unit circle in the complex plane to the inverion on the unit phere in R m+ more about it propertie and application in hypercomplex analyi can be found in [8]. The following propoition how the connection between the Kelvin tranform applied to the polynomial P m k and the hypercomplex derivative of E m. Propoition. Let P m k and E m be the function defined by 3 and 4 repectively. Then E k m x = k m k I[P m k ]x. 7

5 Carla Cruz, M.I. Falcão and H.R. Malonek 5 Proof. The factorization of the fundamental olution in the form m+ x m+ = x allow the ue of Leibniz differentiation rule in order to obtain E k m x = k = x k x m+ k = k = k k! k m + k x m+k+ = = k m k x m+k+ k x x m m+ k k! m k T k m k x = m m+ +k x m + x x! k x = k m k x m+k+ Pm k 8 On the other hand, recalling that the polynomial Pk m and applying the Kelvin tranform 6 we obtain I[P m k ]x = and the final reult follow now at once. x m+ Pm k have the property of being homogeneou of degree k x = x m+k+ Pm k 9 For m =, i.e. in the complex cae, we have I[z k ] = z k+ and E k z = z k and the factor in 7 i nothing ele than k k!. Notice that thi correpond to the form of the claical Joukowki tranformation which we are uing. Moreover, formula 5 how a generalization of the hypercomplex Joukowki tranformation tudied in [5] and [6] by mean of the fundamental olution E m. Propoition and, in particular, formula 9 implifie the neceary calculation ince we can rely on the well tudied tructure of the polynomial Pk m. In what follow we focu on the cae m =, i.e. R 3, and write briefly Pk x = P kx, Jk x = J kx and c = c. Thi mean, that we conider now for arbitrary k the generalized hypercomplex Joukowki tranformation in the form J k x = α k P k x + k E k x = α k P k x I[P k ]x. k! Then the image of the unit phere S = {x = x + x : x = } under the mapping J k i given by: where k k J k S = α k c = k k x α k x x c x k = α k c k + k c k x k + Ak,, k { [ ] } k k k Ak, = α k c + + c x k x. = = x k x Here, a well a before, the contant α k remain for the moment undefined. Applying now the fact that the coefficient c k in thi pecial cae of m = cf. [7] are equal to { k!! k+!!, if k i odd c k = c k, if k i even

6 6 3D Mapping by Generalized Joukowki Tranformation we ee that the coefficient of x k x, for k in depend on the parity of. Uing together with well known propertie of the binomial coefficient one obtain eaily the following identity c [ k ] k k + + c = Finally the image of the unit phere for k > i given by: J k S k + = α k x k [ k ] l= k c l+ l + c k k+ k, if i odd, if i even. x k l+ x l Since x l = l x l the um in i real and therefore the unit phere i mapped onto the hyperplane w =, or equivalently, J k S i a paravector with vanihing calar part, i.e. a pure vector. The ame i true for arbitrary m 3, but the correponding proof relie on more difficult expreion of the c k m ee [7] and for thi reaon ha been omitted here. For k =,, 3 we have the following explicit expreion for the image of the unit phere S under the mapping J k : J S 3 = α x, J S 5 = α x x, J 3 S 7 5 = α 3 3 x 8 x Until now we did not dicu the role of the contant α k in the general expreion of 5. But we have already mentioned the form of the generalized hypercomplex Joukowki tranformation for k = conidered in [6], where it value i α = 3. The reaon for uch a choice wa the tandardization of the mapping in uch a way, that the image of S would be the unit circle S in the hyperplane w =. Obviouly, thi correpond to the unit interval [ i, i] a the image of S in the complex cae. Writing now briefly in the form with B k x = B k x, x := k + k J k S = α k xb k x [ k ] l= k c l+ l + x k l+ x l, we ee that the problem of determining α k for each value of k in the previouly mentioned way i olved by α k := max x = B kx, x. 4 From Sect. we recall the ue of modified polar coordinate in the complex plane cae for decribing the claical Joukowki tranformation with the purely imaginary interval [ i, i] a the image of the unit circle. They lead to the application of geographic pherical coordinate in the 3-dimenional pace and allow to decribe eaily the mapping propertie of J a explained in [5] and [6]. Therefore, let ρ, ϕ, θ be radiu, latitude, and longitude repectively, o that we work with x = ρ co ϕ co θ, x = ρ co ϕ in θ, x = ρ in ϕ where ρ >, π < θ π and π ϕ π. Thu, from 3 we have in term of pherical coordinate J S = α 3 co ϕ. Thi how that the unit dik S in the hyperplane w = a the image of the unit phere S in R 3 i obtained if α i choen equal to 3 [5]. Analogouly, for k = we have that J S 5 = α co ϕ in ϕ = α in ϕ. 5 4

7 Carla Cruz, M.I. Falcão and H.R. Malonek 7 Again, it i eay to ee that in thi cae α = 4 5 guarantee the deired mapping. In the ame way it i, in principle, poible to determine for every k the correponding value of α k in form of 4. Since the olution of algebraic equation of higher order become involved, it will obviouly be more complicated than in thoe lower dimenional cae. 4 3D Mapping by Generalized Joukowki Tranformation After the dicuion of the baic analytical apect of generalized hypercomplex Joukowki tranformation, we now focu on ome baic geometric mapping apect of the tranformation J and J in R 3. Our pecial concern are propertie imilar or not to thoe of the complex cae. Uing, a referred in the previou ection, the normalization factor α = /3 in 5 we obtain the component of J = w + w e + w e in term of pherical coordinate in the form of w = ρ 3 3 ρ in ϕ, 5 w = 3 + ρ 3 ρ co ϕ co θ, 6 w = 3 + ρ 3 ρ co ϕ in θ. 7 One can eaily oberve that J map phere of radiu ρ into pheroid with equatorial radiu a = 3 ρ + ρ and polar radiu b = 3 3 ρ ρ. But if ρ = then 5-7 implie that 3 J S ha a real part identically zero and atifie w + w = co ϕ which mean that the two-fold unit dik in the hyperplane w = i the image of the unit phere. Figure 5 how the image of both hemiphere with radiu equal to one under the mapping J. The relation between the polar and the equatorial radiu how alo that the correponding image, for value of ρ < 3 4, i an oblate pheroid and for value of ρ > 3 4 i a prolate pheroid, wherea the value of ρ = 3 4 correpond to a phere. Moreover, a example for comparion with the complex cae in Sect., Fig. 6 how the image of phere with different radii greater than one, in particular the phere obtained a image of a phere with ρ = 3 4. Thi limit cae between image of an oblate and a prolate pheroid wa for the firt time determined in [5]. Thee picture reveal the imilarity between the complex and the hypercomplex cae, but different from the complex cae where the type of ellipe remain the ame for all ρ > ee Fig.. Conider now the cae k = for which we ue the generalized hypercomplex Joukowki tranformation with α = 4 5 a explained in the previou ection, i.e. J x = 4 5 P x + E x = 4 x + x x + 5 x + 5 x x It real component have, in term of pherical coordinate, the following expreion: w = ρ 5 5 ρ + 3 in ϕ, 8 w = ρ 5 ρ in ϕ co ϕ co θ, 9 w = ρ 5 ρ in ϕ co ϕ in θ, A we expected, phere in R 3 with radiu ρ are tranformed into pheroid, but thi time, we obtain a -fold mapping. It i alo poible to detect another new property, different from the previou cae k =, namely the effect that the center of the pheroid doe not anymore remain on the origin. The hift of the x 3

8 8 3D Mapping by Generalized Joukowki Tranformation Cae R 3 - Order Figure 5: The image of both hemiphere of S Figure 6: The image of phere of radiu ρ =.3, ρ = 3 4 and ρ = 3 center from the origin occur in direction of the real w -axi and i equal to 5 ρ ρ. Therefore the 5 polar radiu i given by b = 3 5 ρ ρ and the equatorial radiu by a = 5 5 ρ + 3 ρ, o that we have: 5 [w 5 ρ ρ 5 ] [ 3 5 ρ ρ 5 ] + w [ 5 ρ + 3 ρ 5 ] + w [ 5 ρ + 3 ρ 5 ] =. The following propoition ummarize ome propertie of the mapping J : Propoition.. Sphere with radiu < ρ < 5 6 are -folded tranformed into oblate pheroid.. The phere with radiu ρ = 5 6 i -folded tranformed into the phere with center 3. Sphere with radiu 5 6 < ρ are -folded tranformed into prolate pheroid.,, The unit phere S i 4-folded mapped onto the unit circle including it interior in the hyperplane w =. Figure 7 how the image of four zone of the unit phere under the mapping J a conequence of the 4-fold mapping of the unit phere S to S cf. with the -fold mapping in the cae k = in Fig. 5. Analogouly to k = where we have hown image of phere in Fig. 6, the image in Fig. 8 are the reult of mapping one of the hemiphere with everal radiu greater than one. Similar to the cae k = the value ρ = 5 6 give a phere but now not centered at the origin. 5 Final Remark on Joukowki Type 3D Airfoil The claical Joukowki tranformation, or equivalently, play in Aerodynamic an important role in the tudy of flow around o-called Joukowki airfoil, ince it map circle with center ufficiently near to the origin into airfoil. In the claical Dictionary of Conformal Repreentation [5], for example, or the more recent book Computational Conformal Mapping [4], pecially dedicated to computational apect, one can

9 Carla Cruz, M.I. Falcão and H.R. Malonek 9 Cae R 3 - Order Figure 7: The image of the unit phere Figure 8: The image of hemiphere of radiu ρ =.3, ρ = 5 6 and ρ = 3 find a lot of detail about thoe ymmetric or unymmetric airfoil. They are, for k =, image in the w-plane obtained by mapping function of the form of circle centered at a point ufficiently near to the origin and paing through or. For example, Fig. 9 how two ymmetric airfoil that are image of circle centered at point different from the origin on the imaginary axi in the complex plane. An unymmetric and more intereting for tudie in aerodynamic Joukowki airfoil a image of a circle centered at a point in the firt quadrant i hown in Fig.. In the hypercomplex cae, the paper [6] analogouly include image produced with Maple of phere in R 3 centered at point of one of the axe x or x with a mall diplacement and paing through the endpoint of the unit vector e and e, repectively. We reproduce them in Fig.. Here we how in Fig. the image of a phere with ρ > in a more general poition. More concretely, it center i choen in.,., and the point /, /, i the correponding fixpoint of the mapping. Though both cae are image of dilocated from the origin phere of the ame radiu ρ >, it eem that the direction of the diplacement - only along one axi or not, for example - lead to lightly different image, a the figure ugget. Particularly one can recognize different curvature of the urface. It eem to u not preumptuou to interpret thoe figure a ome kind of ymmetric Joukowki airfoil generalized to 3D and extended in different direction. Figure 3 which how ome cut of the domain illutrated in Fig. parallel to the hyperplane w = w i in our opinion a very clear illutration of thi ituation. If the diplacement of the center of the phere i alo done in all three direction unymmetrically with three different value of the center coordinate, then we get a mapping like the one preented in the Fig. 4. It could be interpreted a ome kind of unymmetric Joukowki airfoil in 3D. Finally we compare ome mapping for the cae k = in D and 3D. Due to the higher order of ingularitie in the origin we hould alo be aware of more complicated image of circle and phere, repectively, with radii different from ρ = Figure 5-6. Neverthele, we would not exclude the poibility, that they could be ueful for mathematical model working with more complicated geometric configuration with ome ingularitie, particularly in R 3. Reuming thi tep toward a more ytematic tudy of 3D mapping realized by generalized hypercomplex Joukowki tranformation we would like to mention that hypercomplex method eem to u in general a promiing tool for quai-conformal mapping in R 3 []. We could produce by monogenic function ome mapping from the unit phere in R 3 that remind ignificant imilaritie in our opinion with one-wing object reported in connection with an an alternative airframe deign, called Blended Wing Body, or BWB ee [], which include intereting image of ongoing contruction of one-wing airplane. There one find the remark that: The advantage of the BWB approach are efficient high-lift wing and a wide airfoil-haped body. With Applied to our modified form in Sect. they hould pa through i and i. Of coure, thee fixpoint are only choen for ome normalization of the mapping and are not eential for it global behavior.

10 3D Mapping by Generalized Joukowki Tranformation Complex Cae - Order k =.6.6 Figure 9: The image of dik with radiu ρ =. and ρ =.6 centered at d = ρ i.... Figure : The image of the dik with radiu ρ = + d and centered at d =. +.i Hypercomplex Cae - Order k = Figure : The image of a phere of radiu ρ = + d and center d =.e Figure : The image of a phere of radiu ρ = + d and center d =.e +.e

11 Carla Cruz, M.I. Falcão and H.R. Malonek Figure 3: Cut parallel to the hyperplane w = w Hypercomplex Cae - Order k = Figure 4: The image of a phere of radiu ρ = + d and center d =.5e +.e +.e Complex Cae - Order k = Figure 5: The image of a dik with radiu ρ =. and center d = ρ i Hypercomplex Cae - Order k = Figure 6: The image of a phere of radiu ρ = + d and center d =.e +.e thi final remark we leave it a a funny exercie of imagination to the reader to peculate about a poible application of generalized hypercomplex Joukowki tranformation in practical circumtance. Acknowledgment Financial upport from Center for Reearch and Development in Mathematic and Application of the Univerity of Aveiro, through the Portuguee Foundation for Science and Technology FCT, i gratefully acknowledged. The reearch of the firt author wa alo upported by the FCT under the fellowhip SFRH/BD/44999/8. Moreover, the author would like to thank the anonymou referee for their helpful comment and uggetion which improved greatly the final manucript.

12 3D Mapping by Generalized Joukowki Tranformation Reference [] Barran, M., Hakuri, H.: On two new functional equation for generalized Joukowki tranformation. Annale Polon. Math. 56, [] Bock, S., Falcão, M. I., Gürlebeck K., Malonek, H.: A 3-Dimenional Bergman Kernel Method with Application to Rectangular Domain. Journal of Computational and Applied Mathematic, Vol. 89, [3] Brackx, F., Delanghe, R., Sommen, F.: Clifford Analyi. Pitman 76, London, 98 [4] Cruz, J., Falcão, M. I., Malonek H.: 3D-mapping and their approximation by erie of power of a mall parameter. In: Gürlebeck, K., Könke, C. ed. Proc. 7th Int. Conf. on Appl. of Comp. Science and Math. in Architecture and Civil Engineering, Weimar pp.6 [5] De Almeida, R., Malonek, H.R.: On a Higher Dimenional Analogue of the Joukowki Tranformation. In: 6th International Conference on Numerical Analyi and Applied Mathematic, AIP Conf. Proc. Vol. 48, Melville, NY, 8 [6] De Almeida, R., Malonek, H.R.: A note on a generalized Joukowki tranformation. Applied Mathematic Letter. 3, [7] Falcão, M.I, Malonek, H. R.: Generalized exponential through Appell et in R n+ and Beel function, In: 5th International Conference on Numerical Analyi and Applied Mathematic, AIP Conf. Proc. Vol. 936, Melville, NY, 7 [8] Gürlebeck, K., Habetha, K., Spröeig, W.: Holomorphic Function in the plane and n-dimenional pace. Birkäuer, Boton 8 [9] Gürlebeck, K., Malonek, H.: A hypercomplex derivative of monogenic function in R n+ and it application. Complex Variable, Theory Appl. 39, [] Gürlebeck K., Morai, J.: Geometric characterization of M-conformal mapping, Proc. 3rd Intern. Conf. on Appl. of Geometric Algebra in Computer Science and Engineering AGACSE 7 pp. 8 [] Gürlebeck, K., Morai, J.: On mapping propertie of monogenic function, CUBO A Mathematical Journal, Vol., No., 73 9 [] Gürlebeck, K., Morai, J.: Local propertie of monogenic mapping, AIP Conference Proceeding, Numerical analyi and applied mathematic, In: 7th International Conference on Numerical Analyi and Applied Mathematic, AIP Conf. Proc. Vol. 68, Melville, NY, 9 [3] Haruki, H.: A new functional equation characterizing generalized Joukowki tranformation. Aequatione Math. 3, no. -3, [4] Kythe, P. K.: Computational Conformal Mapping. Birkhäuer, Boton, 998 [5] Kober, H.: Dictionary of Conformal Repreentation. Dover Publication, New York 957 [6] Malonek, H.: A new hypercomplex tructure of the euclidean pace R m+ and the concept of hypercomplex differentiability. Complex Variable, Theory Appl. 4, [7] Malonek, H.: Contribution to a geometric function theory in higher dimenion by Clifford analyi method: monogenic function and M-conformal mapping. In: Brackx F., Chiholm, J. S. R., Souček V. ed. Clifford Analyi and It Application, 3. Kluwer, Dordrecht [8] Malonek, H.: Selected topic in hypercomplex function theory. In: Erikon, S.-L. ed. Clifford algebra and potential theory, Report Serie 7, -5. Univerity of Joenuu 4 [9] Malonek, H., Falcão, M.I.: 3D-mapping by mean of monogenic function and their approximation. Math. Method Appl. Sci. 33, 43 43

13 Carla Cruz, M.I. Falcão and H.R. Malonek 3 [] Väiälä, J.: Lecture on n-dimenional quaiconformal mapping, Lecture Note in Mathematic, 9, Springer, Berlin 97 [] Aircraft configuration and Wing deign,