Riemann-Hilbert problems for monogenic functions in axially symmetric domains
- Fuli He^{1},
- Min Ku^{2}Email author,
- Uwe Kähler^{2},
- Frank Sommen^{3} and
- Swanhild Bernstein^{4}
Received: 9 October 2015
Accepted: 12 January 2016
Published: 25 January 2016
Abstract
We consider Riemann-Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric monogenic functions defined in axial symmetric domains. This is done by constructing a method to reduce the RHBVPs for axially symmetric monogenic functions defined in four-dimensional axial symmetric domains into the RHBVPs for analytic functions defined over the complex plane. Then we derive solutions to the corresponding Schwarz problem. Finally, we generalize the results obtained to null-solutions of \((\mathcal{D}-\alpha)\phi=0\), \(\alpha\in\mathbb{R}\), where \(\mathbb{R}\) denotes the field of real numbers.
Keywords
quaternion analysis generalized Cauchy-Riemann operator axial symmetry Riemann-Hilbert boundary value problems variable coefficientsMSC
30E25 35Q15 31A25 31B20 31B10 35J56 35J581 Introduction
The classic theory of Riemann-Hilbert boundary value problems (for short RHBVPs) for analytic functions is closely connected with the theory of singular integral equations and has a wide range of applications in other fields, such as the theory of elasticity, quantum mechanics, statistical physics, the theory of orthogonal polynomials, and asymptotic analysis (cf. [1–7]). Over the years this type of boundary value problems has been systematically investigated by many authors, e.g., [1–4, 8–10]. Therefore, it is natural to attempt to generalize this type of boundary value problem to higher dimensions. This will be not only a purely theoretical question, since such problems are closely linked to the physical applications like transport problems or problems in hydro-dynamic mechanics.
To our knowledge, there are two principal ways to generalize complex analysis to higher dimensions, the theory of several complex variables and Clifford analysis. The latter, in particular quaternionic analysis, is the theory of so-called monogenic functions and a refinement of real harmonic analysis in the sense that the principal operator in Clifford analysis, the Dirac or generalized Cauchy-Riemann operator, factorizes the higher Laplace operator. As principal references for this theory we mention [11–14]. Furthermore, in the context of Clifford analysis RHBVPs, particularly Riemann boundary value problems, with constant coefficients were widely discussed; see, e.g., [15–22]. Here, a direct application of the properties of the Cauchy integral operator and existing power series expansions allows one to represent the solution in terms of integral operators and Taylor series expansions. This kind of Riemann boundary value problems is also closely connected with other types of boundary value problems for different partial differential equations in higher dimensions; see, e.g., [23–26]. However, there are almost no results with respect to RHBVPs with variable coefficients even after more than 30 years of research in this direction. The obstacles lie in the following facts: while monogenic functions retain many of the properties of analytic function, there are two drawbacks. First, the product of monogenic functions is in general not monogenic and second the composition of monogenic is in general not monogenic. While the latter property is not so important for our problem at hand the former is a major problem. Moreover, even in the simplest case of quaternionic analysis, we have several major problems, such as the lack of a function which has the same properties as the logarithmic function of one complex variable. These make it very difficult to solve RHBVPs with variable coefficients for monogenic functions in higher dimensions by directly employing classic methods from complex analysis. But, as a particular interesting case, it was first introduced by Fueter in the case of quaternionic analysis in [11] that the product of monogenic functions of axial type is still a monogenic function of axial type, and later on this class was studied in detail by Sommen et al. (cf. [27, 28]). This makes this class more similar to the classic case of analytic functions in the plane and, therefore, if one wants to generalize any property or fact from complex analysis to quaternionic or Clifford analysis, the standard approach is to see if it can be done for axially monogenic functions. So, instead of trying to solve this problem directly, we are going to show that RHBVPs with variable coefficients for monogenic functions with axial symmetry can be solved in the context of quaternionic analysis. This represents an initial and very crucial step to solve RHBVPs for monogenic functions in the future.
Our idea is to transform a RHBVP with variable coefficients for monogenic functions with axial symmetry in \(\mathbb{R}^{4}\) into a RHBVP for analytic functions over the complex plane, use the classic results from complex analysis, and transfer it back. In this way we can obtain the solution in an explicit form. As a special case we present the solution to the Schwarz problem for monogenic functions with axial symmetry. Furthermore, we adapt the approach above and generalize to the case of null-solutions to \((\mathcal{D}-\alpha)\phi=0\), \(\alpha \in\mathbb{R}\), where \(\mathbb{R}\) denotes the field of real numbers (see Section 3.3). Here, the main contribution that we emphasize lies in constructing a way to solve a RHBVP with variable coefficients in four-dimensional spaces. Moreover, by using the possibility to decompose a monogenic functions into a sum of monogenic functions with axial symmetry, we plan to extend this idea to the case of RHBVPs with variable coefficients for monogenic functions.
The paper is organized as follows. In Section 2, we will recall the necessary facts about quaternion analysis. In Section 3, we will focus on RHBVPs with variable coefficients for monogenic functions with axial symmetry. We will derive the solution to the RHBVP with boundary data belonging to a Hölder space in terms of an integral representation. Moreover, we give the solutions to the Schwarz problem for monogenic functions and null-solutions to \((\mathcal {D}-\alpha)\phi=0\), \(\alpha\in\mathbb{R}\) with axial symmetry, where \(\mathbb{R}\) denotes the field of real numbers.
2 Preliminaries
Good introductions to Clifford algebras, Quaternions, and Clifford analysis can be found in [12–14, 26, 27].
An arbitrary \(x\in\mathbb{H}\) can be written as \(x=x_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3} \triangleq x_{0}+\underline{x}\), where \(\operatorname{Sc}(x)\triangleq x_{0}\) and \(\operatorname{Vec}(x)\triangleq \underline{x}\) are the scalar and vector part of \(x\in\mathbb{H}\), respectively. Elements \(x\in\mathbb{R}^{4}\) can be identified with quaternions \(x\in\mathbb{H}\). The conjugation is defined by \(\bar{x}=\sum^{3}_{j=0}x_{j}\bar{e}_{j}\) with \(\bar{e}_{0}=e_{0}\) and \(\bar{e}_{j}=-e_{j}\), \(j=1,2,3\), and hence, \(\overline{x y}=\bar{y}\bar{x}\). The algebra of quaternions possesses an inner product \((y,x)= \operatorname{Sc}(y\bar{x}) =(y\bar{x})_{0}\) for all \(x,y\in\mathbb{H}\).
In this paper we will consider the generalized Cauchy-Riemann operator \(\mathcal{D}=\sum^{3}_{j=0}e_{j}\partial_{x_{j}}\) in \(\mathbb{R}^{4}\). The generalized Cauchy-Riemann operator factorizes the Laplacian in the following sense: \(\overline{\mathcal{D}}\mathcal{D}=\sum^{3}_{j=0}\partial^{2}_{x_{j}} =\Delta\), where Δ denotes the Laplacian in \(\mathbb{R}^{4}\).
We also need to define axially symmetric open sets.
Definition 2.1
(Axially symmetric open set)
Let Ω be a non-empty open subset of \(\mathbb {R}^{4}\). We say that Ω is axially symmetric if for any \(x\in \Omega\), the subset \([x]\) is contained in \(\Omega\subset\mathbb{R}^{4}\).
Remark 1
The unit ball of \(\mathbb{R}^{4}\) and the upper half space \(\mathbb{R}^{4}_{+}=\{x\in\mathbb{R}^{4}|x_{0}>0\}\) are examples of axially symmetric domains.
Let Ω be an axially symmetric domain of \(\mathbb{R}^{4}\) with smooth boundary ∂Ω. An \(\mathbb{H}\)-valued function \(\phi= \sum_{j=0}^{3} \phi _{j}e_{j} \) is continuous, Hölder continuous, p-integrable, continuously differentiable and so on if all components \(\phi_{j}\) have that property. The corresponding function spaces, considered as either right-Banach or right-Hilbert modules, are denoted by \(C(\Omega,\mathbb{H})\), \(H^{\mu}(\Omega,\mathbb{H})\) (\(0<\mu\leqslant 1\)), \(L_{p}(\Omega,\mathbb{H})\) (\(1< p<+\infty\)), \(C^{1}(\Omega,\mathbb{H})\), respectively.
Throughout this paper any functions defined on \(\Omega\cup \partial\Omega\subset\mathbb{R}^{4}\) with values in \(\mathbb{H}\) are supposed to be of axial type unless otherwise stated.
Definition 2.2
A function \(\phi\in C^{1} (\Omega,\mathbb{H})\) is called (left-) monogenic if and only if \(\mathcal{D}\phi=0\). A monogenic function of axial type is called axially monogenic. The set of all axially monogenic functions defined in Ω forms a right-module, denoted by \(M(\Omega ,\mathbb{H})\).
Definition 2.3
For a function of axial type \(\phi:\Omega\rightarrow\mathbb{H}\), we define the real part as \(\operatorname{Re} \phi=A\).
Remark 2
In [27, 28] it is shown that the equation \(\mathcal{D}\phi=0\) for functions of axial type is equivalent to a special kind of Vekua system [13].
In dimension two, i.e., the case of \(\mathbb{R}_{0,1}= \{x=x_{0}+x_{1}e_{1}: e^{2}_{1}=-1,x_{j}\in\mathbb{R} \ (j=0,1)\}\cong\mathbb{C}\), the field of complex numbers, we have \(\operatorname{Re} {\phi}=\operatorname{Sc}(\phi)\), which exactly corresponds to the usual understanding in complex analysis. In the following \(D\subset\mathbb{C}_{+}\) is the projection of the axially symmetric sub-domain \(\Omega\subset\mathbb{R}^{4}\) into the \((x_{0},r)\)-plane, where \(\mathbb{C}_{+}\) is the upper half of the \((x_{0},r)\)-plane.
3 Riemann-Hilbert boundary value problems
In this section we discuss RHBVPs with variable coefficients for monogenic functions. In this case we will restrict ourselves to functions which have axial symmetry and are defined over axial domains of \(\mathbb{R}^{4}\). We first solve the RHBVP in \(\mathbb{R}^{4}\) by transforming it into a RHBVP for complex analytic functions. In the case of boundary values belonging to a Hölder space we represent its solution in terms of an explicit integral representation formula. As a corollary we obtain the solution to the corresponding Schwarz problem in quaternion analysis.
3.1 A lemma
Let us start with the following lemma.
Lemma 3.1
In fact the surjective map φ is coming from Fueter’s theorem [14, 27, 28].
Proof
3.2 RHBVPs for monogenic functions
The idea behind this lemma allows us to link RHBVPs for axially monogenic functions with RHBVPs for analytic functions over the complex plane. Let us start the discussion with the following problem.
Problem I
Let us first transform the above problem to a RHBVP for analytic functions. The result is given in the following theorem.
Theorem 3.1
Proof
Now, let \(h=\partial_{r}f\) then we have \(\partial_{\bar {z}}h(z)=0\), \(z\in D\) since f is analytic in D.
The above theorem allows us to study the solvability of our original RHBVP by studying the equivalent RHBVP over the complex plane.
Theorem 3.2
- (i)If \(\lambda\in H^{\mu}(\partial\Omega,\mathbb{R})\), and \(\lambda\neq0\) for arbitrary \(z\in\partial D\) then the Riemann-Hilbert boundary value problem (⋆) is solvable and its solution is given bywhere \(\operatorname{Re}(f)\) and \(\operatorname{Im}(f)\) denote the real and imaginary part of the complex-valued function f, respectively. f itself is given by$$ \phi(x)=\Delta\bigl(\operatorname{Re}(f) \bigl(x_{0},\vert \underline{x}\vert \bigr)+\underline{\omega}\operatorname{Im}(f) \bigl(x_{0},\vert \underline{x}\vert \bigr)\bigr),\quad x\in\Omega, $$with \(\tilde{g}=\lambda^{-1}\frac{r}{2}g\), \(t_{n}\in\mathbb{C}\).$$ f(z)=\frac{1}{2\pi} \int^{z}_{a} \int_{\partial D}\frac{\tilde {g}(\zeta)}{\zeta-\xi}\, d\zeta\, d\xi+\sum ^{+\infty }_{n=0}t_{n}(z-a)^{n},\quad z\in D, $$
- (ii)Suppose \(\lambda=\Pi\hat{\lambda}\) with \(\Pi(x)=\Pi ^{m}_{i=1}(x-\hat{\alpha}_{i})^{\nu_{i}}\), \(\hat{\alpha}_{i}\in\partial\Omega\) and \(\nu_{i}\in\mathbb {N}\). Furthermore, if \(\hat{\lambda}\in H^{\mu}(\partial \Omega,\mathbb{R})\) and \(\hat{\lambda}\neq0\) for arbitrary \(x\in\partial\Omega\) then the Riemann-Hilbert boundary value problem (⋆) is solvable, and its solution is given again bywhere f is given by$$ \phi(x)=\Delta\bigl(\operatorname{Re}(f) \bigl(x_{0},\vert \underline{x}\vert \bigr)+\underline{\omega}\operatorname{Im}(f) \bigl(x_{0},\vert \underline{x}\vert \bigr)\bigr),\quad x\in\Omega, $$with \(\hat{g}=\hat{\lambda}^{-1}\frac{r}{2}g\), \(l_{n}\in \mathbb{C}\).$$ f(z)=\frac{1}{2\pi} \int^{z}_{a} \int_{\partial D}\frac{1}{\Pi(\xi )}\frac{\hat{g}(\zeta)}{\zeta-\xi}\, d\zeta\, d\xi+ \sum^{+\infty}_{n=0} \int^{z}_{a}\frac{l_{n}(\xi-a)^{n}}{\Pi(\xi)}\, d\xi, \quad z\in D, $$
Proof
As a special case of Problem I, we can consider the following Schwarz problem.
Schwarz problem
From Theorem 3.2 we can deduce the following theorem.
Theorem 3.3
Remark 3
Theorem 3.2 actually gives us a method to solve the RHBVPs with variable coefficients for axially monogenic functions in \(\mathbb{R}^{4}\) by transferring it to the study of the corresponding RHBVP for analytic functions over the complex plane.
When the space dimension is 2, then trivially RHBVP (⋆) reduces to the RHBVP for analytic functions on the complex plane [1, 3, 5]. Moreover, when the coefficient λ is a constant equal to 1 for all \(x\in\partial\Omega\) our RHBVP (♯) is just the Schwarz problem for analytic functions over the complex plane [1, 2, 8].
3.3 RHBVPs for null-solutions to \((\mathcal {D}-\alpha)\phi=0\), \(\alpha\in\mathbb{R}\)
The above approach can be adapted and generalized to other cases. In the following we will discuss RHBVPs with variable coefficients for functions of axial type defined over an axial domain of \(\mathbb {R}^{4}\) which are null-solutions to the equation \((\mathcal {D}-\alpha)\phi=0\), \(\alpha\in\mathbb{R}\). We begin with the following extension of Problem I.
Problem II
Theorem 3.4
- (i)If \(\lambda\in H^{\mu}(\partial\Omega,\mathbb{R})\), and \(\lambda\neq0\) for arbitrary \(z\in\partial D\) then the Riemann-Hilbert boundary value problem (⋆) is solvable and its solution is given bywhere \(\operatorname{Re}(f)\) and \(\operatorname{Im}(f)\) denote the real and imaginary part of the complex-valued function f, respectively. f itself is given by$$ \phi(x)= e^{\alpha x_{0}}\Delta\bigl(\operatorname{Re}(f) \bigl(x_{0}, \vert \underline{x}\vert \bigr)+\underline{\omega}\operatorname{Im}(f) \bigl(x_{0},\vert \underline{x}\vert \bigr)\bigr),\quad x\in\Omega, $$with \(\tilde{g} =\lambda^{-1}\frac{r}{2}g\), \(t_{n}\in\mathbb {C}\).$$ f(z)=\frac{1}{2\pi} \int^{z}_{a} \int_{\partial D}\frac{\widetilde {e^{-\alpha x_{0}}g}(\zeta)}{\zeta-\xi}\,d\zeta \,d\xi+\sum ^{+\infty}_{n=0}t_{n}(z-a)^{n},\quad z\in D, $$
- (ii)Suppose \(\lambda=\Pi\hat{\lambda}\) with \(\Pi(x)=\Pi ^{m}_{i=1}(x-\hat{\alpha}_{i})^{\nu_{i}}\), \(\hat{\alpha}_{i}\in\partial\Omega\), and \(\nu_{i}\in\mathbb {N}\). Furthermore, if \(\hat{\lambda}\in H^{\mu}(\partial \Omega,\mathbb{R})\) and \(\hat{\lambda}\neq0\) for arbitrary \(x\in\partial\Omega\) then the Riemann-Hilbert boundary value problem (⋆) is solvable, and its solution is given again bywhere f is given by$$ \phi(x) = e^{\alpha x_{0}}\Delta\bigl(\operatorname{Re}(f) \bigl(x_{0}, \vert \underline{x}\vert \bigr)+\underline{\omega}\operatorname{Im}(f) \bigl(x_{0},\vert \underline{x}\vert \bigr)\bigr), \quad x\in \Omega, $$with \(\hat{g}=\hat{\lambda}^{-1}\frac{r}{2}g\), \(l_{n}\in \mathbb{C}\).$$ f(z)=\frac{1}{2\pi} \int^{z}_{a} \int_{\partial D}\frac{1}{\Pi(\xi )}\frac{\widehat{e^{-\alpha x_{0}} g}(\zeta)}{\zeta-\xi}\,d\zeta \,d\xi + \sum^{+\infty}_{n=0} \int^{z}_{a}\frac{l_{n}(\xi -a)^{n}}{\Pi(\xi)}\,d\xi, \quad z\in D, $$
Proof
Noting that \(D=\{z:|z-a|<1\}\subset\mathbb{C}_{+}\) with \(z=x_{0}+ir\), \(a=a_{0}+ia_{1}\in\mathbb{C}_{+}\) is the projection of Ω onto the \((x_{0},r)\)-plane, associating with \(g\in H^{\mu}(\partial\Omega,\mathbb{R})\), we have \(e^{-\alpha x_{0}}g\in H^{\mu}(\partial\Omega,\mathbb{R})\).
We apply Theorem 3.2.
Remark 5
When the space dimension is 2, Problem II reduces to the Riemann-Hilbert boundary value problem for meta-analytic functions defined over the complex plane [9, 10].
Similar to the Schwarz problem in the previous subsection, let \(\lambda=1\) in Problem II, we can take it in account in the following problem.
Problem III
Thanks to Theorem 3.4 we can derive the following theorem.
Theorem 3.5
Remark 6
If the boundary data are of Hölder class, we consider the Riemann-Hilbert boundary value problems I, II, III with variable coefficients for monogenic functions of axial type in \(\mathbb{R}^{4}\) and null-solutions to \((\mathcal{D}-\alpha )\phi=0\), \(\alpha\in\mathbb{R}\). Note that Problem II reduces to Problem I when α equals 0 while Problem III reduces to Problem II when λ equals 1. Following the same argument, all results can be extended to the Riemann-Hilbert boundary value problems of monogenic functions of axial type in \(\mathbb{R}^{4}\), with boundary data belonging to \(L_{p}\) (\(1< p<+\infty\)).
Declarations
Acknowledgements
This work was supported in part by the Portuguese Foundation for Science and Technology (‘FCT-Fundação para a Ciência e a Tecnologia’) through the CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013, and by the Postdoctoral Foundation from FCT (Portugal) under Grant No. SFRH/BPD/74581/2010. The authors cordially thank two anonymous referees for their valuable comments, which lead to the improvement of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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References
- Muskhelishvili, NI: Singular Integral Equations. Noordhoff, Leyden (1977) View ArticleGoogle Scholar
- Lu, J-K: Boundary Value Problems for Analytic Functions. World Scientific, Singapore (1993) MATHGoogle Scholar
- Fokas, AS: A Unified Approach to Boundary Value Problems. SIAM, Philadelphia (2008) View ArticleMATHGoogle Scholar
- Balk, MB: On Poly-Analytic Functions. Akademie Verlag, Berlin (1991) Google Scholar
- Deift, P: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Am. Math. Soc., Providence (2000) Google Scholar
- Chelkak, D, Smirnov, S: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515-580 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Abreu, LD, Feichtinger, HG: Function spaces of poly-analytic functions. In: Harmonic and Complex Analysis and Its Applications. Trends in Mathematics, pp. 1-38 (2014) View ArticleGoogle Scholar
- Begehr, H, Schmersau, D: The Schwarz problem for poly-analytic functions. Z. Anal. Anwend. 24(2), 341-351 (2005) View ArticleMathSciNetMATHGoogle Scholar
- Wang, Y, Du, J: Mixed boundary value problems with a shift for a pair of meta-analytic and analytic functions. J. Math. Anal. Appl. 369(2), 510-524 (2010) View ArticleMathSciNetMATHGoogle Scholar
- Wang, Y: On Hilbert-type boundary-value problem of poly-Hardy class on the unit disc. Complex Var. Elliptic Equ. 58(4), 497-509 (2013) View ArticleMathSciNetMATHGoogle Scholar
- Fueter, R: Die Funktionentheorie der Differentialgleichungen \(\Delta u=0\) und \(\Delta\Delta u=0\) mit vier reellen Variablen. Comment. Math. Helv. 7, 307-330 (1934) View ArticleMathSciNetGoogle Scholar
- Brackx, F, Delanghe, R, Sommen, F: Clifford Analysis. Res. Notes Math., vol. 76. Pitman, London (1982) MATHGoogle Scholar
- Gürlebeck, K, Sprössig, W: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser, Basel (1990) View ArticleMATHGoogle Scholar
- Delanghe, R, Sommen, F, Souček, V: Clifford Algebra and Spinor-Valued Functions. Kluwer Academic, Dordrecht (1992) View ArticleMATHGoogle Scholar
- Gong, Y, Du, J: A kind of Riemann and Hilbert boundary value problem for left monogenic functions in \(\mathbb{R}^{m}\) (\(m\geqslant 2\)). Complex Var. Theory Appl. 49(5), 303-318 (2004) MathSciNetMATHGoogle Scholar
- Abreu-Blaya, R, Bory-Reyes, J, Peña-Peña, D: Jump problem and removable singularities for monogenic functions. J. Geom. Anal. 17(1), 1-13 (2007) View ArticleMathSciNetMATHGoogle Scholar
- Bu, Y, Du, J: The RH boundary value problem for the k-monogenic functions. J. Math. Anal. Appl. 347, 633-644 (2008) View ArticleMathSciNetMATHGoogle Scholar
- Gürlebeck, K, Zhongxiang, Z: Some Riemann boundary value problems in Clifford analysis. Math. Methods Appl. Sci. 33, 287-302 (2010) MathSciNetMATHGoogle Scholar
- Ku, M, Kähler, U, Wang, D: Riemann boundary value problems on the sphere in Clifford analysis. Adv. Appl. Clifford Algebras 22(2), 365-390 (2012) View ArticleMATHGoogle Scholar
- Ku, M, Kähler, U: Riemann boundary value problems on half space in Clifford analysis. Math. Methods Appl. Sci. 35(18), 2141-2156 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Cerejeiras, P, Kähler, U, Ku, M: On the Riemann boundary value problem for null solutions to iterated generalized Cauchy-Riemann operator in Clifford analysis. Results Math. 63(3-4), 1375-1394 (2013) View ArticleMathSciNetMATHGoogle Scholar
- Ku, M, Fu, Y, Kähler, U, Cerejeiras, P: Riemann boundary value problems for iterated Dirac operator on the ball in Clifford analysis. Complex Anal. Oper. Theory 7(3), 673-693 (2013) View ArticleMathSciNetMATHGoogle Scholar
- Ku, M, Wang, D: Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis. J. Math. Anal. Appl. 374(2), 442-457 (2011) View ArticleMathSciNetMATHGoogle Scholar
- Ku, M, Wang, D: Solutions to polynomial Dirac equations on unbounded domains in Clifford analysis. Math. Methods Appl. Sci. 34, 418-427 (2011) MathSciNetMATHGoogle Scholar
- Ku, M, Wang, D, Dong, L: Solutions to polynomial generalized Bers-Vekua equations in Clifford analysis. Complex Anal. Oper. Theory 6, 407-424 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Ku, M, Du, J: On the integral representation of spherical k-regular functions in Clifford analysis. Adv. Appl. Clifford Algebras 19(1), 83-100 (2009) View ArticleMathSciNetMATHGoogle Scholar
- Sommen, F: On a generalization of Fueter’s theorem. Z. Anal. Anwend. 19, 899-902 (2000) View ArticleMathSciNetMATHGoogle Scholar
- Colombo, F, Sabadini, I, Sommen, F: The Fueter mapping theorem in integral form and the \(\mathcal {F}\)-functional calculus. Math. Methods Appl. Sci. 33, 2050-2066 (2010) View ArticleMathSciNetMATHGoogle Scholar