On Structure of Octonion Regular Functions
aa r X i v : . [ m a t h . C A ] J a n On Structure of Octonion Regular Functions
Janne KauhanenMathematicsTampere UniversityFI-33014 Tampere UniversityFinland [email protected]
Heikki OrelmaCivil EngineeringTampere UniversityFI-33014 Tampere UniversityFinland [email protected]
January 10, 2019
Abstract
In this paper we study octonion regular functions and the structural differences betweenregular functions in octonion, quaternion, and Clifford analyses.
Mathematics Subject Classification (2010) . 30G35, 15A63
Keywords . Octonions, Cauchy-Riemann operators, regular functions
In our recent papers [5, 6], we started to study octonion algebraic methods in analysis. This paperis a continuation of our studies in this fascinating field. Over the years, many results of octonionanalysis have been published and studied since the fundamental paper of Dentoni and Sce [2]. Onething which has remained unclear to us is that what is octonion analysis all about? A consensushas been that octonion, quaternion and Clifford analyses are similar from a theoretical point ofview, and maybe for this reason octonion analysis has been left to less attention. Our aim is toprove that octonion analysis and Clifford analysis are different theories from the point of view ofregular functions. Thus, octonion analysis is a completely independent research topic.We start by recalling preliminaries of octonions and Clifford numbers and their connections viatriality. We define our fundamental function classes, i.e., left-, right- and bi-regular functions. Wegive chararacterizations for function classes in biaxial quaternion analysis and in Clifford analysis.The classical Riesz system of Stein and Weiss is used as a familiar reference to clearly see thedifferences.The topic of this paper is highly technical, but we have tried to write everything in as a simpleway as possible. Hopefully we have succeeded in this job. Many questions remain open and thereader may find a lot of open research problems between the lines. Hopefully we can answer someof these questions when the saga continues.
In this algebraic part of the paper, we first recall briefly the basic definitions and notations relatedto the octonion and Clifford algebras. Then we study their connections in detail. In the wholepaper, our principle is to consider the standard orthonormal basis { e , e , . . . , e } for R , and by1efining different products between the elements, we obtain different algebras. We will denote theoctonion product by e i ◦ e j , and the Clifford product by e i e j . The algebra of octonions O is the non-commutative and non-associative -dimensional algebra withthe basis { , e , . . . , e } and the multiplication given by the following table. ◦ e e e e e e e e e e e e e e e e − e − e e − e − e e e e − e − e e e − e − e e e e − e − e − e e − e e e − e − e − e − e e e e e e − e e − e − − e e e e e e − e − e e − − e e e − e e e − e − e e − Let us point out that there are possible ways to define an octonion product such that e = 1 .Our choise is historically maybe the most used and traditional, and for this reason we may call itthe canonical one, but e.g. Lounesto uses a different definition for octonion multiplication in hisfamous book [8].For ≤ i, j ≤ we have e i ◦ e i = e i = − , and e i ◦ e j = − e j ◦ e i if i = j. An element x ∈ O may be represented in the forms x = x + x e + x e + x e + x e + x e + x e + x e = x + x = ( x + x e + x e + x e ) + ( x + x e + x e + x e ) ◦ e = u + v ◦ e = ( u + u ) + ( v + v ) ◦ e . Here, x , ..., x ∈ R , x is the real part , x is the vector part , and u and v ∈ H are quaternions. Thelast form is called the quaternion form of an octonion. The conjugate of x is denoted and definedby x = x − x . We see that the element e plays a kind of a role of the ”imaginary unit”. Theproduct of two octonions can be written as x ◦ y = ( x + x ) ◦ ( y + y ) = X i,j =0 x i y j e i ◦ e j = X i =0 x i y i e i + X i,j =0 i = j x i y j e i ◦ e j = x y − X i =1 x i y i + x X i =1 y i e i + y X i =1 x i e i + X i,j =1 i = j x i y j e i ◦ e j = x y − x · y + x y + y x + x × y, (1)where x · y is the dot product and x × y the cross product of vectors x and y .Denote the quaternion forms of octonions x and y by x = ( u + u ) + ( v + v ) ◦ e ,y = ( a + a ) + ( b + b ) ◦ e . (2)In Lemma 2.3 we will return the cross product x × y of octonion vector parts x and y to crossproducts of the vector parts u , v , a , and b of quaternions, which are classical -dimensional crossproducts (see, e.g., [3, 8, 10]). 2 emma 2.1 (See, e.g., [5, Lemma 2.10]) . Let u, v ∈ H . Then e ◦ u = u ◦ e ,e ◦ ( u ◦ e ) = − u, ( u ◦ e ) ◦ e = − u,u ◦ ( v ◦ e ) = ( v ◦ u ) ◦ e , ( u ◦ e ) ◦ v = ( u ◦ v ) ◦ e , ( u ◦ e ) ◦ ( v ◦ e ) = − v ◦ u. Lemma 2.2. If x, y ∈ O be as in (2), then u × e = u ◦ e ,e × a = − a ◦ e ,u × ( b ◦ e ) = − ( u × b ) ◦ e − ( u · b ) e ,e × ( b ◦ e ) = b, ( v ◦ e ) × a = − ( v × a ) ◦ e + ( a · v ) e , ( v ◦ e ) × e = − v, ( v ◦ e ) × ( b ◦ e ) = − v × b. Proof.
Lemma 2.1 implies e i ◦ ( e j ◦ e ) = ( e j ◦ e i ) ◦ e ,e ◦ ( e j ◦ e ) = e j , ( e i ◦ e ) ◦ e j = − ( e i ◦ e j ) ◦ e , ( e i ◦ e ) ◦ e = − e i , ( e i ◦ e ) ◦ ( e j ◦ e ) = e j ◦ e i for ≤ i, j ≤ . Then u × ( b ◦ e ) = X i,j =1 u i b j e i ◦ ( e j ◦ e ) = X i,j =1 u i b j e j ◦ e i ◦ e = − X i,j =1 i = j u i b j e i ◦ e j − X i =1 u i b i ◦ e = − ( u × b ) ◦ e − ( u · b ) e ,e × ( b ◦ e ) = X j =1 b j e ◦ ( e j ◦ e ) = X j =1 b j e j = b, ( v ◦ e ) × a = X i,j =1 v i a j ( e i ◦ e ) ◦ e j = − X i,j =1 v i a j e i ◦ e j ◦ e = − X i,j =1 i = j v i a j e i ◦ e j + X i =1 v i a i ◦ e = − ( v × a ) ◦ e + ( a · v ) e , ( v ◦ e ) × e = X i =1 v i ( e i ◦ e ) ◦ e = − X i =1 v i e i = − v, ( v ◦ e ) × ( b ◦ e ) = X i,j =1 i = j v i b j ( e i ◦ e ) ◦ ( e j ◦ e ) = X i,j =1 i = j v i b j e j ◦ e i = − X i,j =1 i = j v i b j e i ◦ e j = − v × b. Lemma 2.3.
Denote the quaternion representations of the vectors x and y ∈ O as in (2) . Thenthe cross product in quaternion form is x × y = v b − vb + u × a − v × b ∈ span { e , e , e } + ( v · a − u · b ) e ∈ span { e } + ( ub − v a − u × b − v × a ) ◦ e ∈ span { e , e , e } Proof.
By Lemma 2.2, we compute x × y = u × a + u × ( b e ) + u × ( b ◦ e )+ ( v e ) × a + ( v e ) × ( b e ) + ( v e ) × ( b ◦ e )+ ( v ◦ e ) × a + ( v ◦ e ) × ( b e ) + ( v ◦ e ) × ( b ◦ e )= u × a + ub ◦ e − ( u × b ) ◦ e − v a ◦ e + 0 + v b − ( v × a ) ◦ e − vb − v × b = u × a − v × b + v b − vb + ( v · a − u · b ) e + ( ub − v a − u × b − v × a ) ◦ e . Corollary 2.4. If x, y ∈ O be as in (2), then x ◦ y = u a − v b − u · a − v · b ∈ R + u a + a u + v b − vb + u × a − v × b ∈ span { e , e , e } + ( u b + a v + v · a − u · b ) e ∈ span { e } + ( u b + a v + ub − v a − u × b − v × a ) ◦ e ∈ span { e , e , e } C ℓ , and triality Since the dimension of octonions and Clifford paravectors is , they behave similarly as vectorspaces. Moreover, we may ask if there is a connection between the octonion and the Cliffordproduct? The answer is given by Pertti Lounesto in his book [8]. We will recall his ideas here indetail. Let us recall the basic definitions and properties of Clifford algebras.We continue working with the basis { e , e , ..., e } for R . The Clifford product is defined by e i e j + e j e i = − δ ij , i, j = 1 , ..., , where δ ij is the Kronecker delta symbol. Here, e = 1 . Then, similarly than in the case ofoctonions, e = 1 , and e j = − for all j = 1 , . . . , . The Clifford product e i e j is not necessarya vector or a scalar. This product generates an associative algebra, called the Clifford algebra,denoted by C ℓ , . The dimension of this Clifford algebra is , and an element a ∈ C ℓ , may berepresented as a sum a = X j =0 [ a ] j of a scalar part [ a ] , generated by , a -vector part [ a ] , generated by e j ’s, -vector part [ a ] ,generated by the products e i e j , where ≤ i < j ≤ , etc. Clifford numbers of the form [ a ] are4alled vectors and [ a ] , = [ a ] + [ a ] paravectors . The set of paravectors may be identified with R .The Clifford product of two paravectors x and y can be written xy = ( x + x )( y + y ) = X i,j =0 x i y j e i e j = X i =0 x i y i e i + X i,j =0 i = j x i y j e i e j = x y − X i =1 x i y i + x X i =1 y i e i + y X i =1 x i e i + X i,j =1 i = j x i y j e i e j = x y − x · y + x y + y x + x ∧ y, (3)where x ∧ y is the wedge product of vectors x and y . In particular, xy = x ∧ y − x · y .The reader can see that formally the octonion and the Clifford products are similar, and a reason-able question is, how they are connected? We would like to construct the octonion product usingthe Clifford algebra C ℓ , . Let us consider the octonion product of the basis elements e i and e j ,where ≤ i, j ≤ , i = j : e i ◦ e j = e k . Then ≤ k ≤ , and i = k = j . The corresponding Clifford product e i e j may be mapped to e k bymultiplying it by the trivector e j e i e k , i.e., ( e i e j )( e j e i e k ) = e i e j e k = e k , and by the same trivector e j e i is mapped to − e k . If a and b are vectors, then a b ( e j e i e k ) = ( a i b j − a j b i ) e k + [ a b ( e j e i e k )] + [ a b ( e j e i e k )] . Picking the -vector part [ a b ( e j e i e k )] = ( a i b j − a j b i ) e k gives us a part of the k th component of the octonion product a ◦ b . Using this idea, we may expressthe octonion product a ◦ b as the paravector part of the Clifford product ab (1 − W ) , where W is asuitable -vector. Lemma 2.5 ([8, Sec 23.3], [12, Lem 4.1]) . Define W = e + e + e + e + e + e + e . Let a = a + a and b = b + b be paravectors. Then a ◦ b = [ ab (1 − W )] , and in particular, a × b = − [( a ∧ b ) W ] . Lounesto states Lemmas 2.5 and 2.7 without proofs at pages 303–304 in [8], and for a differentmultiplication table of octonions. Venäläinen gives a proof for Lemma 2.5 in her licentiate thesis[12]. For the convenience of the reader, we give a short proof of Lemma 2.5 here.
Proof.
We compute [ ab (1 − W )] , = [ ab ] , − [ abW ] = a b − a · b + a b + b a − [ abW ] . By (1) and (3), it is enough to show that a × b = − [( a ∧ b ) W ] . Consider the triplets ν = 123 , , , , , , . e i e j e ν is a vector only if the pair of indices ij belongs to the triplet ν . Since the crossand the wedge products a × b = X i,j =1 i A function f : R → O is of the form f = f + f e + · · · + f e = f + f , where f j : R → R . Wedefine the Cauchy–Riemann operator D x = ∂ x + e ◦ ∂ x + ... + e ◦ ∂ x . The vector part of it D x = e ◦ ∂ x + ... + e ◦ ∂ x is called the Dirac operator . If the coordinate functions of f have partial derivatives, then D x operates on f from the left and from the right as D x f = X i,j =0 e i ◦ e j ∂ x i f j and f D x = X i,j =0 e j ◦ e i ∂ x i f j . D x f = ∂ x f − D x · f + ∂ x f + D x f + D x × f and (6) f D x = ∂ x f − D x · f + ∂ x f + D x f − D x × f , (7)where ∂ x f − D x · f is the divergence of f and D x × f is the rotor of f .If D x f = 0 (resp. f D x = 0 ), then f is called left (resp. right ) regular .In Clifford analysis one studies functions f : R → C ℓ , . We define the Cauchy-Riemann operatorsimilarly as in octonion analysis: ∂ x = ∂ x + e ∂ x + ... + e ∂ x = ∂ x + ∂ x . Functions satisfying ∂ x f = 0 (resp. f ∂ x = 0 ) on R are called left (resp. right ) monogenic . In thispaper we only need to consider paravector valued functions f = f + f e + ... + f e . Comparing the real and vector parts in (6) and (7) yields the following well known results. Proposition 3.1. A function f : R → O is left regular if and only if it satisfies the Moisil–Teodorescu type system ∂ x f − D x · f = 0 ,∂ x f + D x f + D x × f = 0 , (8) or componentwise ∂ x f − ∂ x f − . . . − ∂ x f = 0 ,∂ x f + ∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f − ∂ x f + ∂ x f = 0 ,∂ x f + ∂ x f − ∂ x f + ∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f = 0 ,∂ x f + ∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f − ∂ x f + ∂ x f = 0 ,∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f + ∂ x f = 0 ,∂ x f + ∂ x f + ∂ x f − ∂ x f − ∂ x f + ∂ x f + ∂ x f − ∂ x f = 0 ,∂ x f + ∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f − ∂ x f + ∂ x f = 0 ,∂ x f + ∂ x f − ∂ x f + ∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f = 0 . (9)We will denote the space of left regular functions by M ( ℓ ) , and similary, right regular functions by M ( r ) . Proposition 3.2. A function f : R → O is both left and right regular if and only if it satisfiesthe system ∂ x f − D x · f = 0 ,∂ x f + D x f = 0 ,D x × f = 0 , (10) or componentwise ∂ x f − ∂ x f − . . . − ∂ x f = 0 ,∂ x f i + ∂ x i f = 0 , i = 1 , . . . , ,∂ x f − ∂ x f + ∂ x f − ∂ x f − ∂ x f + ∂ x f = 0 , − ∂ x f + ∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f = 0 ,∂ x f − ∂ x f + ∂ x f − ∂ x f − ∂ x f + ∂ x f = 0 , − ∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f + ∂ x f = 0 ,∂ x f − ∂ x f − ∂ x f + ∂ x f + ∂ x f − ∂ x f = 0 ,∂ x f − ∂ x f + ∂ x f − ∂ x f − ∂ x f + ∂ x f = 0 , − ∂ x f + ∂ x f + ∂ x f − ∂ x f + ∂ x f − ∂ x f = 0 . (11)7e will call functions satisfying (11) B-regular , and denote the space of such functions by M B .Naturally M B = M ( ℓ ) ∩ M ( r ) . The fundamental difference between octonion and Clifford analyses is that in Clifford analysis theparavector valued null solutions to the Cauchy-Riemann operator satisfies the Riesz system andare at the same time left and right monogenic, which is not true in octonion analysis. The followingwell-known proposition follows from the definitions similarly as in octonion analysis by comparingthe scalar parts, -vector parts, and -vector parts. Proposition 3.3. Suppose f : R → C ℓ , is a paravector valued function. Then ∂ x f = 0 if andonly if f ∂ x = 0 , and this is equivalent to f satisfying the Riesz-system ∂ x f − ∂ x · f = 0 ,∂ x f + ∂ x f = 0 ,∂ x ∧ f = 0 , (12) or componentwise ∂ x f − ∂ x f − . . . − ∂ x f = 0 ,∂ x f i + ∂ x i f = 0 , i = 1 , . . . , ,∂ x i f j − ∂ x j f i = 0 , i, j = 1 , . . . , , i = j, (13)Functions satisfying (13) are called R-regular , and the space of such functions is denoted by M R .To convince the reader about the existence of these function classes, we recall the following classicalmethod from Clifford analysis. Remark 3.4 (Cauchy–Kovalevskaya extension) . If f : Ω → R is a real analytic function defined onan open Ω ⊂ R ∼ = O ∩ { x = 0 } we may construct its Cauchy-Kovalevskaya extension analogouslyto Clifford analysis (see [1]) by definingCK [ f ]( x ) = e − x D x f ( x ) . It is easy to see that since f is real valued, D x CK [ f ] = CK [ f ] D x = 0 , i.e., CK [ f ] ∈ M B . Since O is an alternative division algebra, that is x ( xy ) = x y for all x, y ∈ O , the Cauchy–Kovalevskayaextension may be extended to octonion valued real analytic functions. A necessary condition forCK [ f ] ∈ M ( ℓ ) is that f is an octonion valued real analytic function with f = 0 . It is not a sufficientcondition since, e.g., CK [ x ]( x ) = 7 x + x belongs to M B .We may conclude that although Clifford and octonion analyses have formally very similar defini-tions, the corresponding function spaces are different. Proposition 3.5. M R ( M B ( M ( ℓ ) Proof. The inclusions follow from Propositions 3.1–3.3. The examples showing that the inclusionsare strict: if f = x e − x e , then D x f = f D x = 0 , but ∂ x f − ∂ x f = 1 = 0 , and if f = x − x e ,then D x f = 0 , but f D x = 2 e = 0 .This result is crucial in understanding the fundamental character of octonion analysis and thestructural differences between octonion, quaternion, and Clifford analyses. Remark 3.6 (Quaternion analysis) . If we make the corresponding definitions for quaternion reg-ular function classes by considering the Cauchy-Riemann operator D x = ∂ x + e ◦ ∂ x + e ◦ ∂ x + e ◦ ∂ x acting on quaternion valued functions f = f + f e + f e + f e , then by comparing (11)and (13) we observe immediately, that M R = M B ( M ( ℓ ) . emark 3.7 (Clifford analysis) . If we make the corresponding definitions for paravector valuedmonogenic functions, then by Proposition 3.3, M R = M B = M ( ℓ ) . In the preceding section we gave characterizations for left-, B -, and R -regular functions usingcomponentwise and vector forms. In this section we write the three systems of Section 3.2 inquaternion forms. The use of the quaternion forms of the function and the Cauchy–Riemannoperator is called the biaxial quaternion analysis .Write the Cauchy–Riemann operator D x and the function f : R → O in the quaternion forms: D x = ∂ u + ∂ u + ( ∂ v + ∂ v ) ◦ e ,f = g + g + ( h + h ) ◦ e . According to Corollary 2.4, we can write D x f = ∂ u g − ∂ v h − ∂ u · g − ∂ v · h + ∂ u g + ∂ u g + ∂ v h − ∂ v h + ∂ u × g − ∂ v × h + ( ∂ u h + ∂ v g + ∂ v · g − ∂ u · h ) e + ( ∂ u h + ∂ v g + ∂ u h − ∂ v g − ∂ u × h − ∂ v × g ) ◦ e . This implies the quaternion forms of the Moisil–Teodorescu type system (8) and the system (10). Proposition 4.1. f : R → O is left regular if and only if it satisfies the system ∂ u h + ∂ v g + ∂ v · g − ∂ u · h = 0 ,∂ u g − ∂ v h − ∂ u · g − ∂ v · h = 0 ,∂ u g + ∂ u g + ∂ v h − ∂ v h + ∂ u × g − ∂ v × h = 0 ,∂ u h + ∂ v g + ∂ u h − ∂ v g − ∂ u × h − ∂ v × g = 0 . (14) Proposition 4.2. f : R → O is left and right regular if and only if it satisfies the system ∂ u h + ∂ v g + ∂ v · g − ∂ u · h = 0 ,∂ u g − ∂ v h − ∂ u · g − ∂ v · h = 0 ,∂ u g + ∂ u g + ∂ v h − ∂ v h = 0 ,∂ u h + ∂ v g + ∂ u h − ∂ v g = 0 ,∂ u × g − ∂ v × h = 0 ,∂ u × h + ∂ v × g = 0 . (15)One example of the use of the biaxial quaternion analysis is the proof of the following vectorcalculus identity in the octonionic case. Lemma 4.3. Let the coordinates of f and g : R → O have partial derivatives. Then D x · ( f × g ) = ( D x × f ) · g − f · ( D x × g ) . Proof. We use quaternion decompositions D x = ∂ u + ∂ v e + ∂ v ◦ e ,f = f + F e + F ◦ e ,g = g + G e + G ◦ e . 9n the left-hand side we apply Lemma 2.3 to the cross product f × g , and use the classical vectorcalculus identity ∇ · ( u × v ) = ( ∇ × u ) · v − u · ( ∇ × v ) for u, v : R → R : D x · ( f × g )= (cid:0) ∂ u + ∂ v e + ∂ v ◦ e (cid:1) · (cid:0) F G − F G + f × g − F × G + ( F · g − f · G ) e + ( f G − F g − f × G − F × g ) ◦ e (cid:1) = ∂ u · ( F G ) − ∂ u · ( F G ) + ∂ u · ( f × g ) − ∂ u · ( F × G )+ ∂ v ( F · g ) − ∂ v ( f · G )+ ∂ v · ( f G ) − ∂ v · ( F g ) − ∂ v · ( f × G ) − ∂ v · ( F × g )= ( ∂ u F ) · G + F ( ∂ u · G ) − ( ∂ u · F ) G − F · ( ∂ u G )+ ( ∂ u × f ) · g − f · ( ∂ u × g ) − ( ∂ u × F ) · G + F · ( ∂ u × G )+ ( ∂ v F ) · g + F · ( ∂ v g ) − ( ∂ v f ) · G − f · ( ∂ v G )+ ( ∂ v · f ) G + f · ( ∂ v G ) − ( ∂ v F ) · g − F ( ∂ v · g ) − ( ∂ v × f ) · G + f · ( ∂ v × G ) − ( ∂ v × F ) · g + F · ( ∂ v × g ) . On the right-hand side we apply Lemma 2.3 to the rotors D x × f and D x × g : ( D x × f ) · g = (cid:0) ∂ v F − ∂ v F + ∂ u × f − ∂ v × F + ( ∂ v · f − ∂ u · F ) e + ( ∂ u F − ∂ v f − ∂ u × F − ∂ v × f ) ◦ e (cid:1) · (cid:0) g + G e + G ◦ e (cid:1) = ( ∂ v F ) · g − ( ∂ v F ) · g + ( ∂ u × f ) · g − ( ∂ v × F ) · g + ( ∂ v · f ) G − ( ∂ u · F ) G + ( ∂ u F ) · G − ( ∂ v f ) · G − ( ∂ u × F ) · G − ( ∂ v × f ) · G , and f · ( D x × g )= (cid:0) f + F e + F ◦ e (cid:1) · (cid:0) ∂ v G − ∂ v G + ∂ u × g − ∂ v × G + ( ∂ v · g − ∂ u · G ) e + ( ∂ u G − ∂ v g − ∂ u × G − ∂ v × g ) ◦ e (cid:1) = f · ( ∂ v G ) − f · ( ∂ v G ) + f · ( ∂ u × g ) − f · ( ∂ v × G )+ F ( ∂ v · g ) − F ( ∂ u · G )+ F · ( ∂ u G ) − F · ( ∂ v g ) − F · ( ∂ u × G ) − F · ( ∂ v × g ) . Remark 4.4 (Regular functions is not a module) . In quaternion analysis ∂ u g = 0 implies ∂ u ( g ◦ a ) = 0 for all a ∈ H (see Lemma 4.5). The same does not hold in octonion analysis. For example,define g : H → H , g ( x ) = x − x e . Then D x g = e − e e = 0 , but D x ( g ◦ e ) = D x ( x e − x e ) = e e − e e = 2 e .For quaternion functions we have the product rules (16) and (17) for the Cauchy–Riemann operator.Remark 4.4 suggests that we do not have any kind of a non-trivial product rule for octonion valuedfunctions. To compute D x ( f g ) for octonion valued functions in practice, one way is to use biaxialquaternion analysis, and then to apply (16)–(24).10 emma 4.5. [3, Thm 1.3.2] Let the coordinates of f and g : H → H have partial derivatives. Then ∂ u ( f ◦ g ) = ( ∂ u f ) ◦ g + f ◦ ( ∂ u g ) − f · ∂ u ) g (16) and ( f ◦ g ) ∂ u = ( f ∂ u ) ◦ g + f ◦ ( g∂ u ) − g · ∂ u ) f. (17) Here, ( f · ∂ u ) g = P i =1 f i ∂ x i g . Corollary 4.6. Let the coordinates of f and g : H → H have partial derivatives. Then ∂ u (( f ◦ e ) ◦ g ) = [( f ∂ u ) ◦ g + f ◦ ( g∂ u ) + 2( g · ∂ u ) f ] ◦ e (18) ∂ u ( f ◦ ( g ◦ e )) = [( g∂ u ) ◦ f + g ◦ ( f ∂ u ) − f · ∂ u ) g ] ◦ e (19) ∂ u (( f ◦ e ) ◦ ( g ◦ e )) = − ( ∂ u g ) ◦ f − g ◦ ( ∂ u f ) − g · ∂ u ) f (20) ( ∂ v ◦ e )( f ◦ g ) = [( ∂ v g ) ◦ f + g ◦ ( ∂ v f ) + 2( g · ∂ v ) f ] ◦ e (21) ( ∂ v ◦ e )(( f ◦ e ) ◦ g ) = − ( g∂ v ) ◦ f − g ( f ∂ v ) − f · ∂ v ) g (22) ( ∂ v ◦ e )( f ◦ ( g ◦ e )) = − ( f ∂ v ) ◦ g − f ◦ ( g∂ v ) − g · ∂ v ) f (23) ( ∂ v ◦ e )(( f ◦ e ) ◦ ( g ◦ e )) = [ − ( ∂ v f ) g − f ( ∂ v g ) − f · ∂ v ) g ] ◦ e (24) Proof. Apply Lemmas 2.1 and 4.5, and use the fact f g = g f . In this last section, we study the classes of left-, B -, and R -regular functions using Clifford analysis.We begin with the following algebraic lemma. Lemma 5.1. Let I = I − be the primitive idempotent (4) , and let a = a + a and b = b + b ∈ C ℓ , be paravectors. Then abI ] = a b − a · b, (25) abI ] = a b + ab − [( a ∧ b ) W ] , (26) abI ] = a ∧ b − [( a b + ab ) W ] + [( a ∧ b ) W e ··· ] , (27) abI ] = − ( a b − a · b ) W + [( a b + ab ) W e ··· ] − [( a ∧ b ) W ] , (28) abI ] = ( a b − a · b ) W e ··· − [( a b + ab ) W ] + [( a ∧ b ) W e ··· ] , (29) abI ] = [( a b + ab ) W e ··· ] − [( a ∧ b ) W ] − ( a ∧ b ) e ··· , (30) abI ] = − ( a b + ab ) e ··· + [( a ∧ b ) W e ··· ] , (31) abI ] = − ( a b − a · b ) e ··· , (32) and [ abI ] k = 0 ⇔ [ abI ] − k = 0 , k = 0 , , . . . , . (33) If [ abI ] = 0 , then the conditions [ abI ] j = 0 , j = 2 , , , , are pairwise equivalent. In particular, if [ abI ] , , = 0 , then abI = 0 .Proof. Write the real part and - and -vector parts of ab using (3), and expand the definition (4)of I using the fact e ··· = 1 : ab = ( a b − a · b ) + ( a b + ab ) + a ∧ b, I = 1 − W + W e ··· − e ··· . Here, W is a -vector and W e ··· is a -vector. Then, for example, aW only contains - and -vector parts, and therefore [ aW ] = 0 . This kind of reasoning implies (25)–(32).11ow, (33) follows from the facts that for any c ∈ C ℓ , , c = 0 ⇔ ce ··· = 0 , and [ c ] k e ··· = [ ce ··· ] − k , k = 0 , , . . . , . To prove the last claim, it is now enough to show that in the case [ abI ] = 0 , [ abI ] = 0 if andonly if [ abI ] = 0 . This can be seen by computing abI ] =( a b + a b + a b − a b + a b − a b − a b + a b )( e + e − e )+( a b − a b + a b + a b + a b + a b − a b − a b )( − e + e + e )+( a b + a b − a b + a b + a b − a b + a b − a b )( e + e − e )+( a b − a b − a b − a b + a b + a b + a b + a b )( − e − e − e )+( a b + a b − a b + a b − a b + a b − a b + a b )( e − e e )+( a b + a b + a b − a b − a b + a b + a b − a b )( e + e − e )+( a b − a b + a b + a b − a b − a b + a b + a b )( − e + e + e ) , and in the case [ abI ] = 0 , abI ] =( a b + a b + a b − a b + a b − a b − a b + a b )( e − e − e − e )+( a b − a b + a b + a b + a b + a b − a b − a b )( − e + e + e − e )+( a b + a b − a b + a b + a b − a b + a b − a b )( e + e − e + e )+( a b − a b − a b − a b + a b + a b + a b + a b )( e − e + e − e )+( a b + a b − a b + a b − a b + a b − a b + a b )( − e − e − e + e )+( a b + a b + a b − a b − a b + a b + a b − a b )( e + e − e − e )+( a b − a b + a b + a b − a b − a b + a b + a b )( − e + e + e + e ) . We infer that left-, B-, and R-regularity can be studied by considering paravector-spinor valuedfunctions f I . Theorem 5.2. Suppose f : R → R is a paravector valued function such that the coordinatefunctions have partial derivatives.(a) f is left-regular if and only if [ ∂ x f I ] j = 0 for j = 0 , . (34) (b) f is B-regular if and only if [ ∂ x f I ] j = 0 for j = 0 , , and [ ∂ x f W ] = 0 . (35) Proof. (a) follows using Lemma 2.7: D x f = 16[ ∂ x f I ] + 16[ ∂ x f I ] . (b) We compute, using (3) and (26), [ ∂ x f W ] = [( ∂ x f − ∂ x · f ) W ] + [( ∂ x f + ∂ x f ) W ] + [( ∂ x ∧ f ) W ] = [( ∂ x ∧ f ) W ] = − ∂ x f I ] + ∂ x f + D x f . Since D x × f = − [( ∂ x ∧ f ) W ] (Lemma 2.5), the claim now follows from (a) and Propositions3.1–3.2. Remark 5.3. If ∂ x f = 0 , then (trivially) [ ∂ x f I ] j = 0 for all j = 0 , , . . . , . The converse does nothold. This follows from the fact that the equation aI = 0 does not have a unique solution a = 0 inthe Clifford algebra. Hence, paravector spinor valued solutions to the Cauchy-Riemann equationsforms a bigger function class, and the class of R-regular solutions is M R ( { f : ∂ x f I = 0 } = { f : [ ∂ x f I ] j = 0 , j = 0 , , . . . , } = { f : [ ∂ x f I ] j = 0 , j = 0 , , } . Equality of the latter two function classes follows from Lemma 5.1. An example showing that theinclusion is strict: if f = x e − x e , then ∂ x f = e e − e e , but [ ∂ x f I ] j = 0 for j = 0 , , .12 onclusion The idea of this paper is to study differences between octonion and Clifford analyses. This leads usto observe the fundamental difference between octonion regular and Clifford monogenic functions.The structure of octonion regular functions is studied by comparing left-, right-, B -, and R -regularfunctions. The existence of these classes is a consequence of different algebraic properties of thealgebras. In the heart of octonion analysis is the study of the properties of these function classesand their relations, which distinguishes it essentially from Clifford analysis. References [1] Delanghe, R., Sommen, F., Souček, V., Clifford algebra and spinor-valued functions . Mathe-matics and its Applications, 53. Kluwer Academic Publishers Group, Dordrecht, 1992.[2] Dentoni, P., Sce, M., Funzioni regolari nellalgebra di Cayley , Rend. Semin. Mat. Univ. Padova50, 251–267 (1973)[3] Gürlebeck, K., and Sprössig, W., Quaternionic analysis and elliptic boundary value problems. International Series of Numerical Mathematics, 89. Birkhäuser Verlag, Basel, 1990.[4] Harvey, F. Spinors and calibrations. Perspectives in Mathematics, 9. Academic Press, Boston,1990.[5] Kauhanen, J., and Orelma, H., Cauchy-Riemann operators in octonionic analysis. Adv. Appl.Clifford Algebr. 28 (2018), no. 1, Art. 1.[6] Kauhanen, J., and Orelma, H., Some theoretical remarks of octonionic analysis. 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