Monogenic functions in finite-dimensional commutative associative algebras
aa r X i v : . [ m a t h . C V ] M a r Monogenic functions in finite-dimensionalcommutative associative algebras
V. S. Shpakivskyi
Let A mn be an arbitrary n -dimensional commutative associative algebra over the field of complexnumbers with m idempotents. Let e = 1 , e , . . . , e k with ≤ k ≤ n be elements of A mn whichare linearly independent over the field of real numbers. We consider monogenic (i. e. continuous anddifferentiable in the sense of Gateaux) functions of the variable P kj =1 x j e j , where x , x , . . . , x k are real,and obtain a constructive description of all mentioned functions by means of holomorphic functions ofcomplex variables. It follows from this description that monogenic functions have Gateaux derivatives ofall orders. The present article is generalized of the author’s paper [1], where mentioned results are obtainedfor k = 3 . Apparently, W. Hamilton (1843) made the first attempts to construct an algebraassociated with the three-dimensional Laplace equation ∆ u ( x, y, z ) := (cid:18) ∂ ∂x + ∂ ∂y + ∂ ∂z (cid:19) u ( x, y, z ) = 0 (1)meaning that components of hypercomplex functions satisfy the equation (1). Heconstructed an algebra of noncommutative quaternions over the field of real numbers R and made a base for developing the hypercomplex analysis.C. Segre [2] constructed an algebra of commutative quaternions over the field R that can be considered as a two-dimensional commutative semi-simple algebraof bicomplex numbers over the field of complex numbers C . M. Futagawa [3] andJ. Riley [4] obtained a constructive description of analytic function of a bicomplexvariable, namely, they proved that such an analytic function can be constructed withan use of two holomorphic functions of complex variables.F. Ringleb [5] and S. N. Volovel’skaya [6, 7] succeeded in developing a functiontheory for noncommutative algebras with unit over the real or complex fields, bypursuing a definition of the differential of a function on such an algebra suggestedby Hausdorff in [8]. These definitions make the a priori severe requirement thatthe coordinates of the function have continuous first derivatives with respect to thecoordinates of the argument element. Namely, F. Ringleb [5] considered an arbitraryfinite-dimensional associative (commutative or not) semi-simple algebra over thefield R . For given class of functions which maps the mentioned algebra onto itself, heobtained a constructive description by means of real and complex analytic functions.. N. Volovel’skaya developed the Hausdorff’s idea defining the monogenic functi-ons on non-semisimple associative algebras and she generalized the Ringleb’s resultsfor such algebras. In the paper [6] was obtained a constructive description ofmonogenic functions in a special three-dimensional non-commutative algebra overthe field R . The results of paper [6] were generalized in the paper [7] whereVolovel’skaya obtained a constructive description of monogenic functions in non-semisimple associative algebras of the first category over R .A relation between spatial potential fields and analytic functions given incommutative algebras was established by P. W. Ketchum [9] who shown that everyanalytic function Φ( ζ ) of the variable ζ = xe + ye + ze satisfies the equation(1) in the case where the elements e , e , e of a commutative algebra satisfy thecondition e + e + e = 0 , (2)because ∂ Φ ∂x + ∂ Φ ∂y + ∂ Φ ∂z ≡ Φ ′′ ( ζ ) ( e + e + e ) = 0 , (3)where Φ ′′ := (Φ ′ ) ′ and Φ ′ ( ζ ) is defined by the equality d Φ = Φ ′ ( ζ ) dζ .We say that a commutative associative algebra A is harmonic (cf. [9, 10, 11]) ifin A there exists a triad of linearly independent vectors { e , e , e } satisfying theequality (2) with e k = 0 for k = 1 , , . We say also that such a triad { e , e , e } is harmonic .P. W. Ketchum [9] considered the C. Segre algebra of quaternions [2] as anexample of harmonic algebra.Further M. N. Ro¸scule¸t establishes a relation between monogenic functions incommutative algebras and partial differential equations. He defined monogenic functions f of the variable w by the equality df ( w ) dw = 0 . So, in the paper[12] M. N. Ro¸scule¸t proposed a procedure for constructing an infinite-dimensionaltopological vector space with commutative multiplication such that monogenicfunctions in it are the all solutions of the equation X α + α + ... + α p = N C α ,α ,...,α p ∂ N Φ ∂x α ∂x α . . . ∂x α p p = 0 , (4)with C α ,α ,...,α p ∈ R . In particular, such infinite-dimensional topological vectorspace are constructed for the Laplace equation (3). In the paper [13] Ro¸scule¸t findsa certain connection between monogenic functions in commutative algebras andsystems of partial differential equations.I. P. Mel’nichenko proposed for describing solutions of the equation (4) to usehypercomplex functions differentiable in the sense of Gateaux, since in this casethe conditions of monogenic are the least restrictive. He started to implementthis approach with respect to the thee-dimensional Laplace equation (3) (see [10]).Mel’nichenko proved that there exist exactly three-dimensional harmonic algebraswith unit over the field C (see [10, 14, 11]).In the paper [15], the authors develop the Melnichenko’s idea for the equation(4), and considered several examples. he investigation of partial differential equations using the hypercomplexmethods is effective if hypercomplex monogenic (in any sense) functions can beconstructed explicitly. On this way the following results are obtained.Constructive descriptions of monogenic (i. e. continuous and differentiable inthe sense of Gateaux) functions taking values in the mentioned three-dimensionalharmonic algebras by means three corresponding holomorphic functions of thecomplex variable are obtained in the papers [16, 17, 18]. Such descriptions make itpossible to prove the infinite differentiability in the sense of Gateaux of monogenicfunctions and integral theorems for these functions that are analogous to classicaltheorems of the complex analysis (see, e. g., [19, 20]).Furthermore, constructive descriptions of monogenic functions taking values inspecial n -dimensional commutative algebras by means n holomorphic functions ofcomplex variables are obtained in the papers [21, 22].In the paper [1], by author is obtained a constructive description of all monogenicfunctions of the variable x e + x e + x e taking values in an arbitrary n -dimensional commutative associative algebra with unit by means of holomorphicfunctions of complex variables. It follows from this description that monogenicfunctions have Gateaux derivatives of all orders.In this paper we extend the results of the paper [1] to monogenic functions of thevariable k P r =1 x r e r , where ≤ k ≤ n . A mn Let N be the set of natural numbers. We fix the numbers m, n ∈ N such that m ≤ n . Let A mn be an arbitrary commutative associative algebra with unit over thefield of complex number C . E. Cartan [23, p. 33] proved that there exist a basis { I r } nr =1 in A mn satisfying the following multiplication rules:1. ∀ r, s ∈ [1 , m ] ∩ N : I r I s = ( if r = s,I r if r = s ; ∀ r, s ∈ [ m + 1 , n ] ∩ N : I r I s = n P p =max { r,s } +1 Υ sr,p I p ;3. ∀ s ∈ [ m + 1 , n ] ∩ N ∃ ! u s ∈ [1 , m ] ∩ N ∀ r ∈ [1 , m ] ∩ N : I r I s = ( if r = u s ,I s if r = u s . (5)Moreover, the structure constants Υ sr,p ∈ C satisfy the associativity conditions:(A 1). ( I r I s ) I p = I r ( I s I p ) ∀ r, s, p ∈ [ m + 1 , n ] ∩ N ;(A 2). ( I u I s ) I p = I u ( I s I p ) ∀ u ∈ [1 , m ] ∩ N ∀ s, p ∈ [ m + 1 , n ] ∩ N .Obviously, the first m basic vectors { I u } mu =1 are idempotents and form a semi-simple subalgebra of the algebra A mn . The vectors { I r } nr = m +1 form a nilpotent ubalgebra of the algebra A mn . The element P mu =1 I u is the unit of A mn .In the cases where A mn has some specific properties, the following propositionsare true. Proposition 1 [1].
If there exists the unique u ∈ [1 , m ] ∩ N such that I u I s = I s for all s = m + 1 , . . . , n , then the associativity condition (A 2) is satisfied. Thus, under the conditions of Proposition 1, the associativity condition (A 1)is only required. It means that the nilpotent subalgebra of A mn with the basis { I r } nr = m +1 can be an arbitrary commutative associative nilpotent algebra of di-mension n − m . We note that such nilpotent algebras are fully described for thedimensions , , in the paper [24], and some four-dimensional nilpotent algebrascan be found in the papers [25], [26]. Proposition 2 [1].
If all u r are different in the multiplication rule , then I s I p = 0 for all s, p = m + 1 , . . . , n . Thus, under the conditions of Proposition 2, the multiplication table of the ni-lpotent subalgebra of A mn with the basis { I r } nr = m +1 consists only of zeros, and allassociativity conditions are satisfied.The algebra A mn contains m maximal ideals I u := ( n X r =1 , r = u λ r I r : λ r ∈ C ) , u = 1 , , . . . , m, and their intersection is the radical R := n n X r = m +1 λ r I r : λ r ∈ C o . Consider m linear functionals f u : A mn → C satisfying the equalities f u ( I u ) = 1 , f u ( ω ) = 0 ∀ ω ∈ I u , u = 1 , , . . . , m. Inasmuch as the kernel of functional f u is the maximal ideal I u , this functional isalso continuous and multiplicative (see [27, p. 147]). Let us consider the vectors e = 1 , e , . . . , e k in A mn , where ≤ k ≤ n , and thesevectors are linearly independent over the field of real numbers R (see [22]). It meansthat the equality k X j =1 α j e j = 0 , α j ∈ R , holds if and only if α j = 0 for all j = 1 , , . . . , k .Let the vectors e = 1 , e , . . . , e k have the following decompositions with respectto the basis { I r } nr =1 : e = m X r =1 I r , e j = n X r =1 a jr I r , a jr ∈ C , j = 2 , , . . . , k. (6) et ζ := k P j =1 x j e j , where x j ∈ R . It is obvious that ξ u := f u ( ζ ) = x + k X j =2 x j a ju , u = 1 , , . . . , m. Let E k := { ζ = k P j =1 x j e j : x j ∈ R } be the linear span of vectors e = 1 , e , . . . , e k over the field R .Let Ω be a domain in E k . With a domain Ω ⊂ E k we associate the domain Ω R := n ( x , x , . . . , x k ) ∈ R k : ζ = k X j =1 x j e j ∈ Ω o in R k .We say that a continuous function Φ : Ω → A mn is monogenic in Ω if Φ isdifferentiable in the sense of Gateaux in every point of Ω , i. e. if for every ζ ∈ Ω there exists an element Φ ′ ( ζ ) ∈ A mn such that lim ε → (Φ( ζ + εh ) − Φ( ζ )) ε − = h Φ ′ ( ζ ) ∀ h ∈ E k . (7) Φ ′ ( ζ ) is the Gateaux derivative of the function Φ in the point ζ .Consider the decomposition of a function Φ : Ω → A mn with respect to the basis { I r } nr =1 : Φ( ζ ) = n X r =1 U r ( x , x , . . . , x k ) I r . (8)In the case where the functions U r : Ω R → C are R -differentiable in Ω R , i. e.for every ( x , x , . . . , x k ) ∈ Ω R , U r ( x + ∆ x , x + ∆ x , . . . , x k + ∆ x k ) − U r ( x , x , . . . , x k ) == k X j =1 ∂U r ∂x j ∆ x j + o vuut k X j =1 (∆ x j ) , k X j =1 (∆ x j ) → , the function Φ is monogenic in the domain Ω if and only if the following Cauchy –Riemann conditions are satisfied in Ω : ∂ Φ ∂x j = ∂ Φ ∂x e j for all j = 2 , , . . . , k. (9) Let b := n P r =1 b r I r ∈ A mn , where b r ∈ C , and we note that f u ( b ) = b u , u =1 , , . . . , m . It follows form the Lemmas 1, 3 of [1] that b − = m X u =1 b u I u + n X s = m +1 s − m +1 X r =2 e Q r,s b ru s I s . (10) here e Q r,s are determined by the following recurrence relations: e Q ,s := b s , e Q r,s = s − X q = r + m − e Q r − ,q e B q, s , r = 3 , , . . . , s − m + 1 , (11) e B q,s := s − X p = m +1 b p Υ pq,s , p = m + 2 , m + 3 , . . . , n, (12)and the natural numbers u s are defined in the rule 3 of the multiplication table ofalgebra A mn .In the next lemma we find an expansion of the resolvent ( te − ζ ) − . Лемма 1.
An expansion of the resolvent is of the form ( te − ζ ) − = m X u =1 t − ξ u I u + n X s = m +1 s − m +1 X r =2 Q r,s ( t − ξ u s ) r I s (13) ∀ t ∈ C : t = ξ u , u = 1 , , . . . , m, where the coefficients Q r,s are determined by the following recurrence relations: Q ,s = T s , Q r,s = s − X q = r + m − Q r − ,q B q, s , r = 3 , , . . . , s − m + 1 , (14) with T s := k X j =2 x j a js , B q,s := s − X p = m +1 T p Υ pq,s , p = m + 2 , m + 3 , . . . , n, (15) and the natural numbers u s are defined in the rule of the multiplication table ofalgebra A mn . Proof.
Taking into account the decomposition te − ζ = m X u =1 ( t − ξ u ) I u − n X r = m +1 k X j =2 x j a js I r , we conclude that the relation (13) follows directly from the equality (10) in whichinstead of b u , u = 1 , , . . . , m it should be used the expansion t − ξ u , and insteadof b s , s = m +1 , m +2 , . . . , n it should be used the expansion k P j =2 x j a js . The lemmais proved.It follows from Lemma 1 that the points ( x , x , . . . , x k ) ∈ R k corresponding tothe noninvertible elements ζ = k P j =1 x j e j form the set M R u : x + k P j =2 x j Re a ju = 0 , k P j =2 x j Im a ju = 0 , u = 1 , , . . . , m n the k -dimensional space R k . Also we consider the set M u := { ζ ∈ E k : f u ( ζ ) =0 } for u = 1 , , . . . , m . It is obvious that the set M R u ⊂ R k is congruent with theset M u ⊂ E k . We say that a domain Ω ⊂ E k is convex with respect to the set of directions M u if Ω contains the segment { ζ + α ( ζ − ζ ) : α ∈ [0 , } for all ζ , ζ ∈ Ω such that ζ − ζ ∈ M u .Denote f u ( E k ) := { f u ( ζ ) : ζ ∈ E k } . In what follows, we make the followingessential assumption: f u ( E k ) = C for all u = 1 , , . . . , m . Obviously, it holds if andonly if for every fixed u = 1 , , . . . , m at least one of the numbers a u , a u , . . . , a ku belongs to C \ R . Лемма 2.
Suppose that a domain Ω ⊂ E k is convex with respect to the setof directions M u and f u ( E k ) = C for all u = 1 , , . . . , m . Suppose also that afunction Φ : Ω → A mn is monogenic in the domain Ω . If points ζ , ζ ∈ Ω suchthat ζ − ζ ∈ M u , then Φ( ζ ) − Φ( ζ ) ∈ I u . (16) Proof.
Inasmuch as f u ( E k ) = C , then there exists an element e ∗ ∈ E k suchthat f u ( e ∗ ) = i . Consider the lineal span E ∗ := { ζ = xe ∗ + ye ∗ + ze ∗ : x, y, z ∈ R } of the vectors e ∗ := 1 , e ∗ , e ∗ := ζ − ζ and denote Ω ∗ := Ω ∩ E ∗ .Now, the relations ( 16 ) can be proved in such a way as Lemma 2.1 [16], in theproof of which one must take Ω ∗ , f u , { αe ∗ : α ∈ R } instead of Ω ζ , f, L , respectively.Lemma 2 is proved.Let a domain Ω ⊂ E k be convex with respect to the set of directions M u , u = 1 , , . . . , m . By D u we denote that domain in C onto which the domain Ω ismapped by the functional f u .We introduce the linear operators A u , u = 1 , , . . . , m , which assign holomorphicfunctions F u : D u → C to every monogenic function Φ : Ω → A mn by the formula F u ( ξ u ) = f u (Φ( ζ )) , (17)where ξ u = f u ( ζ ) ≡ x + k P j =2 x j a ju and ζ ∈ Ω . It follows from Lemma 2 that thevalue F u ( ξ u ) does not depend on a choice of a point ζ for which f u ( ζ ) = ξ u .Now, similar to proof of Lemma 5 [1] can be proved the following statement. Лемма 3.
Suppose that a domain Ω ⊂ E k is convex with respect to the set ofdirections M u and f u ( E k ) = C for all u = 1 , , . . . , m . Suppose also that for anyfixed u = 1 , , . . . , m , a function F u : D u → C is holomorphic in a domain D u and Γ u is a closed Jordan rectifiable curve in D u which surrounds the point ξ u andcontains no points ξ q , q = 1 , , . . . , m , q = u . Then the function Ψ u ( ζ ) := I u Z Γ u F u ( t )( te − ζ ) − dt (18) is monogenic in the domain Ω . емма 4. Suppose that a domain Ω ⊂ E k is convex with respect to the set ofdirections M u and f u ( E k ) = C for all u = 1 , , . . . , m . Suppose also that a function V : Ω R → C satisfies the equalities ∂V∂x = ∂V∂x a u , ∂V∂x = ∂V∂x a u , . . . , ∂V∂x k = ∂V∂x a ku (19) in Ω R . Then V is a holomorphic function of the variable ξ u = f u ( ζ ) = x + k P j =2 x j a ju in the domain D u . Proof.
We first separate the real and the imaginary part of the expression ξ u = x + k X j =2 x j Re a ju + i k X j =2 x j Im a ju =: τ u + iη u (20)and note that the equalities (19) yield ∂V∂η u Im a u = i ∂V∂τ u Im a u , . . . , ∂V∂η u Im a ku = i ∂V∂τ u Im a ku . (21)It follows from the condition f u ( E k ) = C that at least one of the numbers Im a u , Im a u , . . . , Im b u is not equal to zero. Therefore, using (21), we get ∂V∂η u = i ∂V∂τ u . (22)Now we prove that V ( x ′ , x ′ , . . . , x ′ k ) = V ( x ′′ , x ′′ , . . . , x ′′ k ) for points ( x ′ , x ′ , . . . , x ′ k ) , ( x ′′ , x ′′ , . . . , x ′′ k ) ∈ Ω such that the segment that connects these poi-nts is parallel to a straight line L u ⊂ M R u . To this end we use considerations withthe proof of Lemma 2. Since f u ( E k ) = C , then there exists an element e ∗ ∈ E k suchthat f u ( e ∗ ) = i . Consider the lineal span E ∗ := { ζ = xe ∗ + ye ∗ + ze ∗ : x, y, z ∈ R } of the vectors e ∗ := 1 , e ∗ , e ∗ := ζ ′ − ζ ′′ , where ζ ′ := k P j =1 x ′ j e j , ζ ′′ := k P j =1 x ′′ j e j ,and introduce the denotation Ω ∗ := Ω ∩ E ∗ .Now, the relation V ( x ′ , x ′ , . . . , x ′ k ) = V ( x ′′ , x ′′ , . . . , x ′′ k ) can be proved in such away as Lemma 6 [1], in the proof of which one must take Ω ∗ , { αe ∗ : α ∈ R } insteadof Ω ζ , L , respectively. The lemma is proved.Thus, a function V : Ω R → C of the form V ( x , x , . . . , x k ) := F ( ξ u ) , where F ( ξ u ) is an arbitrary function holomorphic in the domain D u , is a general solutionof the system (19). The lemma is proved. Теорема 1.
Suppose that a domain Ω ⊂ E k is convex with respect to the setof directions M u and f u ( E k ) = C for all u = 1 , , . . . , m . Then every monogenicfunction Φ : Ω → A mn can be expressed in the form Φ( ζ ) = m X u =1 I u πi Z Γ u F u ( t )( te − ζ ) − dt + n X s = m +1 I s πi Z Γ us G s ( t )( te − ζ ) − dt, (23) where F u and G s are certain holomorphic functions in the domains D u and D u s ,respectively, and Γ q is a closed Jordan rectifiable curve in D q which surrounds thepoint ξ q and contains no points ξ ℓ , ℓ, q = 1 , , . . . , m , ℓ = q . roof. We set F u := A u Φ , u = 1 , , . . . , m. (24)Let us show that the values of monogenic function Φ ( ζ ) := Φ( ζ ) − m X u =1 I u πi Z Γ u F u ( t )( te − ζ ) − dt (25)belong to the radical R , i. e. Φ ( ζ ) ∈ R for all ζ ∈ Ω . As a consequence of theequality (13), we have the equality I u πi Z Γ u F u ( t )( te − ζ ) − dt = I u πi Z Γ u F u ( t ) t − ξ u dt ++ 12 πi n X s = m +1 s − m +1 X r =2 Z Γ u F u ( t ) Q r,s ( t − ξ u s ) r dt I s I u , from which we obtain the equality f u m X u =1 I u πi Z Γ u F u ( t )( te − ζ ) − dt = F u ( ξ u ) . (26)Operating onto the equality (25) by the functional f u and taking into account therelations (17), (24), (26), we get the equality f u (Φ ( ζ )) = F u ( ξ u ) − F u ( ξ u ) = 0 for all u = 1 , , . . . , m , i. e. Φ ( ζ ) ∈ R .Therefore, the function Φ is of the form Φ ( ζ ) = n X s = m +1 V s ( x , x , . . . , x k ) I s , (27)where V s : Ω R → C , and the Cauchy – Riemann conditions (9) are satisfied with Φ = Φ . Substituting the expressions (6), (27) into the equality (9), we obtain n X s = m +1 ∂V s ∂x I s = n X s = m +1 ∂V s ∂x I s n X r =1 a r I r , ... n X s = m +1 ∂V s ∂x k I s = n X s = m +1 ∂V s ∂x I s n X r =1 a kr I r . (28)Equating the coefficients of I m +1 in these equalities, we obtain the following systemof equations for determining the function V m +1 ( x , x , . . . , x k ) : ∂V m +1 ∂x = ∂V m +1 ∂x a u m +1 , . . . , ∂V m +1 ∂x k = ∂V m +1 ∂x a k u m +1 . t follows from Lemma 4 that V m +1 ( x , x , . . . , x k ) ≡ G m +1 ( ξ u m +1 ) , where G m +1 isa function holomorphic in the domain D u m +1 . Therefore, Φ ( ζ ) = G m +1 ( ξ u m +1 ) I m +1 + n X s = m +2 V s ( x , x , . . . , x k ) I s . (29)Due to the expansion (13), we have the representation I m +1 πi Z Γ um +1 G m +1 ( t )( te − ζ ) − dt = G m +1 ( ξ u m +1 ) I m +1 + Ψ( ζ ) , (30)where Ψ( ζ ) is a function with values in the set (cid:8) P ns = m +2 α s I s : α s ∈ C (cid:9) .Now, consider the function Φ ( ζ ) := Φ ( ζ ) − I m +1 πi Z Γ um +1 G m +1 ( t )( te − ζ ) − dt. In view of the relations (29), (30), Φ can be represented in the form Φ ( ζ ) = n X s = m +2 e V s ( x , x , . . . , x k ) I s , where e V s : Ω R → C .Inasmuch as Φ is a monogenic function in Ω , the functions e V m +2 , e V m +3 , . . . , e V n satisfy the system (28), where V m +1 ≡ , V s = e V s for s = m + 2 , m + 3 , . . . , n .Therefore, similarly to the function V m +1 ( x , x , . . . , x k ) ≡ G m +1 ( ξ u m +1 ) , the functi-on e V m +2 satisfies the equations ∂ e V m +2 ∂x = ∂ e V m +2 ∂x a u m +2 , . . . , ∂ e V m +2 ∂x k = ∂ e V m +2 ∂x a k u m +2 and is of the form e V m +2 ( x , x , . . . , x k ) ≡ G m +2 ( ξ u m +2 ) , where G m +2 is a functionholomorphic in the domain D u m +2 .In such a way, step by step, considering the functions Φ j ( ζ ) := Φ j − ( ζ ) − I m + j πi Z Γ um + j G m + j ( t )( te − ζ ) − dt for j = 2 , , . . . , n − m − , we get the representation (23) of the function Φ . Thetheorem is proved.Taking into account the expansion (13), one can rewrite the equality (23) in thefollowing equivalent form: Φ( ζ ) = m X u =1 F u ( ξ u ) I u + n X s = m +1 s − m +1 X r =2 r − Q r,s F ( r − u s ( ξ u s ) I s ++ n X q = m +1 G q ( ξ u q ) I q + n X q = m +1 n X s = m +1 s − m +1 X r =2 r − Q r,s G ( r − q ( ξ u q ) I q I s . (31) hus, the equalities (23) and (31) specify methods to construct explicitly anymonogenic functions Φ : Ω → A mn using n corresponding holomorphic functions ofcomplex variables.The following statement follows immediately from the equality (31) in which theright-hand side is a monogenic function in the domain Π := { ζ ∈ E k : f u ( ζ ) = D u , u = 1 , , . . . , m } . Теорема 2.
Let a domain Ω ⊂ E k is convex with respect to the set of directions M u and f u ( E k ) = C for all u = 1 , , . . . , m . Then every monogenic function Φ : Ω → A mn can be continued to a function monogenic in the domain Π . The next statement is a fundamental consequence of the equality (31), and it istrue for an arbitrary domain Ω . Теорема 3.
Let f u ( E k ) = C for all u = 1 , , . . . , m . Then for every monogenicfunction Φ : Ω → A mn in an arbitrary domain Ω , the Gateaux r -th derivatives Φ ( r ) are monogenic functions in Ω for all r . The proof is completely analogous to the proof of Theorem 4 [16].Using the integral expression (23) of monogenic function
Φ : Ω → A mn in thecase where a domain Ω is convex with respect to the set of directions M u , u =1 , , . . . , m , we obtain the following expression for the Gateaux r -th derivative Φ ( r ) : Φ ( r ) ( ζ ) = m X u =1 I u r !2 πi Z Γ u F u ( t ) (cid:16) ( te − ζ ) − (cid:17) r +1 dt ++ n X s = m +1 I s r !2 πi Z Γ us G s ( t ) (cid:16) ( te − ζ ) − (cid:17) r +1 dt ∀ ζ ∈ Ω . We note that in the cases where the algebra A mn has some specific properties (forinstance, properties described in Propositions 1 and 2), it is easy to simplify theform of the equality (31). In the case considered in Proposition 1, the following equalities hold: u m +1 = u m +2 = . . . = u n =: η . In this case the representation (31) takes the form Φ( ζ ) = m X u =1 F u ( ξ u ) I u + n X s = m +1 s − m +1 X r =2 r − Q r,s F ( r − η ( ξ η ) I s ++ n X s = m +1 G s ( ξ η ) I s + n X q = m +1 n X s = m +1 s − m +1 X r =2 r − Q r,s G ( r − q ( ξ η ) I s I q . (32)The formula (32) generalizes representations of monogenic functions in boththree-dimensional harmonic algebras (see [16, 17, 18]) and specific n -dimensional lgebras (see [21, 22]) to the case of algebras more general form and to a variable ofmore general form. In the case considered in Proposition 2, the representation (23) takes the form Φ( ζ ) = m X u =1 F u ( ξ u ) I u + n X s = m +1 G s ( ξ u s ) I s + n X s = m +1 T s F ′ u s ( ξ u s ) I s . (33)The formula (33) generalizes representations of monogenic functions in both athree-dimensional harmonic algebra with one-dimensional radical (see [17]) andsemi-simple algebras (see [18, 22]) to the case of algebras more general form andto a variable of more general form. In the case where n = m , the algebra A nn is semi-simple and contains nonilpotent subalgebra. Then the formulae (32), (33) take the form Φ( ζ ) = n X u =1 F u ( ξ u ) I u , because there are no vectors { I k } nk = m +1 . This formula was obtained in the paper[22]. Consider the following linear partial differential equation with constant coefficients: L N U ( x , x , . . . , x k ) := X α + α ... + α k = N C α ,α ,...,α k ∂ N Φ ∂x α ∂x α . . . ∂x α k k = 0 , (34)If a function Φ( ζ ) is N -times differentiable in the sense of Gateaux in everypoint of Ω , then ∂ α + α + ... + α k Φ ∂x α ∂x α . . . ∂x α k k == e α e α . . . e α k k Φ ( α + α + ... + α k ) ( ζ ) = e α e α . . . e α k k Φ ( N ) ( ζ ) . Therefore, due to the equality L N Φ( ζ ) = Φ ( N ) ( ζ ) X α + α + ... + α k = N C α ,α ,...,α k e α e α . . . e α k k , (35)every N -times differentiable in the sense of Gateaux in Ω function Φ satisfies theequation L N Φ( ζ ) = 0 everywhere in Ω if and only if X α + α + ... + α k = N C α ,α ,...,α k e α e α . . . e α k k = 0 . (36)Accordingly, if the condition (36) is satisfied, then the real-valued components Re U k ( x , x , . . . , x k ) and Im U k ( x , x , . . . , x k ) of the decomposition (8) are soluti-ons of the equation (34). n the case where f u ( E k ) = C for all u = 1 , , . . . , m , it follows from Theorem 3that the equality (35) holds for every monogenic function Φ : Ω → A mn .Thus, to construct solutions of the equation (34) in the form of componentsof monogenic functions, we must to find k linearly independent over the field R vectors ( 6 ) satisfying the characteristic equation (36) and to verify the condition: f u ( E k ) = C for all u = 1 , , . . . , m . Then, the formula (23) gives a constructivedescription of all mentioned monogenic functions.In the next theorem, we assign a special class of equations (34) for which f u ( E k ) = C for all u = 1 , , . . . , m . Let us introduce the polynomial P ( b , b , . . . , b k ) := X α + α + ... + α k = N C α ,α ,...,α k b α b α . . . b α k k . (37) Теорема 4.
Suppose that there exist linearly independent over the field R vectors e = 1 , e , . . . , e k in A mn of the form ( 6 ) that satisfy the equality ( 36 ) .If P ( b , b , . . . , b k ) = 0 for all real b , b , . . . , b k , then f u ( E k ) = C for all u =1 , , . . . , m . Proof.
Using the multiplication table of A mn , we obtain the equalities e α = m X u =1 a α u I u + Ψ R , . . . , e α k k = m X u =1 a α k ku I u + Θ R , where Ψ R , . . . , Θ R ∈ R . Now the equality (36) takes the form X α + α + ... + α k = N C α ,α ,...,α k m X u =1 a α u . . . a α k ku I u + e Ψ R ! = 0 , (38)where e Ψ R ∈ R . Moreover, due to the assumption that the vectors e , e , . . . , e k of the form ( 6 ) satisfy the equality ( 36 ) , there exist complex coefficients a jr for j = 1 , , . . . , k, r = 1 , , . . . , n that satisfy the equality (38).It follows from the equality (38) that X α + α + ... + α k = N C α ,α ,...,α k a α u . . . a α k ku = 0 , u = 1 , , . . . , m. (39)Since P ( b , b , . . . , b k ) = 0 for all { b , b , . . . , b k } ⊂ R , the equalities (39) can besatisfied only if for each u = 1 , , . . . , m at least one of the numbers a u , a u , . . . , a ku belongs to C \ R that implies the relation f u ( E k ) = C for all u = 1 , , . . . , m . Thetheorem is proved.We note that if P ( b , b , . . . , b k ) = 0 for all { b , b , . . . , b k } ⊂ R , then C N, , ,..., =0 because otherwise P ( b , b , . . . , b k ) = 0 for b = b = . . . = b k = 0 .Since the function P ( b , b , . . . , b k ) is continuous on R k , the condition P ( b , b , . . . , b k ) = 0 means either P ( b , b , . . . , b k ) > or P ( b , b , . . . , b k ) < forall real b , b , . . . , b k . Therefore, it is obvious that for any equation (34) of elliptictype, the condition P ( b , b , . . . , b k ) = 0 is always satisfied for all { b , b , . . . , b k } ⊂ R . At the same time, there are equations (34) for which P ( b , b , . . . , b k ) > for all { b , b , . . . , b k } ⊂ R , but which are not elliptic. For example, such is the equation ∂ u∂x + ∂ u∂x ∂x + ∂ u∂x ∂x + ∂ u∂x ∂x = 0 onsidered in R . Лiтература [1] V. S. Shpakivskyi,
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