Cauchy-Kowalevski extensions and monogenic. plane waves using spherical monogenics

Transcription

1 Cauchy-Kowalevski extensions and monogenic plane waves using spherical monogenics N. De Schepper a, F. Sommen a a Clifford Research Group, Ghent University, Department of Mathematical Analysis, Faculty of Engineering, Galglaan, B-9000 Gent, Belgium Abstract Clifford analysis may be regarded as a direct and elegant generalization to higher dimensions of the theory of holomorphic functions in the complex plane, centred around the notion of monogenic function, i.e. a null solution of the Dirac operator. This paper deals with axial and biaxial monogenic functions containing spherical monogenics. They are constructed by means of two fundamental methods of Clifford analysis, namely the Cauchy-Kowalevski extension and monogenic plane waves. Key words: Clifford analysis, CK-extension, monogenic plane waves, axial monogenic function, biaxial monogenic function, Cauchy-Riemann equation 99 MSC: 30G35, 33C05, 33C5, 33C45 Corresponding author. Telephone: Fax: addresses: nds@cage.ugent.be N. De Schepper, fs@cage.ugent.be F. Sommen. Preprint submitted to Elsevier Science 8 September 0

2 Introduction Clifford analysis see a.o. [,3] is a theory that offers a natural generalization of complex analysis to higher dimensions. To R m, the Euclidean space in m dimensions, we first associate the Clifford algebra Cl 0,m, generated by the canonical basis e i, i =,..., m. These generators satisfy the multiplication rules e i e j + e j e i = δ ij. The Clifford algebra Cl 0,m can be decomposed as Cl 0,m = m k=0cl k 0,m with Cl k 0,m the space of k-vectors defined by Cl k 0,m = span{e i...i k = e i... e ik, i <... < i k }. More precisely, we have that the space of -vectors is given by Cl 0,m = span{e i, i =,..., m} and it is obvious that this space is isomorphic with R m. The space of so-called bivectors is given explicitly by Cl 0,m = span{e ij = e i e j, i < j}. Moreover, the even subalgebra Cl + 0,m of Cl 0,m is defined by Cl + 0,m = k even Cl k 0,m. Similarly, Cl 0,m = k odd Cl k 0,m is the so-called odd subalgebra. We identify the point x,..., x m in R m with the vector variable x given by x = m j= x j e j. The Clifford product of two vectors splits into a scalar part and a bivector part: xy = x y + x y with x y = x, y = m j= x j y j = xy + yx x y = e j e k x j y k x k y j = xy yx. j<k It is interesting to note that the square of a vector variable x is scalar-valued and equals the norm squared up to a minus sign: x = x, x = x. In a similar way we introduce a first order vector differential operator by x = m j= xj e j. This operator is the so-called Dirac operator. Its square equals, up to a minus sign, the Laplace operator in R m : x = m. In the sequel, we will also consider the so-called Cauchy-Riemann operator x0 + x for which x0 + x x0 x = m+. A function

3 fx,..., x m, respectively fx 0, x,..., x m, defined and continuously differentiable in an open domain of R m, respectively R m+, is called monogenic in that region if x f = 0, respectively x0 + x f = 0. They generally take values in the full Clifford algebra Cl 0,m. The spinor spaces may be interpreted as minimal left ideals of Cl 0,m of the form S = Cl 0,m I, I being the primitive idempotent. Hence solutions of the spinor Dirac operator are in fact monogenic functions of the form f I. In this paper we only consider monogenic functions with values in the full multivector space Cl 0,m. A monogenic homogeneous polynomial P k of degree k k 0 in R m is called a solid inner spherical monogenic of degree k. The set of all solid inner spherical monogenics of degree k will be denoted by M k. A monogenic function is called an axial monogenic of degree k if it is of the form fx 0, x = Ax 0, x + x x Bx 0, x P k x, where A, B are scalar functions and P k is a solid inner spherical monogenic of degree k. The monogenicity condition x0 + x f = 0 then leads to the Vekua-type system for A and B: x0 A r B = k + m B, x0 B + r A = 0, r = x, r the solution of which leads to numerous special monogenic functions see [5]. In general every monogenic function fx 0, x is determined by its restriction f0, x to the hyperplane x 0 = 0 and conversely any given real analytic function fx has a unique monogenic extension fx 0, x called Cauchy-Kowalevski extension abbreviated CK-extension see [4]. Axial monogenics of degree k are also determined by their restriction x k fx 0, x, which turns out to be Ax 0, 0P k ω and consequently, starting from a given function Ax 0 of one variable, one may establish its generalized axial CK-extension Ax0, x + x x Bx 0, x P k x. This generalizes the standard CK-extension that is used in our former paper [4]. If one composes this operation with the restriction to x 0 = 0 we 3

4 arrive at a correspondence Ax 0 A0, x + x B0, x x P k x transforming one dimensional functions into higher dimensional ones. In Section we study this generalized axial CK-extension and the resulting correspondence in detail for some cases, namely Ax 0 = e x 0, Ax 0 = e x 0 / and Ax 0 = x 0 α, α R. In Section 3 we generalize our result to the case of biaxial monogenic functions. These functions originate from the more general splitting R m = R p R q giving rise to two vector variables x = p j= x j e j, y = q j= y j e p+j that anti-commute, and the corresponding Dirac operators x = p j= xj e j, y = q j= yj e p+j. The over-all Dirac operator is x + y and biaxial monogenics are defined as solutions of x + y fx, y = 0 of the special form x = rω, y = ρξ, ω S p, ξ S q Ar, ρ + ωbr, ρ + ξcr, ρ + ωξdr, ρ Pk,l ω, ξ, whereby A, B, C, D are scalar functions and P k,l is a solid inner spherical monogenic of degree k, l in the variables x and y, i.e. P k,l λx, µy = λ k µ l P k,l x, y and x [P k,l x, y] = y [P k,l x, y] = 0. An example of such a solid inner spherical monogenic of degree k, l is P k,l x, y = P k xp l y with P k, respectively P l, a solid inner spherical monogenic of degree k, respectively l, in x, respectively y. Moreover it may be shown that see [3] the monogenicity condition leads to Vekua-type systems for the pairs A, D and B, C: r k r A + ρ + l + q ρ D = 0, r + k + p r D ρ l ρ A = 0 respectively r + k + p r B + ρ + l + q ρ C = 0, r k C ρ l B = 0. r ρ Similar to axial monogenics, biaxial monogenics are determined by their restriction to the first axis R p given by y = 0 and the second axis R q given by x = 0. This leads to a 4

5 correspondence between functions on R p and functions on R q that will be investigated for a number of examples. In the fourth section we will produce special axial and biaxial monogenics as linear superpositions of plane wave type monogenic functions. In the axial case we start from monogenic plane waves of the simple form hx 0 +i x, t it, whereby t S m is a variable unit vector and h is a classical holomorphic function of the complex variable x 0 + i x, t. We then consider the integral hx 0 + i x, t it P k t dt, S m P k being a fixed inner spherical monogenic of degree k, which may be evaluated using Funk-Hecke s theorem and gives, up to a constant, rise to the generalized axial CKextension of h k x 0 see Theorem 4.. Note that in case of the function hx 0 = e x 0 we can link this approach with the generalized axial CK-extension technique of Section, since then h k x 0 = hx 0 see subsection 4.. In the biaxial case we start from plane wave functions F x, y; t, s of either form hi x, t y, s t+is or hi x, t y, s +its with t S p, s S q variable orthogonal unit vectors. After integration over the bi-sphere t, s S p S q of F x, y; t, s P k,l t, s with P k,l an even Clifford algebra-valued inner spherical monogenic of degree k, l in the variable x and y, we obtain biaxial monogenics of odd and of even type respectively. In subsection 4.3 we work out these integrals explicitly for the exponential function exp i x, t y, s, giving rise to integral formulas for the Clifford-Bessel function of biaxial type. The methods used in this section are substantially more complicated than those in [4], due to the use of Funk-Hecke s formula in its most general form. It is clear that this paper deals with simple, though fundamental, examples of special monogenic functions and that many more special functions may be computed using the described methods. 5

6 For an introduction to Clifford analysis we refer to [,3,5,6]. Other papers dealing with axial and biaxial monogenics are [,8,9, 3]. This paper generalizes the results obtained in [4]. For the sake of readability, we have repeated some formulae and calculations, which is then explicitly mentioned. Generalized Cauchy-Kowalevski extensions: the axial framework. General problem We start with explaining the idea of generalized axial CK-extensions. Let us consider an SOm-invariant domain Ω R m+ and let us assume that the intersections of Ω with all subspaces parallel to R m are convex. Putting Ω = Ω R, we have the following result see [3]. Theorem. Let P k x M k be fixed and let W 0 x 0 be an analytic function in Ω. Then there exists a unique sequence W s x 0 s>0 of analytic functions such that the series fx 0, x = x s W s x 0 P k x is convergent in a neighbourhood U R m+ of the domain Ω and its sum f is monogenic in U. The function W 0 x 0 is determined by the relation P k ωw 0 x 0 = x fx 0, x, k ω = x x Sm. The series fx 0, x is the generalized axial CK-extension of the function W 0 x 0. In this section we consider series of the form fx 0, x = c s,k x s F s,k x 0 W x 0 P k x 6

7 with F 0,k x 0 =, the coefficients c s,k satisfying x [c s,k x s P k x] = c s,k x s P k x, c 0,k = and x0 + x [fx 0, x] = 0. In other words, f is monogenic in the whole of R m+ and x k fx 0, x = P k ωw x 0, ω = x x Sm, i.e. fx 0, x is the generalized axial CK-extension of W x 0. The aim is to determine f0, x = c s,k x s F s,k 0W 0 P k x. By means of induction and using the fundamental formula sx s P k x x [x s P k x] = s + m + kx s P k x for s even for s odd, one can prove that the coefficients c s,k satisfying, take the form c l,k = Γ m + k Γ m + k + l l l! and c l+,k = Γ m + k Γ m + k + l + l+ l!. 3. Examples.. W x 0 = e x 0 This example was already considered in [5]. Putting fx 0, x = c s,k x s B s,k x 0 e x 0 P k x with B 0,k x 0 =, the coefficients c s,k satisfying and f being monogenic in R m+, the aim is to determine f0, x. 7

8 The first step is to obtain B s,k x 0. Expressing the monogenicity of f, we find consecutively [ ] x0 + x c s,k x s B s,k x 0 e x 0 P k x = 0 B s,kx 0 e x 0 c s,k x s P k x + B s,k x 0 e x 0 c s,k x s P k x + B s,k x 0 e x 0 c s,k x s P k x = 0 s= B s,kx 0 e x 0 c s,k x s P k x + B s,k x 0 e x 0 c s,k x s P k x + B s+,k x 0 e x 0 c s,k x s P k x = 0, where in the last series the substitution s = s was executed. The functions B s,k x 0 must hence satisfy the recurrence relation B s+,k x 0 = B s,k x 0 B s,kx 0, B 0,k x 0 =, from which it follows that B l,k x 0 = and B l+,k x 0 =. Next, using spherical co-ordinates x = x ω, ω S m, we obtain f0, x = c s,k x s B s,k 0P k x = c l,k x l P k x c l+,k x l+ P k x m = Γ + k l x l Γ m + k + P k x l l l! m + Γ + k l x l Γ m + k + l + xp k x. l l! In view of ν J ν ρ = l ρ l l!γν + l + l 4 with J ν ρ = ρ ν J ν ρ, J ν being the Bessel function of the first kind, we finally arrive at m f0, x = Γ + k m/+k J m/+k x + J m/+k x x P k x. Remark. Note that the full generalized axial CK-extension of e x 0 is given by: m fx 0, x = Γ + k m/+k J m/+k x + J m/+k x x e x 0 P k x. 8

9 .. W x 0 = e x 0 / We now consider the series fx 0, x = c s,k x s H s,k x 0 e x 0 / P k x with H 0,k x 0 =, f being monogenic in R m+ and the coefficients c s,k given by 3. We again want to obtain a closed form for f0, x. In a similar way as for the first example, the monogenicity of f yields: H s+,k x 0 = x 0 H s,k x 0 H s,kx 0, H 0,k x 0 =. From the above recurrence relation it follows that H s,k x 0 are the classical Hermite polynomials on the real line associated with the weight function e x 0 /, which are defined by the Rodrigues formula H s,k x 0 = e x 0 / d s [ e x dx 0 These polynomials satisfy see e.g. [0], p. 50: from which we obtain 0 /]. H l,k 0 = l l! l! l and H l+,k 0 = 0, f0, x = c l,k x l H l,k 0P k x = Taking into account see e.g. [0], p. 3 Γ m + k Γ m l! l x l P + k + l l! k x. Γ l + Γ l l! =, 5 l! we finally arrive at f0, x = Γ m + k Γ l + Γ Γ x l l + m + P k x = F k l! ; m x + k; P k x 9

10 with Kummer s function. F a; c; z = Γc Γa Γa + l z l Γc + l l!..3 W x 0 = x 0 α, α R As a last example, we consider for W x 0 the Gegenbauer weight function. The generalized axial CK-extension now takes a slightly different form, namely fx 0, x = c s,k x s G α s,kx 0 x 0 α s P k x with G α 0,kx 0 = and the coefficients c s,k again satisfying. We indeed have that x k fx 0, x = x 0 α P k ω. First the aim is to determine G α s,kx 0 and then the series c s,k x s G α s,k0p k x. The monogenicity relation x0 + x [fx 0, x] = 0, leads to the recurrence relation G α s+,kx 0 = α sx 0 G α s,kx 0 x 0 G α s,k x 0, G α 0,kx 0 =. 6 The results below concerning the polynomials G α s,kx 0 are already mentioned in [4]. The polynomials of lower degree take the form G α 0,kx 0 = G α,kx 0 = αx 0 G α,kx 0 = αα x 0 α G α 3,kx 0 = 4αα α x 3 0 αα x 0 etc. 0

11 Putting G α t,kx 0 = t i=0 b t,α i x i 0 and G α t+,kx 0 = we can derive an expression for the coefficients b t,α i t i=0 and b t+,α i+. b t+,α i+ x i+ 0, Theorem. For 0 i t, one has b t,α i = t t α t + t i Γα + / t + i t! α 4t + t Γα t + / i t i!i! i+ = t t+ α t t+ t +! α 4t t+ Γα t / i i+ Γα + / t + i t i!i +! b t+,α with α l = αα +... α + l = Γα+l Γα Pochhammer s symbol. Proof. This result is proved by means of induction on the degree of the polynomial and using the recurrence relation 6. Using the above theorem, we can write G α t,kx 0 in terms of the classical Gegenbauer polynomials on the real line. Corollary. One has G α t,kx 0 = t t! α t + t C α t+/ t x 0 α 4t + t G α t+,kx 0 = t+ α t t+ t +! C α t / t+ x 0 α 4t t+ with C λ nx 0 the classical Gegenbauer polynomial on the real line given explicitly by see e.g. [0], p. 9 Cnx λ 0 = n Γλ i=0 i Γλ + n i x 0 n i, λ > 0. 7 i!n i! Proof. This follows immediately from Theorem. and the explicit expression 7.

12 From the above it follows that G α t+,k0 = 0 and G α t,k0 = b t,α 0 = t t t! t! In view of see e.g. [0], p. 3: Γα + Γα 4t + Γα + / t Γα t + Γα t + Γα t + /. Γz = π / z ΓzΓ z +, 8 the expression for G α t,k0 can be simplified to We thus obtain G α t,k0 = t t! t! Γα + Γα t +. c s,k x s G α s,k0p k x = c l,k x l G α l,k0p k x Γ m = + k Γ l! Γα + m + k + l l x l l! l! Γα l + P kx. 9 Taking into account 5 and see e.g. [0], p. : Γα + l = l Γα + Γ α Γl α, expression 9 becomes c s,k x s G α s,k0p k x = Γ m + k Γ m + k + l Γ l + Γ Γl α Γ α = F /, α; m/ + k; x P k x x l P k x l! with F a, b; c; z = the hypergeometric function. Γa + l Γa Γb + l Γb Γc Γc + l z l l!

13 3 Generalized Cauchy-Kowalevski extensions: the biaxial framework In this section we consider functions of two vector variables x = p j= x j e j and y = q j= y j e p+j. The sum x + y represents a vector variable in R p+q = R m equipped with the orthonormal basis e,..., e p, e p+,..., e p+q. The corresponding Dirac operators take the form x = p j= xj e j and y = q j= yj e p+j. Note that the vector variables and Dirac operators anti-commute, i.e. xy = yx and x y = y x and that the vector variables are orthogonal, i.e. x, y = 0. In the following, P k x and P l y represent solid inner spherical monogenics of degree k in the variable x and of degree l in the variable y. This means that P k x and P l y are homogeneous polynomials taking values in the Clifford algebra spanned by respectively {e,..., e p } and {e p+,..., e p+q }; they satisfy the relations x [P k x] = 0, P k tx = t k P k x and y [P l y] = 0, P l ty = t l P l y. We also assume P k x and P l y to be even Clifford algebra-valued, which implies that P k x and P l y commute. Let us first introduce the concept of generalized CK-extensions in this biaxial framework. Consider an SOq-invariant domain Ω R m and assume that the intersections of Ω with all subspaces parallel to R q are convex. Putting Ω = Ω R p, the following holds see [3]. Theorem 3. Let P l y M l q be fixed and let W 0 x be a fixed analytic function in Ω. Then there exists a unique sequence W s x s>0 of analytic functions such that the series fx, y = y s P l yw s x is convergent in a neighbourhood U R m of the domain Ω and its sum f is monogenic in U. The function W 0 x is determined by the relation P l ξw 0 x = fx, y, y 0 y l y = y ξ. The function fx, y is the so-called generalized biaxial CK-extension of W 0 x. 3

14 3. First example: W 0 x = e x / P k x Given with H 0,k x =, fx, y = c s,l y s P l yh s,k xe x / P k x y [c s,l y s P l y] = c s,l y s P l y, c 0,l = and x + y [fx, y] = 0, the question is to determine x k fx, y = c s,l y s P l yh s,k 0P k ω. In a similar way as in the previous section, we obtain that the coefficients c s,l are c l,l = Γ q + l Γ q + l + l l l! and c l+,l = Γ q + l Γ q + l + l + l+ l!. From the monogenicity of f, we obtain consecutively [ ] x + y c s,l y s P l yh s,k xe x / P k x = 0 c s,l s y s P l y xh s,k xp k x + x [H s,k xp k x] e x / + c s,l y s P l yh s+,k xe x / P k x = 0. The functions H s,k x thus satisfy the recurrence relation H s+,k xp k x = s x x [H s,k xp k x], H 0,k x =, from which we observe that H l,k x = l H GCH l,k x and H l+,k x = l H GCH l+,k x with Hs,k GCH x the so-called generalized Clifford-Hermite polynomials introduced by Som- men in [5]. Taking into account the expression of these polynomials in terms of the 4

15 generalized Laguerre polynomials on the real line Hl,k GCH x = l l!l p/+k l x and H GCH l+,k x = l l!l p/+k l x x, and the relation see e.g. [0], p. 40: L α n0 = α+n n!, we have that This finally yields H l,k 0 = l l Γ p + k + l x k fx, y = Γ p + k and H l+,k 0 = 0. c l,l y l P l yh l,k 0P k ω = Γ q + l Γ p + Γ p + k + l k Γ q + l + l y l P l yp k ω l! = F p/ + k ; q/ + l ; y P l yp k ω. 3. Second example: W 0 x = J k +p/ x + x J k +p/ x P k x In this subsection we determine the generalized biaxial CK-extension of the Clifford-Bessel function without exponential factor, introduced by Sommen in [5]. We now put fx, y = c s,l y s P l y s x [ J k +p/ x + x J k +p/ x P k x ] where the coefficients c s,l are taken such that c s,l y s P l y s x is monogenic, i.e. x + y [ c s,l y s P l y s x] = 0. This leads to the equation c s,l y [y s P l y] = s c s,l y s P l y. As y 0 y l fx, y = W 0xP l ξ, we also have that c 0,l =. By means of induction, one can prove that in this case the coefficients c s,l take the form c l,l = l Γ q + l Γ q + l + l l l! and c l+,l = l Γ q + l Γ q + l + l + l+ l!. 0 5

16 The question is to determine x k fx, y. Hereto we start with the following result. Theorem 3. One has s x [ J k +p/ x + x J k +p/ x P k x ] = s J k +p/ x + x J k +p/ x P k x. Proof. By means of the series expansion 4 for the Bessel function we have that J k +p/ x + x J k +p/ x P k x x l P = p/ k k x l!γ p + k + l + p/ k l x l+ P k x l!γ p + k + l +. l Using the fundamental formula and executing the substitution l = l in the first series, we have consecutively x [ J k +p/ x + x J k +p/ x P k x ] x l P = p/ k k x l= l!γ p + k + l l + p + k x l P k x p/ k l l!γ p + k + l + l x l + P = p/ k k x l =0 l!γ p + k + l + p/ k + x l P k x l l!γ p + k + l l = J k +p/ x + x J k +p/ x P k x, from which the desired result follows. We thus have that fx, y = c s,l y s P l y s J k +p/ x + x J k +p/ x P k x 6

17 which leads to x k fx, y = c s,l y s P l y s p/ k Γ p + P k ω k = p/ k Γ p + c l,l y l c l+,l y l+ P l yp k ω k Γ q = + p/ k l Γ y l p + k Γ q + l + l l l! y l y Γ q + l + l + P l yp k ω. l+ l! The above calculations lead to the following result. Theorem 3.3 One has that x k fx, y Γ q = + q p/+l k l Γ p + P k ωp l y J q/+l i y J q/+l i y y k = q p/+l k Γ q + l Γ p + k P k ωp l y y q/ l I q/+l y y y I q/+l y with I ν z the modified Bessel function of the first kind. Proof. This theorem is proved using the series expansion 4 and the relations see e.g. [0] J ν z = z ν J ν z, J ν iz = i ν I ν z. Remark 3. During the above calculations we have also obtained a closed form for the 7

18 full generalized biaxial CK-extension: q fx, y = q/+l Γ + l P l y J i y q/+l J q/+l i y y J p/+k x + x J p/+k x P k x q = q/+l Γ + l P l y y q/ l I q/+l y y y I q/+l y x p/ k J p/+k x + x x J p/+k x P k x. Note that from the monogenicity of f it follows that the even Clifford algebra-valued part: [fx, y] e = q/+l Γ q + l J p/+k x J q/+l i y + xy J k +p/ x J q/+l i y P k x P l y and the odd Clifford algebra-valued part [fx, y] o = q/+l Γ q + l J p/+k x J q/+l i y x J k +p/ x J q/+l i y y P k x P l y are also monogenic. 3.3 Third example: W 0 x = F a; c; x Pk x, a, c R Similar to the previous subsection, we start from the following expression for the generalized biaxial CK-extension fx, y = c s,l y s P l y s x [F a; c; x P k x ] where the coefficients c s,l take the form 0. We start with calculating the derivatives of W 0 x. 8

19 Proposition 3. For n N one has n x [F a; c; x P k x ] = n i=0 a n i p i c n i + k + n i i n i F a + n i; c + n i; x x n i P k x n+ x [F a; c; x P k x ] = n i=0 a n+ i p i c n+ i + k + n i + F a + n + i; c + n + i; x i n i x n+ i P k x. Proof. This result is proved by induction on n using the relations [ ] x F α; β; x = αβ F α + ; β + ; x x and l x l P k x for l even x [ x l P k x] = l + p + k x l P k x for l odd. The differential equations in Proposition 3. lead to n x k x n+ x k x ] [F a; c; x P k x = a n x=0 ] [F a; c; x P k x = 0. x=0 Hence, we obtain the following boundary value: x k fx, y = = n=0 p n c n + k c n,l y n P l y a n p n c n + k a n p + k n c n q + l n! n=0 n = F a, p + k ; c, q + l ; y n n P k ω y n P l yp k ω P l yp k ω. P k ω 9

20 Remark 3. The above calculations can be generalized to: W 0 x = k F l a,..., a k ; c,..., c l ; x P k x with a i R i =,..., k, c j R j =,..., l. In a similar way as above, we obtain p fx, y = k+ F l+ x k + k, a,..., a k ; q + l, c,..., c l ; y P l yp k ω. 4 Monogenic plane waves 4. General method For this method we refer to [6]. In what follows, we will frequently use the following Funk-Hecke theorem see for e.g. [7]. Theorem 4. Funk-Hecke theorem Let S k be a spherical harmonic of degree k, then S m f ω, η S k ω dsω = A m S k η ft t m 3/ P k,m t dt where P k,m t denotes the Legendre polynomial of degree k in R m, expressed as follows in terms of the Gegenbauer polynomials P k,m t = k!m 3! k + m 3! Cm / k t, and A m = πm / Γ m the area of the unit sphere Sm in R m. 4.. The axial case Let hz be a holomorphic function in z C. Consider hx 0 +i x, t it with x 0 R, x R m and t S m. We have that x0 + x [hx 0 + i x, t it] = h x 0 + i x, t + it it = 0. 0

21 In the sequel, we consider the following integral: fx 0, x = C hx 0 + i x, t it P k t dt S m with C a constant and P k t a fixed inner spherical monogenic of degree k. Naturally, we have that x0 + x [fx 0, x] = 0. Let us now determine x k fx 0, x. Hereto, we first expand hx 0 + i x, t into a Taylor series: i x, t l hx 0 + i x, t = l! h l x 0. Hence, we have that S m hx 0 + i x, t it P k t dt = h l x 0 i l x, t l! S l it P k tdt. m Let us consider the integral over S m in the case when l < k. Application of the Funk- Hecke theorem yields x, t l it P k tdt S m = πm / Γ r l k!m 3! P m k ω k + m 3! k +!m 3! iωp k ω k + m! t l t m 3/ C m / k tdt t l t m 3/ C m / k+ tdt where we have used spherical coordinates x = rω, r = x, ω S m. Taking into account the orthogonality of the Gegenbauer polynomials:, we immediately obtain that in case of l < k: We thus have that S m hx 0 + i x, t it P k t dt = t l C λ k t t λ / dt = 0 if l < k, S m x, t l it P k t dt = 0. l=k h l x 0 i l x l ω, t l! S l it P k tdt m

22 and also x C k S m hx 0 + i x, t it P k t dt = C h l x 0 i l l=k l! x l k S m ω, t l it P k tdt, from which we immediately see that x fx 0, x = C hk x 0 i k k k! S m ω, t k it P k tdt. Applying once more the Funk-Hecke theorem gives x fx 0, x = C hk x 0 i k k k! π m / Γ m P k ω iωp k ω t k t m 3/ P k,m t dt t k t m 3/ P k+,m t dt. The second integral is zero in view of the orthogonality relation. Furthermore, applying the following relation see [7], p. 88, ex. 4 ft P k,m t t m 3/ dt = k Γ m Γ k + m with f a function which is k times differentiable, we arrive at t k+m 3/ f k t dt x fx 0, x = C ik k+ π m / k Γ h k x k + m 0 P k ω t k+m 3/ dt. From the orthonormality relation of the Gegenbauer polynomials we derive that C k+m / 0 t C k+m / 0 t t k+m 3/ dt = π3 k m Γk + m k + m Γ k + m where we have used 8. We thus finally obtain x fx 0, x = C ik k π m/ k Γ k + m The above calculations lead to the following result. = πγ k + m Γ k + m h k x 0 P k ω.

23 Theorem 4. Let hz be a holomorphic function in z C, x 0 R, x R m and P k t a fixed inner spherical monogenic of degree k. Then the function fx 0, x = Γ k + m i k k π m/ S m hx 0 + i x, t it P k t dt satisfies i ii x0 + x fx 0, x = 0 x k fx 0, x = h k x 0 P k ω, ω = x x. In other words, fx 0, x is the generalized axial CK-extension of h k x 0. Remark 4. Note that we can only link this approach with the generalized CK-extension approach in the axial framework see Section in case of the function hx 0 = e x 0, since only then hx 0 = h k x The biaxial case In the biaxial case monogenic plane waves F x, y; t, s take the form hi x, t y, s t+ is or hi x, t y, s + its, t S p, s S q. We consider integrals of the form S p S q F x, y; t, s P k t P l s dt ds, where, as in Section 3, P k x and P l y represent solid inner spherical monogenics of degree k in the variable x and of degree l in the variable y. Again we assume them to be even Clifford algebra-valued, which implies that they commute. Integrals of the form lead to biaxial monogenic functions of the form [A x, y + ωb x, y + ξc x, y + ωξd x, y ] P k ω P l ξ, x = x ω, y = y ξ. 3

24 The evaluation of integrals of the above type is based on applying Funk-Hecke s theorem twice. Let us first consider the restriction to x = 0 of the even Clifford algebra-valued biaxial monogenic: f e x, y = S p hi x, t y, s + its P k t P l s dtds. S q Like in the axial case, we start with expanding hi x, y y, s into a Taylor series: i x, t l hi x, t y, s = h l y, s, l! which, after application of the Funk-Hecke theorem, leads to f e i l 4π p+q/ { x, y = l! x l Γ p Γ q P k ω t l t p 3/ P k,ptdt P l ξ h l y s s q 3/ P l,qsds + iωp k ω t l t p 3/ P k +,ptdt ξp l ξ h l y s s q 3/ P l +,qsds }. From the orthogonality it follows that all terms with l < k are zero. Moreover, it is then easily seen that f e x, y = ik x k k! 4π p+q/ Γ q Γ p P l ξ P k ω t k t p 3/ P k,ptdt h k y s s q 3/ P l,qsds. 3 Next, in a similar way as in the axial case, we find that t k t p 3/ P k,ptdt = k k! πγ p Γ, k + p 4

25 hence the boundary value 3 becomes x k f e x, y = ik k π p+q / Γ q Γ k + P p k ωp l ξ h k y s s q 3/ P l,qsds. 4 The Taylor series expansion: yields in an analogous way y, s l hi x, t y, s = h l i x, t, l! y 0 y l f e x, y = l l π p+q / Γ P p Γ l + q k ωp l ξ h l i x t t p 3/ P k,ptdt. 5 The it values of the odd Clifford algebra-valued biaxial monogenic: f o x, y = S p hi x, t y, s t + is P k t P l s dtds S q take the form x k f o x, y + = ik k π p+q / Γ P q Γ k + p k ωξp l ξ h k y s s q 3/ P l +,qsds 6 and y 0 y l f o x, y = l l π p+q / Γ ωp p Γ l + q k ωp l ξ h l i x t t p 3/ P k +,ptdt. 7 5

26 4. Illustration of monogenic plane wave integrals in the axial case From Theorem 4. it follows that fx 0, x = Γ k + m i k k e x 0 π m/ S m e i x,t it P k t dt 8 is the generalized axial CK-extension of e x 0. From the generalized axial CK-extension approach see subsection.., Remark. we already know that fx 0, x takes the form m fx 0, x = Γ + k m/+k J m/+k x x m/ +ωj m/+k x x P m/ k ωe x 0. We further calculate 8 using the Funk-Hecke theorem 9 Sm e i x,t it P k t dt = πm / Γ P m k ω e i x t t m 3/ P k,m tdt iωp k ω e i x t t m 3/ P k+,m tdt. By comparing now the scalar and vector part of 8 and 9 we obtain the integral equations J m/+k x = J m/+k x = i k m/ x πγ m ik+ πγ m e i x t t m 3/ P k,m tdt m/ x e i x t t m 3/ P k+,m tdt which are an application of the following integral expression which is derived in [7], p : J n+m/ z = i n z m/ πγ m e izt P n,m t t m 3/ dt. 0 6

27 4.3 Illustration of monogenic plane wave integrals in the biaxial case We will now illustrate the monogenic plane wave approach in the biaxial framework for hz = e z in order to obtain integral formulas for the Clifford-Bessel function of biaxial type see subsection 3., Remark 3.: q f Bes x, y = q/+l Γ + J l i y q/+l J q/+l i y y J k +p/ x + x J k +p/ x P k x P l y. Applying Funk-Hecke twice, we obtain for the odd Clifford algebra-valued biaxial monogenic f o x, y = S p = 4πp+q/ Γ p P l ξ + i e i x,t y,s t + is P k t P l s dt ds S q Γ q ω P k ω e i x t t p 3/ P k +,pt dt 4π p+q/ Γ p Γ q ξ P l ξ e y s s q 3/ P l,qs ds Next, in view of 0 we become P k ω e i x t t p 3/ P k,pt dt e y s s q 3/ P l +,qs ds. f o x, y = i k + l π p+q/ J k +p/ x J l +q/ i y x J k +p/ x J l +q/i y y P k x P l y. In a similar way, we find for the even Clifford algebra-valued biaxial monogenic: f e x, y = e i x,t y,s + its P k t P l s dt ds S p S q = i k l π p+q/ J k +p/ x J l +q/ i y + xy J k +p/ x J l +q/i y P k x P l y. 7

28 From the above calculations and subsection 3., we obtain the following result. Theorem 4.3 The odd and even Clifford algebra-valued part of the Clifford-Bessel function of biaxial type q f Bes x, y = q/+l Γ + J l i y q/+l J q/+l i y y J k +p/ x + x J k +p/ x P k x P l y. can be written as plane wave integrals: [f Bes x, y] o = l i k + Γ q + l +p/ l π p+q/ [f Bes x, y] e = l i k Γ q + l +p/ l π p+q/ One thus also has that S p S p S q e i x,t y,s t + is P k t P l s dt ds S q e i x,t y,s + its P k t P l s dt ds. f Bes x, y = l i k Γ q + l +p/ l π p+q/ S p S q e i x,t y,s it+s+its P k t P l s dt ds. The boundary values 4, 5, 6 and 7 can easily be calculated using see 0: e izt t m 3/ P n,m tdt = m m/ πγ i n J n+m/ z. z We find y 0 y 0 x k f e x, y = l y l f e x, y = l i k k +q/ π p+q/ Γ J k + p l +q/ i y P k ωp l y i k l +p/ π p+q/ Γ J l + q k +p/ x P k xp l ξ f o x, y = l + i k + k +q/ π p+q/ x k Γ k + p i k + l +p/ π p+q/ y l f o x, y = l Γ l + q J l +q/i y P k ωyp l y J k +p/ x xp k xp l ξ. 8

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