Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 2, 349–360

On Operators and Elementary Functions in Clifford Analysis

W. Sprössig

Dedicated to Prof. L. von Wolfersdorf on the occasion of his 65th birthday

Abstract. In this paper a survey on the construction of Clifford regular elementary functions by Fueter’s mapping is given. Furthermore, using a suitable decomposition of the Dirac operator an application of the \overline{∂}-problem is lined out.

Keywords: Clifford regular elementary functions, Fueter’s method, \overline{∂}-problem

AMS subject classification: 30 G 35

1. Indroduction

The effective application of methods of Clifford analysis to partial differential equations needs systems of elementary functions which are Clifford regular and still satisfy most of the properties which we know from the complex plane. The reason for writing this paper is to show presumed users of Clifford-valued elementary functions the considerable similarity to classical complex elementary functions and to reduce the shyness in applying higher-dimensional elementary functions. In this paper we will give a review on Fueter’s method and will construct some important Clifford regular elementary functions which knowledge seems to be useful in transform analysis.

We will mention here important authors which deliver contributions in this field. In his early papers [5 - 7] R. Fueter, a follower of D. Hilbert, formulated a method to transfer complex analytic functions to Clifford regular ones. Later M. Sce [18], A. Sudbury [22], M. Imaeda [9], F. Sommen [20, 21], P. Lounesto and P. Bergh [15], G. Jank and F. Sommen [11], H. Leutwiler [12 - 14], M. Marinov [16] and K. Nono [17] made attempts in generalizing classical elementary functions in a hypercomplex sense. Further, the reader can find contributions to this subject in [1, 2, 4].

Using the power series expansion of an exponential function with a paravector argument we deduce a ”full” class of so-called radially regular elementary functions including a paravector-valued logarithm. Most of the expected properties could be maintained. We continue with a suitable decomposition of the Dirac operator and study the operators which there occur. In this part we use results of J. de Graaf [3] and N. van Acker [23]. On the basis of papers by M. Sce [19] and T. Qian [18] we are able to describe at least in the case of quaternions the transform of radially regular function to quaternionic regular functions. Finally, we apply these results for the construction of kernel functions of the \overline{∂}-problem.

W. Sprössig: Univ. of Mining & Techn., Dept. Math. & Comp. Sci., D-09695 Freiberg ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

2. Preliminaries

Let ℝ^{0,n} be the anti-Euclidean space with the basis {e_{1}, ..., e_{n}} and the quadratic form Q(x) = − ∑_{i=1}^{n} x_{i}^{2}. We consider the 2^{n}-dimensional real Clifford algebra Cℓ_{0,n} which is generated by {e_{1}, ..., e_{n}} and contains copies of ℝ and ℝ^{0,n}. The multiplication rule is defined by e_{i}e_{j}+e_{j}e_{i} = −2δ_{ij}. In Cℓ_{0,n} a basis is given by e_{0}, e_{1}, …, e_{n}, e_{1}e_{2}, … e_{n−1}e_{n}, …, e_{1} ⋯ e_{n}. Elements of the type x = x_{1···k}e_{ℓ_{1}} ⋯ e_{ℓ_{k}} (1 ≤ ℓ_{1} < … < ℓ_{k} ≤ n) are called k-vectors.

For k-vektors the conjugation is defined by \overline{x} = (−1)^{\frac{k(k+1)}{2}} x. Elements x = x_{0} + \underline{x} with \underline{x} = ∑_{i=1}^{n} x_{i}e_{i} are called paravectors. Furthermore, x_{0} is called scalar part, x_{0} := Sc x, \underline{x} is called vector part, \underline{x} := Vec x, and x−x_{0} is called imaginary part, x−x_{0} := Im x. Obviously, x\overline{x} = ∑_{i=1}^{n} x_{i}^{2} for x = x_{0} + \underline{x}. Denote ω(x) := \underline{x}/|\underline{x}| ∈ S^{n}, where S^{n} is the unit sphere in ℝ^{n}.

3. Paravector-valued elementary functions

Let x be a paravector in Cℓ_{0,n}. The following statement can be easily shown.

Proposition 3.1. For an arbitrary ε > 0 it is always possible to find a sufficiently large number N such that for any r, s > N

|∑_{k=r}^{s} \frac{x^{k}}{k!} | ≤ ∑_{k=r}^{s} \frac{K^{k}|x|^{k}}{k!} < ε  (3.1)

holds where K is a constant which only depends on n and satisfies the inequality |xy| < K(n)|x| |y|.

Inequality (3.1) gives us the possibility to define elementary functions similarly to the case of one complex variable. First of all we have to introduce an exponential function:

Definition 3.2. For a paravector x ∈ Cℓ_{0,n} the exponential function e^{x} is defined by e^{x} = ∑_{k=0}^{∞} \frac{x^{k}}{k!} .

Similarly to the complex case one can prove the following properties.

Theorem 3.3. Let x be a paravector in Cℓ_{0,n}. Then we have:

(i) e^{x} = e^{x_{0}}(cos |\underline{x}| + ω(x) sin |\underline{x}|).

(ii) e^{x} = lim_{m→∞} (1 + \frac{x}{m})^{m}.

(iii) e^{x+y} = e^{x}e^{y} if xy = yx.

Proof. We will only prove property (ii). Because of (\frac{1}{k!} − \frac{( {m \atop k})}{m^{k}}) > 0 we obtain

|e^{x} − (1 + \frac{x}{m})^{m}| ≤ ∑_{k=0}^{∞} (\frac{1}{k!} − \frac{\binom{m}{k}}{m^{k}})(K |x|)^{k} = e^{|Kx|} − (1 + \frac{|Kx|}{m})^{m}.

For m → ∞ this difference tends to zero Corollary 3.4. Furthermore, we have:

(i) e^{x} ≠ 0. (ii) |e^{x}| = e^{x_{0}}. (iii) e^{−x}e^{x} = 1. (iv) e^{k\underline{x}} = (e\underline{x})^{k} (Moivre’s Formula)

(v) e^{ω(x)π} = −1. Proof. This is a straightforward consequence of Theorem 3.3 Let x be a paravector in Cℓ_{0,k}. Then hyperbolic and trigonometric functions are

defined by

sinh x = \frac{e^{x} − e^{−x}}{2} and cosh x = \frac{e^{x} + e^{−x}}{2} .

and, for |\underline{x}| ≠ 0,

sin x = \frac{e^{xω(x)} − e^{−xω(x)}}{2} ω(x)  and  cos x = \frac{e^{xω(x)} + e^{−xω(x)}}{2} .

Corollary 3.5. Immediately the representations

cosh x = cos hx_{0} cos |\underline{x}| + ω(x) sin hx_{0} sin |\underline{x}|
sinh x = sin hx_{0} cos |\underline{x}| + ω(x) cos hx_{0} sin |\underline{x}|
cos x = cos x_{0} cos h|\underline{x}| + ω(x) sin h|\underline{x}| sin x_{0}
sin x = sin x_{0} cos h|\underline{x}| + ω(x) sin h|\underline{x}| cos x_{0}

follow. The right-hand sides of sin x and cos x can be used by definition in the case of |\underline{x}| = 0.

Proof. For the proof we refer to our book [8: pp. 53 - 55] Definition 3.6. Let x be a paravector in Cℓ_{0,n} with x ≠ x_{0} < 0. Then log x is

defined by

log x = ln |x| + ω(x) arccos \frac{x_{0}}{|x|} (|\underline{x}| ≠ 0 or |\underline{x}| = 0, x_{0} > 0).

Theorem 3.7. Let x be a paravector in Cℓ_{0,n} with x ≠ x_{0} < 0. Then we have:

(i) e^{log x} = x and log e^{x} = x. (ii) log 1 = 0 and log e_{i} = \frac{π}{2} (i = 1, ..., n). (iii) 1 − \frac{1}{|x|} − arctan \frac{|\underline{x}|}{|x_{0}|} ≤ log |x| ≤ |x| − 1 + arctan \frac{|\underline{x}|}{|x_{0}|}. (iv) log(xy) = log x + log y if xy = yx. Proof. The proof follows immediately from Definition 3.6

352 W. Sprössig

Finally, it is also possible to introduce a general power function.

Definition 3.8. Let α be a real number. The general power function x^{α} is defined by x^{α} = e^{α log x}.

The following example will confirm this definition.

Example 3.9. Let x = \underline{x} and α = \frac{1}{n}. We obtain after a straightforward calculation

\underline{x}^{\frac{1}{n}} = ^{n} \sqrt{|\underline{x}|} [cos (\frac{π}{2n} + \frac{2ki}{n}) + ω(x) sin (\frac{π}{2n} + \frac{2kπ}{n} )]

for k = 0, 1, ..., n − 1.

4. On the Cauchy-Fueter operator

In order to prepare differentiability properties of the elementary functions introduced above we need a suitable decomposition of the so-called Cauchy-Fueter operator

∂ = ∂_{0} + D where ∂_{0} = \frac{∂}{∂x_{0}} and D = ∑_{i=1}^{n} e_{i}∂_{i}, ∂_{i} = \frac{∂}{∂x_{i}} .

With the denotations

L = ∑_{i=1}^{n} e_{i}L_{i}(x) with L_{i}(x) = |\underline{x}|∂_{i} − x_{i}ℓ_{ω}  and  ℓ_{ω} = ∑_{i=1}^{n} ω_{i}∂_{i}

we will obtain the decomposition

∂ = \frac{1}{2}(∂_{0} − \frac{1}{|\underline{x}|}L + ωℓ_{ω}).

For a better understanding it is useful to deduce properties of the decomposition operators L and ℓ_{ω}.

Proposition 4.1. Let f ∈ C^{1}(ℝ^{n}) be a paravector-valued function. Then:

(i) l_{ω}\underline{x} = ω. (ii) ℓ_{ω}ω = 0. (iii) ℓ_{ω}f = \frac{d}{d|\underline{x}|}f. (iv) |\underline{x}|∂_{j}ω_{k} = δ_{jk} − ω_{k}ω_{j} = L_{j}ω_{k}.

(v) ∑_{i=1}^{n} ω_{i}L_{i} = Sc ωL = 0.

Proof. These relations are simple consequences from the definition of the operators L_{i} and ℓ_{ω} Proposition 4.2.

(i) Let f ∈ C^{1}(ℝ) and f = f(|\underline{x}|) a paravector-valued function. Then Lf = 0. (ii) Let φ ∈ C^{1}(ℝ) and φ = φ(ω) a scalar-valued function. Then

Lφ = |\underline{x}| [∑_{k=1}^{n} ∂_{ω_{k}}φe_{k} − ∑_{k=1}^{n} ω_{k}∂_{ω_{k}}φ] = |\underline{x}|[grad_{ω}φ − ω(grad_{ω}φ · ω)].

(iii) ℓ_{ω}φ(ω) = 0 is valid. Proof. Statement (i): For j = 1, ..., n we obtain by definition L_{j}(|\underline{x}|) = |\underline{x}|∂_{j}|\underline{x}| −

x_{j}ℓ_{ω}|\underline{x}| = 0. Furthermore, we have

L_{j}f(|\underline{x}|) = |\underline{x}|∂_{j}f − x_{j} ∑_{i=1}^{n} ω_{i}∂_{i}f = ℓ_{ω}f(|\underline{x}|∂_{j}|\underline{x}| − x_{j}ℓ_{ω}|\underline{x}|) = ℓ_{ω}fL_{j}|\underline{x}| = 0.

Statement (ii): Let φ = φ(ω). Then

Lφ = |\underline{x}|grad_{ω}φ − ω[(grad_{ω}φ) · ω]

where grad_{ω}φ = ∑_{i=1}^{n} e_{i}∂_{ω_{i}} = φ. Indeed, we have

(Lφ)(ω) = ∑_{i=1}^{n} e_{i}L_{i}φ(ω) = |\underline{x}| ∑_{i=1}^{n}[e_{i}∂_{i} − ω_{i}ℓ_{ω}]φ(ω).

Setting now φ^{∗}(x) := φ(ω),

(∂_{j}φ^{∗})(x) = ∑_{j=1}^{n} ∂_{ω_{k}}φ∂_{j}ω_{k} = ∑_{k=1}^{n} ∂_{ω_{k}}φ(δ_{jk} − ω_{j}ω_{k}).

Hence,

∑_{j=1}^{n} e_{j}∂_{j}φ^{∗}(x) = ∑_{j=1}^{n} ∑_{k=1}^{n} e_{j}∂_{ω_{k}}φ∂_{j}ω_{k}

= ∑_{k=1}^{n} ∑_{j=1}^{n} e_{j}∂_{ω_{k}}φδ_{jk} − ∑_{k=1}^{n} ∑_{j=1}^{n} e_{j}ω_{j}∂_{ω_{k}}φω_{k}

= grad_{ω}φ − ω ∑_{k=1}^{n} ∂_{ω_{k}}φω_{k}

= grad_{ω}φ − ω(grad_{ω}φ · ω).

Statement(iii): We have

ℓ_{ω}φ(ω) = ∑_{k=1}^{n} ∂_{k}φℓ_{ω}ω_{k} = grad_{ω}φ · ℓ_{ω}ω = 0.

Thus the proposition is proved Remark. Following [3] we get for L the vector representation

L = ∑_{j=1}^{n} e_{j}(v^{j} · D)

where v^{j} = |\underline{x}|e_{j} − ω_{j}\underline{x}, which can be easily seen by the help of Proposition 3.1 Proposition 4.4. ωL is the spherical Dirac operator. Proof. Because of Proposition 4.1/(iv) we find

ωL = ∑_{j≠k} e_{j}e_{k}ω_{j}L_{k}

= ∑_{j<k} e_{j}e_{k}(ω_{j}L_{k} − ω_{k}L_{j})

= ∑_{j<k} e_{j}e_{k}[(ω_{j}|\underline{x}|∂_{k} − ω_{j}x_{k}ℓ_{ω}) − ω_{k}|\underline{x}|∂_{j} − x_{j}ω_{k}ℓ_{ω}]

= ∑_{j<k} e_{j}e_{k}(x_{j}∂_{k} − x_{k}∂_{j})

and the statement is proved

Theorem 4.5. We have the equality

(ωL + Lω)u = (1 − n)u

for the anti-commutator. Special cases:

(i) If u ≡ u_{0}(|\underline{x}|), where u_{0} a real-valued function, then Lωu_{0} = (1 − n)u_{0}. (ii) If u = ω, then (ωL)ω = (n − 1)ω. Proof. We have

Lω = ∑_{j,k=1}^{n} e_{j}e_{k}L_{j}ω_{k} + ∑_{j,k=1}^{n} e_{j}e_{k}ω_{k}L_{j}.

Using Proposition 4.1/(iii)-(iv) we get

Lω = ∑_{j,k=1}^{n}(δ_{jk} − ω_{k}ω_{j})e_{j}e_{k} − ωL = −n + 1 − ωL

and the statement is proved Proposition 4.6. The Cauchy-Fueter operator permits the decomposition

∂ = \frac{1}{2}(∂_{0} + ωℓ_{ω}) − \frac{1}{2} \frac{1}{|\underline{x}|}L.

Proof. It is easy to see that

ωL = ω ∑_{j=1}^{n} e_{j} [|\underline{x}|∂_{j} − x_{j}ℓ_{ω}] = \underline{x}D + |\underline{x}|ℓ_{ω}.

From this we obtain the assumption by a straightforward calculation Corollary 4.7. Let ξ = \underline{ξ}(|\underline{x}|). Then:

(i) L(ξω) = (n − 3)ξ + 2(ξ, ω)ω. (ii) D(ξω) = \frac{1}{|\underline{x}|}L(ξω) + \frac{d}{d|\underline{x}|}ξ.

Proof. Corollary 4.6, the generalized Leibniz rule [8: p. 40] and Proposition 4.4 deliver

D(ξω) = (Dξ)ω − ξDω − 2 ∑_{j=1}^{n} ξ_{j}∂_{j}ω = ωξ′ω − 2 ∑_{j=1}^{n} ∑_{k=1}^{n} e_{k}ξ(∂_{j}ω_{k}) + \frac{n − 1}{|\underline{x}|}ξ.

Furthermore, it follows

D(ξω) = ω(\frac{d}{d|\underline{x}|}ξ)ω + \frac{(n − 1)}{|\underline{x}|}ξ − \frac{2}{|\underline{x}|} ∑_{j=1}^{n} ∑_{k=1}^{n} e_{k}ξ_{j}[δ_{jk} − ω_{j}ω_{k}]

= ω(\frac{d}{d|\underline{x}|}ξ)ω + \frac{n − 1}{|\underline{x}|}ξ − \frac{2}{|\underline{x}|}ξ + \frac{2}{|\underline{x}|}(ξ, ω)ω

= ω(\frac{d}{d|\underline{x}|}ξ)ω + \frac{1}{|\underline{x}|}(n − 3)ξ + \frac{2(ξ, ω)}{|\underline{x}|}ω.

Obviously, we get from ℓ_{ω}(ξω) = (\frac{d}{d|\underline{x}|}ξ)ω and Corollary 4.6

L(ξω) = −|\underline{x}|(D(ξω) − ω(\frac{d}{d|\underline{x}|}ξ)ω) = (n − 3)ξ + 2(ξ, ω)ω

and the proof is finished

5. Fueter’s mapping

The elementary functions introduced above are not Clifford regular. Above all this is caused by the occurence of the operator L and his action

L(u_{0}ω) = (1 − n)u_{0}

(cf. Theorem 4.5). Such a cross-mapping which acts from the vector part to the real part is disturbing the structure of this simple type of elementary functions. In order to maintain such differentiability properties it seems to be useful to introduce the ”reduced” operator ∂_{ra} = \frac{1}{2}(∂_{0} −ωℓ_{ω}). Analogously to the complex case we abbreviate ∂_{ra}u =: u′.

Definition 5.1. Let

f = f_{0} + ω(x)f_{1}, f_{i} : ℝ^{n} ⊕ ℝ^{1} → ℝ^{1}, f_{i} = f_{i}(x_{0}, |\underline{x}|) (i = 1, 2), h = h_{0}ω(x)|\underline{h}|.

Such a paravector-valued function f is called radially differentiable or radially Clifford regular if

lim_{h→0} \frac{[f(x + h) − f(x)]\overline{h}}{|h|^{2}} = A_{f}(x)

exists.

Corollary 5.2. For radially differentiable functions A_{f}(x) = f′(x) holds.

Corollary 5.3. The paravector-valued function f = f_{0} + ωf_{1} is radially differentiable if and only if \overline{∂}_{ra}f = 0

Corollary 5.4. All above defined elementary functions are radially Clifford regular. Hence it follows:

(i) (e^{x})′ = e^{x}.

(ii) (sin x)′ = cos x and (cos x)′ = − sin x. (iii) (sinh x)′ = cosh x and (cosh x)′ = sinh x. (iv) (log x)′ = \frac{1}{x}.

(v) x^{α} = αx^{α−1} (α ∈ ℝ).

Proof. The proof follows straightforward by using the elementary relations ℓ_{ω}ω = 0, ℓ_{ω}|x| = \frac{|\underline{x}|}{|x|} and ε = ℓ_{ω}\underline{x} = ω

Now we will demonstrate how to transform these radially differentiable elementary functions to Clifford regular functions. The key-idea goes back to R. Fueter and was later generalized by M. Sce [19], F. Sommen [20, 21] and T. Qian [18]. Let u = u(x_{0}, |\underline{x}|) and v = v(x_{0}, |\underline{x}|) real-valued functions and h = u + ωv. We will denote by τ_{n} the mapping

q = τ_{n}(h) = κ_{n}Δ^{\frac{(n−1)}{2}} h,

where Δ denotes the Laplacian and κ_{n} a normalization factor. For even n the operator Δ^{\frac{n−1}{2}} has to be considered as Fourier multiplier operator induced by the symbol (2πi|ξ|)^{n−1}.

Theorem 5.5 (Qian [18]). Let h = u + ωv be radially differentiable in ℝ^{n+1} and n = 2k + 1. Then for any k ∈ ℕ we find

τ_{n}(h) = \frac{1}{2^{k}k} Δ^{k}h = (k − 1)! [(\frac{1}{|\underline{x}|}ℓ_{ω})^{k} u + ω (ℓ_{ω} \frac{1}{|\underline{x}|})^{k} v] .  (5.2)

Corollary 5.6. Let k = 1. A radially quaternionic regular function h = u + ωv fulfils the partial differential equation

|\underline{x}|^{2}Δh − 2|\underline{x}|ℓ_{ω}h + 2Vec h = 0.

Remark. A componentwise consideration of equation (5.2) leads for the scalar part to the Laplace-Beltrami equation in the hyperbolic metric and for the vector components to the Laplace-Beltrami equation to the eigenvalue -2.

Now we will give a lot of examples which will emphasize how effectively Fueter’s mapping τ_{n} is working. For abbreviation we write

b_{k}(x) = \frac{Γ (\frac{1}{2} − k)}{Γ (2k + 1))} (\frac{2}{|\underline{x}|})^{\frac{k−1}{2}} \frac{1}{k 2^{k}} .

Example 5.7 (Exponential function). Let n = 2k + 1 and h(x) = e^{x}. We obtain by applying Fueter’ s mapping

EXP_{k}x := b_{k}(x) [J_{k− \frac{1}{2}} (|\underline{x}|) + ωJ_{k+ \frac{1}{2}} (|\underline{x}|)] e^{x_{0}}

where

J_{λ}(|\underline{x}|) = \frac{(\frac{|\underline{x}|}{2})λ}{Γ (\frac{1}{2}) Γ (\frac{1}{2} + λ)} ∫_{−1}^{1} e^{i|\underline{x}|t}(1 − t^{2})^{λ− \frac{1}{2}} dt (λ ∈ ℝ)

are Bessel functions of first kind. Indeed, we obtain from [10: p. 740] the equalities

J_{k+ \frac{1}{2}} (|\underline{x}|) = (−1)^{k+1}\sqrt{\frac{2}{π}} |\underline{x}|^{k+ \frac{1}{2}} (\frac{1}{|\underline{x}|}ℓ_{ω})^{k+1} cos |\underline{x}| = (−1)^{k}\sqrt{\frac{2}{π}} (ℓ_{ω} \frac{1}{|\underline{x}|})^{k} sin |\underline{x}|.

A straightforward calculation leads to the above defind exponential function EXP_{k}x. In the special case for k = 1 we have

b_{1}(x) = −\frac{Γ (−\frac{1}{2})}{2!}\sqrt{\frac{2}{|\underline{x}|}} = \sqrt{\frac{2π}{|\underline{x}|}}

and

J_{\frac{1}{2}} (|\underline{x}|) = \sqrt{\frac{2}{π|\underline{x}|}} sin |\underline{x}|

J_{\frac{3}{2}} (|\underline{x}|) = \sqrt{\frac{2}{π|\underline{x}|}} (\frac{sin |\underline{x}|}{|\underline{x}|} − cos |\underline{x}|) .

Hence,

EXP_{1}x = e^{x_{0}} [\frac{sin |\underline{x}|}{|\underline{x}|} + ω (\frac{sin |\underline{x}| − |\underline{x}| cos |\underline{x}|}{|\underline{x}|^{2}})] = e^{x_{0}} (sinc |\underline{x}| − ωsinc′|\underline{x}|)

where sinc |\underline{x}| = \frac{sin |\underline{x}|}{|\underline{x}|} and sinc′|\underline{x}| := \frac{d}{d|\underline{x}|}(sinc |\underline{x}|).

In [21] F. Sommen obtained by using of a similarity principle for Vekua systems a class of hypercomplex exponential functions which can be seen as generalization of Fueter type mappings.

Example 5.8 (Hyperbolic functions). Let n = 2k+1. Further, let h(x) the radially hyperbolic sine function. We get

SINH x := τ_{n}(sinh x) = b_{k}(x)[ sinh x_{0}J_{k− \frac{1}{2}} (|\underline{x}|) + ω cosh x_{0}J_{k+ \frac{1}{2}} (|\underline{x}|)],

and in the special case k = 1 we have

SINH x := sinh x_{0}sinc |\underline{x}| − ω cos x_{0}sinc′|\underline{x}|.

358 W. Sprössig

Let h(x) the radially cosine function. We obtain

COSH x := τ_{n}(cosh x) = b_{k}(x)[ cosh x_{0}J_{k− \frac{1}{2}} (|\underline{x}|) + ω sinh x_{0}J_{k+ \frac{1}{2}} (|\underline{x}|)], and in the special case k = 1 we have

COSH x := cosh x_{0} sinc |\underline{x}| − ω sinh x_{0} sinc′|\underline{x}|. Example 5.9 (Trigonometric functions). Let n = 2k + 1. It is easy to see that

y = ω(x)x = −|\underline{x}| + \frac{\underline{x}x_{0}}{|\underline{x}|} .

Hence, |\underline{y}| = |x_{0}| and ω(y) = ω(x)\frac{x_{0}}{|x_{0}|}. It follows that

EXP_{k}y = b_{k}(x_{0}) [J_{k− \frac{1}{2}} (|\underline{x}|_{0}) + ω(x)\frac{x_{0}}{|x_{0}|}J_{k+ \frac{1}{2}} (|x_{0}|)] e^{−|x_{0}|}.

We can now define a Clifford regular SIN-function by

SIN x :=\frac{1}{2} (EXP_{k}y + EXP_{k}(−y)) ω(y)
=b_{k}(x) [cosh |\underline{x}|J_{k+ \frac{1}{2}} (|x_{0}|) + ω(x)sinh |\underline{x}|J_{k− \frac{1}{2}} (|x_{0}|)\frac{x_{0}}{|x_{0}|}] .

Analogously we can also define

COS x :=\frac{1}{2} [EXP_{k}y + EXP_{k}(−y)]
=b_{k}(x_{0}) [cosh |\underline{x}|J_{k− \frac{1}{2}} (|x_{0}|) − ω(x) sinh |\underline{x}|J_{k+ \frac{1}{2}} (|x_{0}|)\frac{x_{0}}{|x_{0}|}] .

In the special case k = 1 we have

SIN x := − cosh |\underline{x}| sinc′|x_{0}| + ω(x)sinh|\underline{x}| sinc x_{0} \frac{x_{0}}{|x_{0}|}
COS x := cosh |\underline{x}| sinc |x_{0}| + ω(x) sinh |\underline{x}| sinc′|x_{0}|\frac{x_{0}}{|x_{0}|}.

Now we will give an application of this conception of elementary functions.

Theorem 5.10. Let n = 3. The exponential function EXP_{3}x fulfils the property \overline{∂}EXP_{3}(λx) = λ EXP_{3}x (λ ∈ ℂ).

Proof. Using the decomposition D = |\underline{x}|^{−1}L + ωℓ_{ω} we obtain

(|\underline{x}|^{−1}L + ωℓ_{ω}) [\frac{sin λ|\underline{x}|}{λ|\underline{x}|} − ω(x)ℓ_{ω} (\frac{sin λ|\underline{x}|}{λ|\underline{x}|})]

= \frac{1}{λ} [|\underline{x}|^{−1}L (\frac{sin λ|\underline{x}|}{|\underline{x}|}) + ωℓ_{ω} (\frac{sin λ|\underline{x}|}{|\underline{x}|})]

− [ω(x)ℓ_{ω}ω(x)ℓ_{ω} (\frac{sin λ|\underline{x}|}{|\underline{x}|}) − |\underline{x}|^{−1}L (ω(x)ℓ_{ω} \frac{sin λ|\underline{x}|}{|\underline{x}|})] .

On Operators and Elementary Functions 359

From Proposition 4.2/(i) immediately

L (\frac{sin λ|\underline{x}|}{|\underline{x}|}) = 0

and

\frac{1}{λ|\underline{x}|}L (ωℓ_{ω} \frac{sin λ|\underline{x}|}{|\underline{x}|}) = −\frac{2}{λ|\underline{x}|}ℓ_{ω} (\frac{sin λ|\underline{x}|}{|\underline{x}|}) = −\frac{2 sin λ|\underline{x}|}{λ|\underline{x}|^{3}} + \frac{cos λ|\underline{x}|}{|\underline{x}|^{2}}

follows. It remains to consider

\frac{1}{λ}ωℓ_{ω} (−ωℓ_{ω} \frac{sin λ|\underline{x}|}{|\underline{x}|}) = \frac{1}{λ}ℓ_{ω}^{2} (\frac{sin λ|\underline{x}|}{|\underline{x}|}) .

A straightforward computation delivers

\frac{1}{λ}ℓ_{ω}^{2} (\frac{sin λ|\underline{x}|}{|\underline{x}|}) = \frac{λ sin λ|\underline{x}|}{|\underline{x}|} − 2\frac{cos λ|\underline{x}|}{|\underline{x}|^{2}} + \frac{2 sin λ|\underline{x}|}{λ|\underline{x}|^{3}}.

Finally, we have DEXP_{3}(λx) = −λEXP_{3}(λx) and so \overline{∂}EXP_{3}(λx) = λEXP_{3}(λx)

From the definition of the exponential function it follows now

Corollary 5.11. We have |EXP_{3}x|^{2} > 0 and lim_{\underline{x}→0} EXP_{3}x = e^{x_{0}}.

We consider now the so-called \overline{∂}-problem and obtain the following result.

Corollary 5.12. Let L_{n} = a_{n}\overline{∂}^{n} + ... + a_{1}\overline{∂} + a_{0} (a_{k} ∈ ℝ). Further, let λ_{k} be the roots of the algebraic equation a_{n}λ^{n} + ... + a_{1}λ + a_{0} = 0. Then

u_{k} = EXP_{3}λ_{k}x (k = 1, ..., n)

belong to ker L_{n}, e.g. we have constructed a set of solutions of the n-th order linear partial differential equation L_{n}u = 0 relatively to the operator \overline{∂}.

Proof. We have to make the ansatz u = EXP_{3}λx. The result follows by using Theorem 5.10

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Received 02.07.1998; in revised form 24.01.1999