Abstract
We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called \(\alpha \)-hyperbolic harmonic, that are harmonic with respect to the Riemannian metric
in the upper half space \({\mathbb {R}}_{+}^{4}=\{\left( x_{0},x_{1},x_{2} ,x_{3}\right) \in {\mathbb {R}}^{4}:x_{3}>0\}\). If \(\alpha =2\), the metric is the hyperbolic metric of the Poincaré upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function \(x^{m}\,(m\in {\mathbb {Z}})\), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental \(\alpha \)-hyperbolic harmonic functions, depending only on the hyperbolic distance and \(x_{3}\), we verify a Cauchy type integral formula for conjugate gradient of \(\alpha \)-hyperbolic harmonic functions. We also compare these results with the properties of paravector valued \(\alpha \)-hypermonogenic in the Clifford algebra \({{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\).
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1 Introduction
We study quaternion valued twice continuous differentiable functions \(f\left( x\right) \) defined in an open subset of the full space \({\mathbb {R}}^{4}\) satisfying the following modified Cauchy–Riemann system
Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables [5]. In case \(\alpha =2,\) this system was studied by Hempfling and Leutwiler in [11]. Recently, we verified Cauchy type formulas for these function in [6]. In this paper, we study integral formulas and operators produced by these formulas. The results are interesting, since we are building hyperbolic function theory in the full skew field of quaternions. We also develop the theory of paravector valued \(\alpha \)-hypermonogenic functions in the Clifford algebra \({{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) and find similar integral theorems as in the quaternionic hyperbolic function theory.
2 Preliminaries
The skew-field of quaternions \({\mathbb {H}}\) is four dimensional associative division algebra over reals with an identity \({\mathbf {1}}\). We denote by \({\mathbf {1}}\), \(\varvec{i}\), \(\varvec{j}\) and \(\varvec{k}\) the generating elements of \({\mathbb {H}}\) satisfying the relations
The elements \(\beta {\mathbf {1}}\) and \(\beta \) are identified for any \(\beta \in {\mathbb {R}}\).
Any quaternion x may be represented with respect to the base \(\{{\mathbf {1}},\varvec{i},\varvec{j},\varvec{k}\}\) by
where \(x_{0},x_{1},x_{2}\) and \(x_{3}\) are real numbers. The real vector spaces \({\mathbb {R}}^{4}\) and \({\mathbb {H}}\) may be identified.
We denote the upper half space by
and the lower half space by
We recall that the hyperbolic distance \(d_{h}(x,a)\) between the points x and a in \({\mathbb {R}}_{+}^{4}\) is \(d_{h}(x,a)=\text {arcosh}(\lambda (x,a))\) where
and
(see a proof for example in [12]). Similarly, we may compute the hyperbolic distance between the points x and a in \({\mathbb {R}}_{-}^{4}\).
The following simple calculation rules
are useful.
We recall that the hyperbolic ball \(B_{h}\left( a,r_{h}\right) \) with the hyperbolic center a in \({\mathbb {R}}_{+}^{4}\) and the radius \(r_{h}\) is the same as the Euclidean ball with the Euclidean center
and the Euclidean radius \(r_{e}=a_{3}\sinh \,r_{h}\).
The inner product \(\left\langle x,y\right\rangle \) in \({\mathbb {R}}^{4}\) is defined as usual by
If \(x=x_{0}+x_{1}\varvec{i}+x_{2}\varvec{j}+x_{3}\varvec{k}\) and \(y=y_{0}+y_{1}\varvec{i}+y_{2}\varvec{j}+y_{3}\varvec{k}\) are quaternions their inner product is defined similarly as in \({\mathbb {R}}^{4}\) by
The elements
are called reduced quaternions. The set of reduced quaternions is identified with \({\mathbb {R}}^{3}\).
The involution \(\left( \,\right) ^{\prime }\) in \({\mathbb {H}}\) is the mapping \(x\rightarrow x^{\prime }\) defined by
and it satisfies
for all quaternions x and y. The reversion \(\left( \,\right) ^{*\text { }}\)in \({\mathbb {H}}\) is the mapping \(x\rightarrow x^{*}\) defined by
and the conjugation \(\left( \,\right) ^{^{\_}\text { }}\)in \({\mathbb {H}}\) is the mapping \(x\rightarrow {\overline{x}}\) defined by \({\overline{x}}=\left( x^{\prime }\right) ^{*}=\left( x^{*}\right) ^{\prime }\), that is
These involutions satisfy the following product rules
and
for all \(x,y\in {\mathbb {H}}\).
The prime involution may be computed as
for all quaternions x. This formula shows, in fact, that the involution \(\left( ~\right) ^{\prime }\) is the rotation around the \(x_{3}\) axes. Similarly, the formulas
hold for all quaternions x. Hence we have the identities
and
valid for all quaternions x.
The real part of a quaternion \(x=x_{0}+x_{1}\varvec{i}+x_{2} \varvec{j}+x_{3}\varvec{k}\) is defined by
and the vector part by
if \(\text {Re}\ x=\text {Re}\ y=0\), the product rule
holds, where \(\times \) is the usual cross product.
The mappings \(S:{\mathbb {H}}\rightarrow {\mathbb {R}}^{3}\) and \(T:{\mathbb {H}} \rightarrow {\mathbb {R}}\) are defined by
and
for \(a=a_{0}+a_{1}\varvec{i}+a_{2}\varvec{j}+a_{3}\varvec{k} \in {\mathbb {H}}\). Using the reversion, we compute the formulas
We use the identities
and
valid for all quaternions a, b and c. The term \(\left[ a,b,c\right] \), called a triple product, is defined by
If \(\text {Re}\ a=\text {Re}\ b=\text {Re}\ c=0\), then (see [10])
Notice that the triple product is linear with respect to a, b and c. Moreover,
3 Hyperregular Functions
We define the following hyperbolic generalized Cauchy–Riemann operators \(H_{\alpha }^{l}(x)\) and \(H_{\alpha }^{r}(x)\) for \(x\in \Omega \backslash \{x_{3}=0\}\) as follows
where the parameter \(\alpha \in {\mathbb {R}}\) and
When there is no confusion, we abbreviate \(D_{l}^{q}f\) by \(D^{q}f\) and \(H_{\alpha }^{l}\) by \(H_{\alpha }\).
Definition 3.1
Let \(\Omega \subset {\mathbb {R}}^{4}\) be open. A function \(f:\Omega \rightarrow {\mathbb {H}}\) is called \(\alpha \) -hyperregular, if \(f\in {\mathcal {C}} ^{1}\left( \Omega \right) \) and
for any \(x\in \Omega \backslash \{x_{3}=0\}\).
We emphasize that a function is \(\alpha \)-hyperregular provided that it is continuous differentiable in the total open set \(\Omega \subset {\mathbb {R}}^{4}\) and satisfies the preceding equation for all x with \(x_{3}\ne 0\).
Computing the components of \(H_{\alpha }^{l}f\left( x\right) \) and \(H_{\alpha }^{r}f\left( x\right) \), we obtain
Proposition 3.2
[6] Let \(\Omega \subset {\mathbb {R}}^{4}\) be open and a function \(f:\Omega \rightarrow {\mathbb {H}}\) continuously differentiable. A function \(\mathbb {\ }f\) is \(\alpha \)-hyperregular in \(\Omega \) if and only if
Our operators are connected to the hyperbolic metric via the hyperbolic Laplace operator as follows.
Proposition 3.3
[6] Let \(\Omega \subset {\mathbb {R}}^{4}\) be open, \(x\in \Omega \backslash \{x_{3}=0\}\) and \(f:\Omega \rightarrow {\mathbb {R}}\) a real twice continuously differentiable function. Then
where the operator
is the Laplace–Beltrami operator (see [13]) with respect to the Riemannian metric
Definition 3.4
Let \(\Omega \subset {\mathbb {R}}^{4}\) be open. A twice continuously real differentiable function \(h:\Omega \rightarrow {\mathbb {R}}\) is called \(\alpha \) -hyperbolic harmonic, if
for all \(x\in \Omega \backslash \{x_{3}=0\}\).
We list a couple of simple observations.
Lemma 3.5
Let \(\Omega \ \)be an open subset of \({\mathbb {R}}^{4}\). If \(h:\Omega \rightarrow {\mathbb {R}}\) is \(\alpha \)-hyperbolic on \(\Omega \) and \(h\in {\mathcal {C}}^{3}\left( \Omega \right) \) then the function \(\frac{\partial h}{\partial x_{3}}\) satisfies the equation
for all \(x\in \Omega \). Moreover, a twice continuously differentiable function \(h:\Omega \rightarrow {\mathbb {R}}\) satisfies the preceding equation if and only if the function \(x_{3}^{-\alpha }h(x)\) is \(-\alpha \)-hyperbolic harmonic for any \(x\in \Omega \backslash \{x_{3}=0\}\).
Proof
Assume that \(x\in \Omega \backslash \{x_{3}=0\}\). We just compute as follows
\(\square \)
Real valued \(\alpha \)-hyperbolic functions are especially important, since they produce \(\alpha \)-hyperregular functions.
Theorem 3.6
[6] Let \(\Omega \) be an open subset of \({\mathbb {R}}^{4}\). If h is \(\alpha \)-hyperbolic on \(\Omega \) then the function \(f={\overline{D}}^{q}h\) is \(\alpha \)-hyperregular on \(\Omega \). Conversely, if f is \(\alpha \)-hyperregular on \(\Omega \), there exists locally a \(\alpha \)-hyperbolic function h satisfying \(f={\overline{D}}^{q}h\).
Theorem 3.7
[6] Let \(\Omega \ \)be an open subset of \({\mathbb {R}}^{4}\). If a twice continuously differentiable function \(f:\Omega \rightarrow {\mathbb {H}}\) is \(\alpha \)-hyperregular then the coordinate functions \(f_{n}\) for \(n=0,1,2\) are \(\alpha \)-hyperbolic harmonic and \(f_{3}\) satisfies the equation
for any \(x\in \Omega \).
The following transformation property is proved in [1, 3].
Lemma 3.8
Let \(\Omega \) be an open set contained in \({\mathbb {R}}_{+}^{4}\) or in \({\mathbb {R}}_{-}^{4}\). A function a twice continuously differentiable function \(f:\Omega \rightarrow \) \({\mathbb {R}}\) is \(\alpha \)-hyperbolic harmonic if and only if the function \(g\left( x\right) =x_{3}^{\frac{2-\alpha }{2}}f\left( x\right) \) satisfies the equation
4 Cauchy Type Integral Formulas
We recall the Stokes theorem for T and S-parts proved in [6].
Theorem 4.1
Let \(\Omega \) be an open subset of \({\mathbb {R}}^{4}\backslash \left\{ x_{3}=0\right\} \) and K a 3-chain satisfying \({\overline{K}}\subset \Omega \). Denote \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) the outer unit normal and the corresponding quaternion by \(\nu =\nu _{0}+\nu _{1}\varvec{i}+\nu _{2}\varvec{j}+\nu _{3}\varvec{k}\). If \(f,g\in {\mathcal {C}}^{1}\left( \Omega ,{\mathbb {H}}\right) \), then
where \(d\sigma \) is the surface element and dm the usual Lebesgue volume element in \({\mathbb {R}}^{4}\).
Theorem 4.2
Let \(\Omega \) be an open subset of \({\mathbb {R}}^{4}\backslash \left\{ x_{3}=0\right\} \) and K a 3-chain satisfying \({\overline{K}}\subset \Omega \). Denote \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) the outer unit normal and the corresponding quaternion by \(\nu =\nu _{0}+\nu _{1}\varvec{i}+\nu _{2}\varvec{j}+\nu _{3}\varvec{k}\). If \(f,g\in {\mathcal {C}}^{1}\left( \Omega ,{\mathbb {H}}\right) \), then
where \(d\sigma \) is the surface element and dm the usual Lebesgue volume element in \({\mathbb {R}}^{4}\).
The fundamental \(\alpha \)-hyperbolic harmonic function, that is the fundamental solution of \(\Delta _{\alpha }\), is the following function (see [4, 6, 7]).
Theorem 4.3
Let x and y be points in the upper half space. The fundamental \(\alpha \)-hyperbolic harmonic function is
where the associated Legendre function is defined by
and the hypergeometric function by
for \(\left| x\right| <1\).
We remark that the fundamental \(\alpha \)-hyperbolic harmonic function is unique up to a harmonic function. The reason why we picked the preceding function is that it leads to nice symmetry properties of a kernel, verified after the following theorem.
Theorem 4.4
Denote \(r_{h}=d_{h}\left( x,y\right) \), \(t=\frac{\alpha -2}{2}\) and define
where
The \(\alpha \)-hyperregular kernel is the function
where
and
is 2-hyperregular with respect to x.
The function \(s\left( x,y\right) \) is the kernel computed in [2] and in [3].
Clearly, the function \(h_{\alpha }\left( x,y\right) \) is not symmetrical with respect to x and y. However, it has the following symmetry properties.
Proposition 4.5
The function \(h_{\alpha }\) has the properties
and
for all x and y outside the hyperplane \(\left\{ \left( u_{0},u_{1} ,u_{3},u_{3}\right) \in {\mathbb {R}}^{4}~|~u_{3}=0\right\} \).
Proof
Denote
If \(m=0,1,2\), then
The last properties follow from the tedious calculations
which are done in [7]. \(\square \)
We recall the integral formulas for S- and T-parts verified in [6].
Theorem 4.6
Let \(\Omega \) and be an open subsets of \({\mathbb {R}}_{+}^{4}\) (or \({\mathbb {R}}_{-}^{4})\). Assume that K is an open subset of \(\Omega \) and \({\overline{K}}\) \(\subset \Omega \) is a compact set with the smooth boundary. Let \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) be the outer unit normal and denote the corresponding quaternion by \(\nu =\nu _{0}+\nu _{1}\varvec{i}+\nu _{2}\varvec{j}+\nu _{3}\varvec{k}\). If f is \(\alpha \)-hyperregular in \(\Omega \) and \(a\in K\), then
and
If we combine these formulas we obtain a new formula.
Theorem 4.7
Let \(\Omega \) and be an open subsets of \({\mathbb {R}}_{+}^{4}\) (or \({\mathbb {R}}_{-}^{4})\). Assume that K is an open subset of \(\Omega \) and \({\overline{K}}\) \(\subset \Omega \) is a compact set with the smooth boundary. Let \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) be the outer unit normal and denote the corresponding quaternion by \(\nu =\nu _{0}+\nu _{1}\varvec{i}+\nu _{2}\varvec{j}+\nu _{3}\varvec{k}\). If f is \(k-\)hyperregular in \(\Omega \) and \(a\in K\) , then
where
and
Proof
We combine the preceding integral formulas using the formula
We introduce the following notation
Applying the symmetry properties of the kernels we deduce
Applying the properties (2.4) and (2.5), we obtain
Hence
Using the definition of the triple product we infer
Using the symmetry properties, we obtain
In order to shorten the notations, we abbreviate \(g=Tf\varvec{k}\). Then we simply compute
Symmetry properties imply that
\(\square \)
Corollary 4.8
Let \(\Omega \) be an open subsets of \({\mathbb {R}}_{+}^{4}\) (or \({\mathbb {R}} _{-}^{4})\). Assume that K is an open subset of \(\Omega \) and \({\overline{K}}\) \(\subset \Omega \) is a compact set with the smooth boundary. Let \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) be the outer unit normal and denote the corresponding quaternion by \(\nu =\nu _{0}+\nu _{1}\varvec{i}+\nu _{2}\varvec{j}+\nu _{3}\varvec{k}\). If f is k-hyperregular in \(\Omega \) and \(a\in K\), then the functions
and
are \(\alpha \)-hyperregular and \(f=r_{1}+r_{2}\).
Theorem 4.9
Let \(\Omega \) be an open subsets of \({\mathbb {R}}_{+}^{4}\) (or \({\mathbb {R}} _{-}^{4})\). Assume that K is an open subset of \(\Omega \) and \({\overline{K}}\) \(\subset \Omega \) is a compact set with the smooth boundary. Let \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) be the outer unit normal and denote the corresponding quaternion by \(\nu =\nu _{0}+\nu _{1}\varvec{i}+\nu _{2}\varvec{j}+\nu _{3}\varvec{k}\). If f : \(\partial K\rightarrow {\mathbb {H}}\) is a continuous function then the function
is \(\alpha \)-hyperregular for all \(a\in K\).
Theorem 4.10
Let \(\Omega \) be an open subsets of \({\mathbb {R}}_{+}^{4}\) (or \({\mathbb {R}} _{-}^{4})\). Assume that K is an open subset of \(\Omega \) and \({\overline{K}}\) \(\subset \Omega \) is a compact set with the smooth boundary. Let \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) be the outer unit normal and denote the corresponding quaternion by \(\nu =\nu _{0}+\nu _{1}\varvec{i}+\nu _{2}\varvec{j}+\nu _{3}\varvec{k}\). If f : \(\partial K\rightarrow {\mathbb {H}}\) is a continuous function, then the function
is k-hyperbolic harmonic for all \(a\in K\) and
satisfies the equation
and \(a_{3}^{-\alpha }Tr_{1}\) is \(-\alpha \)-hyperbolic harmonic.
We consider the Teodorescu and Cauchy type operators in subsequent papers. Also the case \(a\in {\mathbb {R}}_{+}^{4}\backslash K\) involves some technical assumptions and left for later work.
5 Comparison of \(\alpha \)-Hyperregular and \(\alpha \)-Hypermonogenic Functions
The universal real Clifford algebra \({{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) is a real associated algebra with a unit \({\mathbf {1}}\) and is generated by \(e_{1},e_{2}\) and \(e_{3}\) satisfying the relation
where \(\delta _{st}\) is the usual Kronecker delta and \(s,t=1,2,3\). We denote \(r{\mathbf {1}}\) briefly by \(r\in {\mathbb {R}}\).
The elements
for \(x_{0},x_{1},x_{2},x_{3}\in {\mathbb {R}}\) are called paravectors. The real number \(x_{0}\) is the real part of the paravector x.
The main involution in \({{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) is the mapping \(a\rightarrow a^{\prime }\) defined by \(e_{s}^{\prime }=-e_{s}\) for \(s=1,\ldots ,3\) and extended to the total algebra by linearity and the product rule \(\left( ab\right) ^{\prime }=a^{\prime }b^{\prime }\). Similarly the reversion is the mapping \(a\rightarrow a^{*}\) defined by \(e_{s}^{*}=-e_{s}\) for \(s=1,\ldots ,3\) and extended to the total algebra by linearity and the product rule \(\left( ab\right) ^{*}=b^{*}a^{*}\). The conjugation is the mapping \(a\rightarrow {\overline{a}}\) defined by \({\overline{a}}=\left( a^{\prime }\right) ^{*}=\left( a^{*}\right) ^{\prime }.\)
Any element w in \({{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) may be written as
where \(e_{mn}=e_{m}e_{n}\) for \(1\le m<n\le 3\) and \(e_{123}=e_{1}e_{2}e_{3}\). The element \(e_{123}\),denoted by I, is commuting with all elements and \(\left( e_{1}e_{2}e_{3}\right) ^{2}=1\).
We recall that \(C\ell _{0,1}\) may be identified with the field of complex numbers. The universal Clifford algebra \({{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,2}\) may be identified with the quaternions, by setting \(\varvec{i}=e_{1}\), \(\varvec{j}=e_{2}\) and \(\varvec{k}=e_{1}e_{2}\). This identification we used in the first section when we defined involutions.
We generalize the imaginary part of a complex number to \( {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) by decomposing any element \(a\in {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) as
for \(b,c\in {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,2}\). The mappings \(P:{{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\rightarrow {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,2}\) and \(Q:{{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\rightarrow {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,2}\) are defined in [9] by
In order to compute the P- and Q- parts we use the involution \(a\rightarrow {\widehat{a}}\) defined by \({\widehat{e}}_{i}=\left( -1\right) ^{\delta _{s3}}e_{i}\) for \(s=1,2,3\) and extended to the total algebra by linearity and the product rule \({\widehat{ab}}={\widehat{a}}{\widehat{b}}\). Then we obtain the formulas
and
The following calculation rules [9] hold
Note that if \(a\in {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\), then
Moreover if \(a\in {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,2}\) then
We consider functions \(f:\Omega \rightarrow {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\), defined on an open subset \(\Omega \) of \({\mathbb {R}}^{4}\), and assume that its components are continuously differentiable. The left Dirac operator (also called the Cauchy–Riemann operator) in \({{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) is defined by
and the right Dirac operator by
Their conjugate operators \(\overline{D_{l}}\) and \(\overline{D_{r}}\) are defined by
The modified Dirac operators \(M_{\alpha }^{l}\), \({\overline{M}}_{\alpha }^{l}\), \(M_{\alpha }^{r}\) and \({\overline{M}}_{\alpha }^{r}\) , introduced in [8, 9], are defined in \(\left\{ \left( x_{0},x_{1},x_{2} ,x_{3}\right) \in \Omega ~|~x_{3}\ne 0\right\} \) by
where \(\left( Qf\right) ^{\prime }=Q^{\prime }f\). The operator \(M_{2}^{l}\) is also abbreviated by M.
Definition 5.1
Let \(\Omega \subset {\mathbb {R}}^{4}\) be open. A function \(f:\Omega \rightarrow C\ell _{0,3}\) is called left \(\alpha \) -hypermonogenic if \(f\in {\mathcal {C}}^{1}\left( \Omega \right) \) and
for any \(x\in \left\{ x\in \Omega ~|~x_{3}\ne 0\right\} \). The right \(\alpha \)-hypermonogenic functions are defined similarly. The 2-left hypermonogenic functions are called hypermonogenic functions. A twice continuously differentiable function \(f:\Omega \rightarrow {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) is called \(\alpha \)-hyperbolic harmonic if \({\overline{M}}_{\alpha }^{l} M_{\alpha }^{l}f=0.\)
Computing the components of \(M_{\alpha }^{l}f\left( x\right) \) and \(M_{\alpha }^{r}f\left( x\right) \), we obtain
Theorem 5.2
Let \(\Omega \subset {\mathbb {R}}^{4}\) be open and a function \(f:\Omega \rightarrow {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) continuously differentiable. If f is paravector valued then f is \(\alpha \)-hypermonogenic in \(\Omega \) if and only if
Applying Proposition we obtain the result.
Theorem 5.3
Let \(\Omega \subset {\mathbb {R}}^{4}\) be open and a function \(f=\left( f_{0},f_{1},f_{2},f_{3}\right) :\Omega \rightarrow {\mathbb {R}}^{4}\) continuously differentiable. Then the function \(f_{0}+f_{1}\varvec{i}+f_{2} \varvec{j}+f_{3}\varvec{k}\) is \(\alpha \)-hyperregular in \(\Omega \) if and only if the \(f_{0}+f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3}\) is \(\alpha \)-hypermonogenic in \(\Omega \).
We recall the Cauchy type formula for \(\alpha \)-hypermonogenic functions.
Theorem 5.4
[7] Let \(\Omega \) be an open subset of \({\mathbb {R}}_{+}^{4}\) and \(K\subset \Omega \) be a smoothly bounded compact set. Denote \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) the outer unit normal and the corresponding paravector by \(\nu =\nu _{0}+\nu _{1}e_{1}+\nu _{2}e_{2}+\nu _{3}e _{3}\). If f is \(\alpha \)-hypermonogenic in \(\Omega \) and \(a\in K\), then
where
and \(h_{\alpha }(a,x)\) and \(a_{3}^{\alpha }h_{-\alpha }(a,x)e_{3}\) are the \(\alpha \)-hypermonogenic kernels with respect to a.
Using this formula we may verify the formula also for paravector valued functions. Before this, we present three preliminary results.
Lemma 5.5
Let \(a\in \Omega \rightarrow {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\). Then
and
Proof
Assume that \(a\in {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\). Since
and \(e_{3}Qa=Q^{\prime }ae_{3}\) then
Noticing that \(\left( \left( Qa\right) ^{*}\right) ^{^{\prime } }={\overline{Qa}}\) we conclude
and therefore
The last formula follows from if we take \(\left( \,\right) ^{*}\) and \(\left( \,\right) ^{\prime }\) from the both side of the equation. \(\square \)
Lemma 5.6
Let a, b be paravectors in \({{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\). Then \(Q\left( ab\right) \) is a paravector.
Proof
We just compute
Since a, b are paravectors, the elements Qa and Qb are scalars, completing the proof. \(\square \)
Lemma 5.7
Let \(\Omega \subset {\mathbb {R}}^{4}\) be open. A function \(f:\Omega \rightarrow {{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}\) is left \(\alpha \)-hypermonogenic if and only if \(f^{*}\)is right \(\alpha \)-hypermonogenic.
Proof
Assume that f is left \(\alpha \)-hypermonogenic then
Since \(\left( a^{*}\right) ^{\prime }={\overline{a}}\), we infer
Using the previous lemma we obtain
Hence \(f^{*}\) is right \(\alpha \)-hypermonogenic. Similarly, we verify that if f is right \(\alpha \)-hypermonogenic then f is left \(\alpha \)-hypermonogenic. \(\square \)
Theorem 5.8
Let \(\Omega \) be an open subset of \({\mathbb {R}}_{+}^{4}\) and \(K\subset \Omega \) be a smoothly bounded compact set. Denote \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) the outer unit normal and the corresponding paravector by \(\nu =\nu _{0}+\nu _{1}e_{1}+\nu _{2}e_{2}+\nu _{3}e _{3}\). If f is right \(\alpha \)-hypermonogenic in \(\Omega \) and \(a\in K\), then
where \(h_{\alpha }(a,x)\) and \(e_{3}a_{3}^{\alpha }h_{-\alpha }(a,x)\) are right \(\alpha \)-hypermonogenic with respect to the variable a.
Proof
If f is right \(\alpha \)-hypermonogenic then \(f^{*}\) is left \(\alpha \)-hypermonogenic and therefore
Taking \(\left( \,\right) ^{*}\) from the both sides we obtain
where \(h_{\alpha }^{*}(a,x)=\left( h_{\alpha }(a,x)\right) ^{*}\). Applying the previous lemma, we infer
and
since f and \(\nu \) are paravectors. Hence we have
Since \(h_{\alpha }^{a}\) is a paravector we infer
completing the proof. \(\square \)
Theorem 5.9
Let \(\Omega \) be an open subset of \({\mathbb {R}}_{+}^{4}\) and \(K\subset \Omega \) be a smoothly bounded compact set. Denote \((\nu _{0},\nu _{1},\nu _{2},\nu _{3})\) the outer unit normal and the corresponding paravector by \(\nu =\nu _{0}+\nu _{1}e_{1}+\nu _{2}e_{2}+\nu _{3}e _{3}\). Then, if f is a paravector valued \(\alpha \)-hypermonogenic in \(\Omega \) and \(a\in K\),
Proof
If f is a paravector valued \(\alpha \)-hypermonogenic in \(\Omega \) and \(a\in K\), then
Since \(P(\nu f)=P\nu Pf+Q\nu Q^{\prime }f\) and f is a paravector we obtain
Similarly we compute
completing the proof. \(\square \)
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This article is part of the Topical Collection on FTHD 2018, edited by Sirkka-Liisa Eriksson, Yuri M. Grigoriev, Ville Turunen, Franciscus Sommen and Helmut Malonek.
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Eriksson, SL., Orelma, H. Hyperbolic Function Theory in the Skew-Field of Quaternions. Adv. Appl. Clifford Algebras 29, 97 (2019). https://doi.org/10.1007/s00006-019-1017-5
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DOI: https://doi.org/10.1007/s00006-019-1017-5