Open AccessArticle Solutions to Time-Harmonic Maxwell Equations via Transmutation Theory 1 Department of Quantitative Methods, Universidad Anáhuac Campus Querétaro, Circuito Universidades I, Kilómetro 7, Fracción 2, El Marqués 76246, Querétaro, Mexico 2 Department of Basic Sciences, TecNM Campus Querétaro, Av. Tecnológico S/N, Querétaro 76000, Querétaro, Mexico Mathematics 2026, 14(12), 2055; https://doi.org/10.3390/math14122055 (registering DOI) Submission received: 17 April 2026 / Revised: 20 May 2026 / Accepted: 2 June 2026 / Published: 9 June 2026 We construct solutions of a time-harmonic Maxwell-type system within the framework of the algebra of complex quaternions. Using quaternionic analysis, we establish a connection between this system and certain first-order differential operators whose kernels consist of monogenic functions. Building on known representations of harmonic and monogenic functions, we develop a constructive procedure based on transmutation operators for generating explicit solutions of the equations ( D ବ୍ଦ λ ) u = 0 , and consequently of the corresponding Maxwell system. This approach provides a systematic method for reconstructing electromagnetic fields from harmonic and monogenic data, yielding an explicit link between quaternionic operator theory, transmutation methods, and the classical formulation of Maxwell equations. transmutation theory; Maxwell equations; Helmholtz equation; harmonic function; complex quaternions MSC: 35A22; 35A35; 35C10; 30G35; 35Q60; 78A25 Within this framework, properties analogous to those of holomorphic functions are obtained: generalized versions of the Cauchy integral formula hold, monogenic solutions are harmonic, and they admit power series representations. 2. Preliminaries We denote by H ( C ) the algebra of complex quaternions, i.e., elements of the form q = q 0 + q 1 e 1 + q 2 e 2 + q 3 e 3 , q j ∈ C , where the quaternionic units e j satisfy e j 2 = − 1 , e j e k = − e k e j , j ≠ k , e 1 e 2 = e 3 , i e j = e j i , ( ie j ) 2 = 1 , i ∈ C denotes the imaginary unit. Each element q ∈ H ( C ) can be uniquely written as q = q 0 + q → , q → = ∑ j = 1 3 q j e j , where q 0 ∈ C is the scalar part and q → is the vector part. We write Sc ( q ) = q 0 , Vec ( q ) = q → . The product of two elements p , q ∈ H ( C ) is given by p q = p 0 q 0 − p → · q → + p 0 q → + q 0 p → + p → ୍ଠ q → , where · and × denote the standard dot and cross products in R 3 . We consider the elements 1 + i e 3 2 , 1 − i e 3 2 , which satisfy 1 ବ୍ଦ i e 3 2 2 = 1 ବ୍ଦ i e 3 2 , 1 + i e 3 2 1 − i e 3 2 = 0 . This is a difference between the classical quaternions H , which form a division algebra and hence contain no zero divisors, and the complex quaternions H ( C ) , where zero divisors do exist. A detailed characterization of zero divisors in H ( C ) can be found in [ 4]. Moreover, the projection operators associated with these elements were introduced in [ 4] and are defined as follows. P ବ୍ଦ : H ( C ) → H ( C ) , P ବ୍ଦ = M ( 1 ବ୍ଦ i e 3 ) / 2 , where M q ( p ) = p q denotes right multiplication by q. The operators P ବ୍ଦ satisfy the following properties: P + + P − = I , P + P − = 0 , ( P ବ୍ଦ ) 2 = P ବ୍ଦ . The norm | q | = | q 0 | 2 + | q 1 | 2 + | q 2 | 2 + | q 3 | 2 does not satisfy | p q | = | p | | q | in general, but it holds that | p q | ≤ 2 | p | | q | . In particular, | P ବ୍ଦ q | ≤ | q | . 2.1. Monogenic Functions From now on, we write x → = ( x 1 , x 2 , x 3 ) = x 1 e 1 + x 2 e 2 + x 3 e 3 ∈ R 3 and let Ω ⊂ R 3 be an open set. The Moisil–Theodorescu operator [ 4] is defined as follows: D = e 1 ∂ 1 + e 2 ∂ 2 + e 3 ∂ 3 acts from the left on functions u = u 0 + u → ∈ C 1 ( Ω , H ( C ) ) . By expanding the expression, we get D u = − ( ∂ 1 u 1 + ∂ 2 u 2 + ∂ 3 u 3 ) + e 1 ( ∂ 1 u 0 + ∂ 2 u 3 − ∂ 3 u 2 ) + e 2 ( ∂ 2 u 0 − ∂ 1 u 3 + ∂ 3 u 1 ) + e 3 ( ∂ 3 u 0 + ∂ 1 u 2 − ∂ 2 u 1 ) , = − div u → + grad u 0 + curl u → . Definition 1.A function u ∈ C 1 ( Ω , H ( C ) ) is called left-monogenic (or simply, monogenic ) if D u = 0 . In this case, we write u ∈ Ker D . Let u = u 0 + u 1 e 1 + u 2 e 2 + u 3 e 3 = u 0 + u → . From the scalar and vector components in ( 3), we deduce the following conditions. D u = 0 ⇔ div u → = 0 , curl u → = − grad u 0 . Also since the Dirac operator satisfies − D 2 = Δ 3 , with Δ 3 the three-dimensional Laplacian, it follows that every component of a monogenic function is harmonic. An important observation is that, in general, if u is monogenic, then left multiplication by a quaternionic constant does not necessarily result in products that are monogenic. For example, consider u ( x → ) = x 2 − x 3 e 1 , which is a monogenic function, i.e., D u = 0 . However, multiplying u from the left by e 3 yields e 3 u = e 3 ( x 2 − x 3 e 1 ) = x 2 e 3 − x 3 e 3 e 1 = x 2 e 3 + x 3 e 2 , since e 3 e 1 = − e 2 . Applying the operator D, we obtain D ( e 3 u ) = D ( x 2 e 3 + x 3 e 2 ) = − 2 , which is nonzero. Hence, e 3 u is not monogenic. This example shows that Ker D is not invariant under left multiplication by arbitrary quaternionic constants. Lemma 1(Theorem 1, [ 4]) . Leibniz rule. Let Ω ⊆ R 3 be a domain and let u , v ∈ C 1 ( Ω , H ( C ) ) . ThenD [ u v ] = D [ u ] v + u ପ୍ତ D [ v ] − 2 ∑ i = 1 3 u i ∂ i v . By applying the Leibniz rule, the following lemma is obtained directly. Lemma 2([ 4]) . The Ker D is a quaternionic right linear space. This means that for u , v ∈ Ker D λ ∈ H ( C ) constant, ( u λ + v ) ∈ Ker D . In this sense, D [ P ବ୍ଦ u ] = P ବ୍ଦ [ D u ] , which shows that the operators P ବ୍ଦ commute with D. 2.2. Monogenic Polynomials Definition 2([ 2]) . The Grigor’ev polynomials P l , m are the homogeneousH -valued polynomials of degreen = l + m in the vector variablex → = ( x 1 , x 2 , x 3 ) ∈ R 3 , defined in [2] recursively byP l , m ( x → ) = P 1 , 0 ( x → ) P l − 1 , m ( x → ) + P 0 , 1 ( x → ) P l , m − 1 ( x → ) , l , m ≥ 0 , with the initial conditionsP 0 , 0 ( x → ) = 1 , P 1 , 0 ( x → ) = x 2 − x 3 e 1 , P 0 , 1 ( x → ) = x 1 + x 3 e 2 . The first cases of Grigor’ev polynomials are in Table 1. The significance of these polynomials is established in the following proposition. Proposition 1([ 2]) . Taylor series expansion. 1. For all l , m ∈ N 0 , the Grigor’ev polynomials P l , m are left-monogenic, that is,D P l , m ( x → ) = 0 . 2. Let u ∈ Ker D in a ball B R ( 0 ) . Then u admits the Taylor expansionu ( x → ) = ∑ n = 0 ∞ ∑ l + m = n P l , m ( x → ) c l , m , which converges uniformly in every ball B R ′ ( 0 ) with R ′ < R . The coefficients are given byc l , m = 1 n ! ∂ n u ( 0 ) ∂ 1 l ∂ 2 m , n = l + m . Definition 3.We say that a set Ω is star-shaped respect to the origin if t x → ∈ Ω whenever x → ∈ Ω t ∈ [ 0 , 1 ] . The next proposition give us a way to construct a monogenic function given an harmonic function. Proposition 2([ 3, 11]) . Let Ω be an open set in R 3 star-shaped with respect to the origin and let u 0 : Ω → R be harmonic in Ω . Then the complex quaternionic functionu ( x → ) = u 0 ( x → ) − Vec ∫ 0 1 t ( D u 0 ) ( t x → ) x → d t is monogenic in Ω . Proof.We prove that u is monogenic, i.e., D u = 0 . By linearity of the Dirac operator, D u ( x → ) = D u 0 ( x → ) − ∫ 0 1 t D ( D u 0 ) ( t x → ) ୍ଠ x → d t . The order of integration and differentiation can be interchanged since u 0 D u 0 have continuous partial derivatives in Ω . Since u 0 is scalar-valued by ( 3), we have D u 0 = ∇ u 0 , and for a vector field F → , D F → = − div F → + curl F → . Thus, D ( D u 0 ) ( t x → ) ୍ଠ x → = − div ∇ u 0 ( t x → ) ୍ଠ x → + curl ∇ u 0 ( t x → ) ୍ଠ x → . Using the identity div ( A → ୍ଠ B → ) = B → · ( curl A → ) − A → · ( curl B → ) , with A → = ∇ u 0 ( t x → ) B → = x → , we obtain − div ∇ u 0 ( t x → ) ୍ଠ x → = 0 , since curl ∇ u 0 = 0 curl x → = 0 . Therefore, D ( D u 0 ) ( t x → ) ୍ଠ x → = curl ∇ u 0 ( t x → ) ୍ଠ x → . Now we compute ∫ 0 1 t curl ∇ u 0 ( t x → ) ୍ଠ x → d t = ∫ 0 1 t ∇ ( x → · ∇ ) u 0 ( t x → ) + ∇ u 0 ( t x → ) − x → Δ u 0 ( t x → ) d t = ∫ 0 1 t ∇ ( x → · ∇ ) u 0 ( t x → ) + ∇ u 0 ( t x → ) d t ( since Δ u 0 = 0 ) = ∇ ∫ 0 1 t ( x → · ∇ ) u 0 ( t x → ) + u 0 ( t x → ) d t = ∇ ∫ 0 1 d d t t u 0 ( t x → ) d t = ∇ t u 0 ( t x → ) 0 1 = ∇ u 0 ( x → ) . Hence, D u ( x → ) = ∇ u 0 ( x → ) − ∇ u 0 ( x → ) = 0 . Therefore, u is monogenic. □ Example 1.Let u 0 ( x → ) = x 1 2 − x 2 2 be a harmonic function. Calculating D u 0 ( t x → ) = 2 t x 1 e 1 − 2 t x 2 e 2 , then Vec ∫ 0 1 t D u 0 ( t x → ) x → d t = Vec ∫ 0 1 t ( 2 t x 1 e 1 − 2 t x 2 e 2 ) ( x 1 e 1 + x 2 e 2 + x 3 e 3 ) d t , = ∫ 0 1 2 t 2 ( − x 2 x 3 e 1 − x 1 x 3 e 2 + 2 x 1 x 2 e 3 ) d t , = 2 3 ( − x 2 x 3 e 1 − x 1 x 3 e 2 + 2 x 1 x 2 e 3 ) . Finally the function u ( x → ) = x 1 2 − x 2 2 + 2 3 ( x 2 x 3 e 1 + x 1 x 3 e 2 − 2 x 1 x 2 e 3 ) is monogenic. 2.3. The λ -Monogenic Functions Let λ ∈ R . A function w ∈ C 1 ( Ω , H ( C ) ) is called λ - monogenic in Ω if it satisfies ( D + λ ) w = 0 in Ω . Using the decomposition of the Dirac operator given in ( 3), the equation ( D + λ ) w = 0 is equivalent to the system: ( D + λ ) w = 0 ⇔ div w → = λ w 0 , curl w → + λ w → = − grad w 0 . If w 0 = 0 , then w is a Beltrami field (or a vectorial λ-monogenic function, see [ 12]). Such fields are closely related to certain Maxwell systems (see Section 4). Define the operator V ବ୍ଦ : Ker ( D ବ୍ଦ λ ) ⟶ Ker ( D ବ୍ଦ λ ) ∩ C ( Ω , Vec H ( C ) ) , V ବ୍ଦ [ u ] = Vec u ବ୍ଦ 1 λ D ( Sc u ) . This operator transforms λ -monogenic functions into Beltrami fields in Ω . Note that if u → is a purely vectorial monogenic function, then V ବ୍ଦ [ u → ] = u → . The following decompositions hold (see [ 4]): Ker ( D + λ ) = P + Ker ( D + M i λ e 3 ) ⊕ P − Ker ( D − M i λ e 3 ) , Ker ( D − λ ) = P + Ker ( D − M i λ e 3 ) ⊕ P − Ker ( D + M i λ e 3 ) . Hence, every u ∈ Ker ( D − λ ) admits a unique representation u = P + u 1 + P − u 2 , with (note that M − i λ e 3 = − M i λ e 3 ) u 1 ∈ Ker ( D − M i λ e 3 ) , u 2 ∈ Ker ( D + M i λ e 3 ) . Interchanging the projected components, u ⟼ P − u 1 + P + u 2 , yields an element of Ker ( D + λ ) . This correspondence explains why it suffices to study the equations ( D + M ବ୍ଦ i λ e 3 ) u = 0 , since solutions of ( D ବ୍ଦ λ ) u = 0 can be reconstructed from them by means of the projections P ବ୍ଦ . An important property of D + λ D − λ is the the factorization of the Helmholtz operator ( Δ + λ 2 ) = ( D + λ ) ( D − λ ) . 3. Transmutation Theory Definition 4([ 13]) . Let X be a linear topological space and let X 1 ⊆ X be a linear subspace (not necessarily closed) and let A , B : X 1 → X be linear operators. A linear invertible operator T : X → X , such that X 1 is T invariant, is said to be a transmutation operator for the pair A B , if the following conditions are fulfilled: 1. Both the operators T and its inverse T − 1 are continuous in X. 2. The following equality is valid in X 1 : A T = T B . The importance of this type of operator is that if we know any solution v ∈ X 1 of B v = 0 , then we have that u = T v satisfies A u = A T v = T B v = 0 . Definition 5([ 6]) . A complex-valued function f ∈ C 2 [ − a , a ] , such that f ( x ) ≠ 0 for all x ∈ [ − a , a ] f ( 0 ) = 1 , will be called simply a nonvanishing coefficient on the real interval [ − a , a ] . Let f be a nonvanishing coefficient. Define the following iterated integrals (see [ 9]) associated to f as the sequences X ( n ) , X ˜ ( n ) constructed as follows: X ( 0 ) ( x ) ≡ X ˜ ( 0 ) ( x ) ≡ 1 , X ( n ) ( x ) = n ∫ 0 x X ( n − 1 ) ( s ) f 2 ( s ) ( − 1 ) n d s , X ˜ ( n ) ( x ) = n ∫ 0 x X ˜ ( n − 1 ) ( s ) f 2 ( s ) ( − 1 ) n − 1 d s . Definition 6([ 6]) . The formal powers φ n , n = 0 , 1 , … , associated to a nonvanishing coefficient f are defined as φ n ( x ) = f ( x ) X ( n ) ( x ) , n odd , f ( x ) X ˜ ( n ) ( x ) , n even . Example 2.Consider the function f ≡ 1 . Then X ˜ ( n ) = X ( n ) X ( n ) = n ∫ 0 x s n − 1 d s = x n ; therefore φ n ( x ) = x n . Theorem 1([ 6]) . Let f be a nonvanishing coefficient. There exists an invertible Volterra operator T f with continuous kernel K f , defined byT f [ v ] ( x ) = v ( x ) + ∫ − x x K f ( x , t ; h ) v ( t ) d t , such that T f [ x n ] = φ n ( x ) , for n = 0 , 1 , … . Further, for any v ∈ C 2 [ − a , a ] , T f satisfies the transmutation property − ∂ 2 + f ″ f T f [ v ] = T f [ − ∂ 2 v ] . In general, it is difficult to obtain an explicit representation of the kernel K f , since it is determined by a partial differential equation of Goursat type (see [ 9]). If f is a nonvanishing function, then 1 / f is also nonvanishing, and one can therefore construct the associated transmutation operator T 1 / f . The operators T f T 1 / f are related as follows: Proposition 3([ 9]) . On C 1 [ − a , a ] we have the following relation for any nonvanishing coefficient f:∂ x 1 f T f = 1 f T 1 / f ∂ x . The kernel is known when we consider the nonvanishing coefficient f ( x ) = e i λ x (see [ 6] example 12) and we have the following transmutation operator: T f [ u ] ( x ) = u ( x ) − λ 2 ∫ − x x x 2 − y 2 J 1 λ x 2 − y 2 u ( y ) x − y − i u ( y ) J 0 λ x 2 − y 2 d y is a continuous operator, where J 0 J 1 are the Bessel functions of the first kind. This satisfies the transmutation relation ( ∂ 2 + λ 2 ) T f [ u ] = T f [ ∂ 2 u ] . Proposition 4([ 12, 14]) . Let f ( x ) = e i λ x ; then, a following relation holds thatT 1 / f = T f ∗ , the * denoting the complex conjugation. AndT f [ x n ] = φ n ( x ) ∀ n ∈ N 0 , where φ n ( x ) = ( 2 k − 1 ) ! ! x λ k λ x j k − 1 ( λ x ) + i j k ( λ x ) , n = 2 k , ( 2 k + 1 ) ! ! x λ k x j k ( λ x ) n = 2 k + 1 , where j k denotes the spherical Bessel’s function of order k. The Table 2 shows the first five formal powers φ n ( x ) . From now, we consider f ( x ) = e i λ x and denote T f = T . Proposition 5.We have the following λ ≠ 0 ∈ R . 1. T [ sin ] ( x ) = sin x λ 2 + 1 λ 2 + 1 ; 2. T [ cos ] ( x ) = cos x λ 2 + 1 + i λ sin x λ 2 + 1 λ 2 + 1 ; 3. Define the exponential function by Exp ( y ) = e y . Then T [ Exp ] ( x ) , = 1 + 1 + i λ λ 2 + 1 e λ 2 − 1 x + 1 − 1 + i λ λ 2 + 1 e − λ 2 − 1 x , λ 2 ≠ 1 , 1 + ( 1 + i λ ) x λ 2 = 1 . Proof.Since we consider f ( x ) = e i λ x sin ( x ) , the Equation ( 27) turns into ( ∂ 2 + λ 2 ) T [ sin ] ( x ) = − T [ sin ] ( x ) This implies the function T [ sin x ] satisfies the differential equation with constant coefficients ( ∂ 2 + ( λ 2 + 1 ) ) T [ sin ] ( x ) = 0 , which implies that T [ sin ] ( x ) = C 1 sin x λ 2 + 1 + C 2 cos x λ 2 + 1 , using the definition of the operator ( 26), which implies that the operator fixes the value of x = 0 , so T [ sin ] ( 0 ) = 0 , and then C 2 = 0 . To determine the value of C 1 , we use Relation ( 25) as follows: T 1 / f [ cos ] ( x ) = e i λ x ∂ x e − i λ x T [ sin ] ( x ) , T ∗ [ cos ] ( x ) = − i λ C 1 sin x λ 2 + 1 + C 1 λ 2 + 1 cos x λ 2 + 1 , evaluating x = 0 , we have that 1 = C 1 λ 2 + 1 . Using the last relation, we can obtain the value of T [ cos ] ( x ) . The remainder relation can be established by an analogous way. □ Another important property of the transmutation operator T is that it enables the construction of solutions to the Helmholtz equation ( Δ + λ 2 ) u = 0 from harmonic functions. Since T acts only in a single spatial variable, this framework can be extended to functions defined on domains of the form [ − x 1 , x 1 ] ୍ଠ [ − x 2 , x 2 ] ୍ଠ [ − x 3 , x 3 ] ⊂ R 3 by applying the scalar transmutation operator in the variable x j . This provides a natural extension of the scalar transmutation theory to higher-dimensional settings while preserving its ability to transform harmonic functions into solutions of the Helmholtz equation. Lemma 3(Theorem 26, [ 6]) . Let λ 1 , λ 2 , λ 3 ∈ R and defineλ 2 = λ 1 2 + λ 2 2 + λ 3 2 . For i = 1 , 2 , 3 , let f i ( x i ) = e i λ i x i , and let u be a harmonic function. Define the operatorT = T f 1 ∘ T f 2 ∘ T f 3 . where T f i denotes the operator acting only on the variable x i , for i = 1 , 2 , 3 . Then T [ u ] satisfies the Helmholtz equation( Δ + λ 2 ) T [ u ] = 0 . Example 3.Consider the function u ( x → ) = x 1 2 − x 2 2 + x 3 be an harmonic function and λ i = λ 3 for i = 1 , 2 , 3 . Then the function T [ u ] ( x → ) = φ 2 ( x 1 ) φ 0 ( x 2 ) φ 0 ( x 3 ) − φ 0 ( x 1 ) φ 2 ( x 2 ) φ 0 ( x 3 ) + φ 0 ( x 1 ) φ 0 ( x 2 ) φ 1 ( x 3 ) , = λ 1 x 1 e i λ 1 x 1 − sin ( λ 1 x 1 ) i λ 1 2 e i λ 2 x 2 e i λ 3 x 3 − e i λ 1 x 1 λ 2 x 2 e i λ 1 x 2 − sin ( λ 2 x 2 ) i λ 2 2 e i λ 3 x 3 + e i λ 1 x 1 e i λ 2 x 2 sin ( λ 3 x 3 ) λ 3 , if we change the parameters λ 1 = λ λ i = 0 for i = 2 , 3 . Then the function T [ u ] ( x → ) = φ 2 ( x 1 ) − φ 0 ( x 1 ) x 2 2 + φ 0 ( x 1 ) x 3 , = λ x 1 e i λ x 1 − sin ( λ x 1 ) i λ 2 − e i λ x 1 x 2 2 + e i λ x 1 x 3 , is another solution of the Helmholtz equation. Since ( Δ + λ 2 ) u = ( D + λ ) ( D − λ ) u . If u satisfies the Helmholtz equation then the quaternionic function v = ( D ବ୍ଦ λ ) u satisfies ( D ∓ λ ) v = 0 . Now with the aid of the complex operator T, which acts only on the variable x 3 in a domain Ω , such that the straight segment from ( x 1 , x 2 , − x 3 ) to ( x 1 , x 2 , x 3 ) also lies in Ω , in [ 10], the authors defined a quaternionic operator T 0 T 1 as follows: T 0 [ u ] ( x → ) = T ∗ [ u 0 ] ( x → ) + T [ u 1 ] ( x → ) e 1 + T [ u 2 ] ( x → ) e 2 + T ∗ [ u 3 ] ( x → ) e 3 , T 1 [ u ] ( x → ) = T [ u 0 ] ( x → ) + T ∗ [ u 1 ] ( x → ) e 1 + T ∗ [ u 2 ] ( x → ) e 2 + T [ u 3 ] ( x → ) e 3 , = T 0 ∗ [ u ] . Since by the definition of ( 26), ∥ T 0 [ v ] ∥ K ≤ N 1 max K | v 0 | + N 1 max K | v 1 | + N 1 max K | v 2 | + N 1 max K | v 3 | , where N 1 depends only on the corresponding kernel of the bounded Volterra operator T, T ∗ ; thus ∥ T 0 ∥ K ≤ 4 N 1 , which implies the continuity of T 0 . Since T 1 is the complex conjugate of T 0 , it is also continuous. Proposition 6([ 10]) . For v ∈ C 1 ( Ω , H ( C ) ) ,D + M i λ e 3 T 0 [ v ] = T 1 [ D v ] , D − M i λ e 3 T 1 [ v ] = T 0 [ D v ] . Since T 0 T 1 are invertible operators, we find that all solutions Equations (33) and (34) can be obtained by transmutation of monogenic functions. 4. Time-Harmonic Maxwell System In this section, we consider the time-harmonic Maxwell equations in a homogeneous medium. These equations describe the behavior of the electric and magnetic fields under the assumption of harmonic time dependence with angular frequency ω . Let E → H → denote the electric and magnetic field intensities, respectively, and let ε μ be the electric permittivity and magnetic permeability of the medium of both constants. The Maxwell system then takes the form curl H → = − i ω ε E → , curl E → = i ω μ H → , div E → = 0 , div H → = 0 . We now transform Systems ( 35)–(38) into an equivalent system involving the operators D + λ D − λ , where λ = ω μ ε . To this end, we multiply ( 35) by i μ and (36) by ε , obtaining curl ( i μ H → ) = ω μ ε E → , curl ( ε E → ) = i ω μ ε H → . Adding and subtracting these identities yields curl ( ε E → + i μ H → ) = λ ( ε E → + i μ H → ) , curl ( ε E → − i μ H → ) = − λ ( ε E → − i μ H → ) . Defining the vector fields u → = ε E → + i μ H → , v → = ε E → − i μ H → , For a purely vectorial monogenic function u → by ( 4), one has D u → = curl u → . Consequently, the system can be written as ( D − λ ) u → = 0 , ( D + λ ) v → = 0 , which implies that u → ∈ Ker ( D − λ ) v → ∈ Ker ( D + λ ) . Finally, solving for E → H → in terms of u → v → gives E → = 1 2 ε ( u → + v → ) , H → = 1 2 i μ ( u → − v → ) . Lemma 4([ 10]) . Every solution of Maxwell system (35)–(38) can be expressed asE → = 1 2 ε ( u → + v → ) , H → = − i 2 μ ( u → − v → ) , = − i ω μ D E → , where λ = ω μ ε , u → ∈ Ker ( D − λ ) v → ∈ Ker ( D + λ ) . Remark 1.There exist several methods for constructing functions belonging to Ker ( D + λ ) . One such approach, described in [15], is based on solutions of the scalar Helmholtz equation. More precisely, let Ψ be a solution ofΔ Ψ + λ 2 Ψ = 0 , and define the vector fieldv → = λ curl ( Ψ a → ) + curl curl ( Ψ a → ) , where a → is a constant unit vector. Then v → ∈ Ker ( D + λ ) . Moreover, it is shown in [16] that, for suitable choices of Ψ , this construction generates a complete basis of solutions. 5. Construction of Solutions In this section, we develop a method for obtaining solutions to Maxwell’s Equation ( 35) using monogenic functions and transmutation operators. Let u 0 be a harmonic function in Ω . By Proposition 2, there exists a vector field u → such that u = u 0 + u → is a monogenic function. Another way to construct a monogenic function via harmonic functions is to consider the function u → = D u 0 , which is monogenic due to the decomposition of the Laplacian operator Δ = − D D . Consider the ball B r = { x → ∈ R 3 : | x → | < r } . Since harmonic functions are analytic, if h is harmonic, then it admits the expansion (see [ 17]). h ( x → ) = ∑ k 1 , k 2 , k 3 ≥ 0 ∂ 1 k 1 ∂ 2 k 2 ∂ 3 k 3 h ( 0 ) k 1 ! k 2 ! k 3 ! x 1 k 1 x 2 k 2 x 3 k 3 . Therefore, it suffices to determine how the operator T f i acts. Recalling that T [ x i k ] = φ k ( x i ) for i = 1 , 2 , 3 T [ 1 ] = e i λ x i , we can explicitly determine the operator T . For example, if we consider the action only of T f 3 , T f 3 ∑ k 1 , k 2 , k 3 ≥ 0 ∂ 1 k 1 ∂ 2 k 2 ∂ 3 k 3 h ( 0 ) k 1 ! k 2 ! k 3 ! [ x 1 k 1 x 2 k 2 x 3 k 3 ] = ∑ k 1 , k 2 , k 3 ≥ 0 ∂ 1 k 1 ∂ 2 k 2 ∂ 3 k 3 h ( 0 ) k 1 ! k 2 ! k 3 ! T [ x 1 k 1 x 2 k 2 x 3 k 3 ] = ∑ k 1 , k 2 , k 3 ≥ 0 ∂ 1 k 1 ∂ 2 k 2 ∂ 3 k 3 h ( 0 ) k 1 ! k 2 ! k 3 ! x 1 k 1 x 2 k 2 φ k 3 ( x 3 ) Let λ i be as it is in Lemma 3. Then we have T [ D h ( x → ) ] = ∑ k 1 , k 2 , k 3 ≥ 0 ∂ 1 k 1 ∂ 2 k 2 ∂ 3 k 3 h ( 0 ) k 1 ! k 2 ! k 3 ! φ k 1 ( x 1 ) φ k 2 ( x 2 ) φ k 3 ( x 3 ) Then the function u = ( D + λ ) T [ D h ] satisfies ( D − λ ) u = 0 due to Lemma 3. Applying the operator D = ∑ j = 1 3 e j ∂ j to h, we obtain D h ( x → ) = ∑ j = 1 3 e j ∂ j h ( x → ) = ∑ k 1 , k 2 , k 3 ≥ 0 ∂ 1 k 1 ∂ 2 k 2 ∂ 3 k 3 h ( 0 ) k 1 ! k 2 ! k 3 ! e 1 k 1 x 1 k 1 − 1 x 2 k 2 x 3 k 3 + e 2 k 2 x 1 k 1 x 2 k 2 − 1 x 3 k 3 + e 3 k 3 x 1 k 1 x 2 k 2 x 3 k 3 − 1 . T 0 [ D h ( x → ) ] = ∑ k 1 , k 2 , k 3 ≥ 0 ∂ 1 k 1 ∂ 2 k 2 ∂ 3 k 3 h ( 0 ) k 1 ! k 2 ! k 3 ! ( e 1 k 1 x 1 k 1 − 1 x 2 k 2 φ k 3 ( x 3 ) + e 2 k 2 x 1 k 1 x 2 k 2 − 1 φ k 3 ( x 3 ) + e 3 k 3 x 1 k 1 x 2 k 2 φ k 3 − 1 ∗ ( x 3 ) ) . Then these functions belong to the kernels of D + M i λ e 3 ( T 0 [ D h ( x → ) ] ) ∗ = T 1 [ D h ( x → ) ] ∈ Ker D − M i λ e 3 . This construction allows us to generate solutions in the kernels of the operators D + λ D − λ , as stated in the following result. For a detailed treatment of the construction of λ -monogenic functions, we refer to [ 1, 12 Lemma 5.Let u 1 , u 2 , v 1 , v 2 be monogenic functions. Then the functionsu → = V − P − T 0 [ v 1 ] + P + T 1 [ v 2 ] , v → = V + P + T 0 [ u 1 ] + P − T 1 [ u 2 ] , belong to Ker ( D − λ ) Ker ( D + λ ) , respectively. Example 4.Consider u i ( x → ) = v i ( x → ) = x 1 + x 3 e 2 to be a monogenic function. Then for i = 1 , 2 we have T 0 [ u i ( x → ) ] = x 1 φ 0 ∗ ( x 3 ) e 1 − φ 1 ( x 3 ) e 3 = x 1 e − i λ x 3 − sin ( λ x 3 ) λ e 3 T 1 [ v i ( x → ) ] = x 1 φ 0 ( x 3 ) e 1 − φ 1 ( x 3 ) e 3 = x 1 e i λ x 3 − sin ( λ x 3 ) λ e 3 Then the functions in (42) and (43) are u → = V − P − T 0 [ u 1 ] + P + T 1 [ u 2 ] = x 1 cos ( λ x 3 ) e 1 − x 1 sin ( λ x 3 ) e 2 − sin ( λ x 3 ) λ e 3 v → = V + P + T 0 [ v 1 ] + P − T 1 [ v 2 ] = x 1 cos ( λ x 3 ) e 1 + x 1 sin ( λ x 3 ) e 2 − sin ( λ x 3 ) λ e 3 , This implies that E → = 1 ε ( x 1 cos ( λ x 3 ) e 1 − sin ( λ x 3 ) λ e 3 ) , H → = i μ x 1 sin ( λ x 3 ) e 2 Proposition 7(Theorem 7.5, [ 10]) . Let v ∈ Ker ( D + λ ) in B R ( 0 ) . Then u → can be expanded into Taylor series in λ-powers of the form v → = ∑ n = 0 ∞ ∑ l + m = n P + T 0 [ P l , m ] + P − T 1 [ P l , m ] c l , m , These series converge uniformly in every B R 1 ( 0 ) with R 1 < R . And if v → ∈ Ker ( D + λ ) in B R ( 0 ) , then u → = ∑ n = 0 ∞ ∑ l + m = n P + T 1 [ P l , m ] + P − T 0 [ P l , m ] d l , m , These series converge uniformly in every B R 2 ( 0 ) with R 2 < R . The coefficients c l , m , d l , m are the same as in the Taylor series (7). Since the series defining u → v → converge normally in balls of radius R 1 R 2 , respectively, and taking into account the relations between E → , u → , and v → , it follows that the electric field admits a Taylor series expansion. Lemma 6.Let u → ∈ Ker ( D + λ ) , v → ∈ Ker ( D − λ ) . Then every electric field E → can be expanded in B R ( 0 ) into powers of the form 2 ε E → = ∑ n = 0 ∞ ∑ l + m = n [ P + T 1 [ P l , m ] c → l , m + T 0 [ P l , m ] d → l , m + P − T 0 [ P l , m ] c → l , m + T 1 [ P l , m ] d → l , m ] , The coefficients are given byc → l , m = 1 n ! ∂ n v → ( 0 ) ∂ x 1 l ∂ x 2 m , d → l , m = 1 n ! ∂ n u → ( 0 ) ∂ x 1 l ∂ x 2 m , where n = l + m . These series converge uniformly in every B R 3 ( 0 ) with R 3 = min { R 1 , R 2 } . Then H → can be recovered and expressed in Taylor series using the relation in (41). When we used Relation ( 19), Equation ( 46) turned into 2 ε E → = ∑ n = 0 ∞ ∑ l + m = n T 0 [ P l , m ] + T 1 [ P l , m ] c → l , m . 6. Conclusions and Discussion In this work, we developed a constructive framework for solving the time-harmonic Maxwell system by combining complex quaternionic analysis with transmutation theory. The central idea is the reformulation of Maxwell’s equations in terms of the first-order operators D ବ୍ଦ λ , which enables the interpretation of electromagnetic fields as superpositions of λ -monogenic functions. We establish a systematic procedure for the explicit construction of solutions: starting from harmonic or monogenic functions, we generate solutions to the equations ( D ବ୍ଦ λ ) u = 0 via transmutation operators, and subsequently reconstruct the electromagnetic fields E → H → . This approach provides a direct and computable pathway from classical function theory to solutions of Maxwell’s system which uses solutions to scalar Hemlholtz equations to find solutions to the problem. The construction developed here combines transmutation operators, λ -monogenic functions and complex quaternionic analysis developed in [ 2, 4, 6, 10, 11, 14] in the context of the spaces Ker ( D ବ୍ଦ λ ) and their relation to the time-harmonic Maxwell system, leading to explicit representations and electromagnetic examples. In particular, the use of formal powers associated with the transmutation operator T leads to Taylor-type expansions of electromagnetic fields in terms of explicitly known basis functions. This representation offers potential advantages for numerical implementation and approximation schemes. Funding This research was supported by the Centro de Investigación of Anáhuac Querétaro University through the institutional research project CI-MC20251008 and SECIHITI via the project CBF-2026-1546. Data Availability Statement No new data were created or analyzed in this study. Acknowledgments The author wishes to express their sincere gratitude to Anáhuac University Querétaro for the invaluable support provided throughout the development of this project, with special thanks to Andrea Oviedo, Chief of the Quantitative Methods Department. Conflicts of Interest The author declares no conflicts of interest. References Table 1. Grigor’ev polynomials of low degree. Table 1. Grigor’ev polynomials of low degree. Polynomial Expression P 0 , 2 x 2 2 − x 3 2 − 2 x 2 x 3 e 1 P 1 , 1 2 x 1 x 2 − 2 x 1 x 3 e 1 + 2 x 2 x 3 e 2 P 2 , 0 x 1 2 − x 3 2 + 2 x 1 x 3 e 2 P 0 , 3 x 2 3 − 3 x 2 x 3 2 + ( x 3 3 − 3 x 2 2 x 3 ) e 1 P 1 , 2 ( 3 x 1 x 2 2 − 3 x 1 x 3 2 ) − 6 x 1 x 2 x 3 e 1 + ( 3 x 2 2 x 3 − x 3 3 ) e 2 P 2 , 1 ( 3 x 1 2 x 2 − 3 x 2 x 3 2 ) − ( 3 x 1 2 x 3 − x 3 3 ) e 1 + 6 x 1 x 2 x 3 e 2 P 3 , 0 ( x 1 3 − 3 x 1 x 3 2 ) + ( 3 x 1 2 x 3 − x 3 3 ) e 2 Table 2. Explicit expressions for the first formal powers. Table 2. Explicit expressions for the first formal powers. Function Expression φ 0 ( x ) e i λ x φ 1 ( x ) sin ( λ x ) λ φ 2 ( x ) ( λ x + i ) sin ( λ x ) − i λ x cos ( λ x ) λ 2 φ 3 ( x ) 3 sin ( λ x ) λ 3 − 3 x cos ( λ x ) λ 2 φ 4 ( x ) 3 λ 4 − i λ 2 x 2 + λ x + 3 i sin ( λ x ) − λ x ( λ x + 3 i ) cos ( λ x ) φ 5 ( x ) − 15 ( λ 2 x 2 − 3 ) sin ( λ x ) + 3 λ x cos ( λ x ) λ 5 Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2026 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. Moreira, P. Solutions to Time-Harmonic Maxwell Equations via Transmutation Theory. Mathematics 2026, 14, 2055. https://doi.org/10.3390/math14122055 Moreira P. Solutions to Time-Harmonic Maxwell Equations via Transmutation Theory. Mathematics. 2026; 14(12):2055. https://doi.org/10.3390/math14122055 Moreira, Pablo. 2026. "Solutions to Time-Harmonic Maxwell Equations via Transmutation Theory" Mathematics 14, no. 12: 2055. https://doi.org/10.3390/math14122055 Moreira, P. (2026). Solutions to Time-Harmonic Maxwell Equations via Transmutation Theory. Mathematics, 14(12), 2055. https://doi.org/10.3390/math14122055
