Conjugate complex harmonic functions

This paper presents several properties and relations that satisfy the components of a bicomplex holomorphic function. It also exhibits several analogies and differences with the case of analytic functions.


Introduction
One of the best-known extensions of complex numbers to four dimensions is the algebra of quaternions, introduced by Hamilton [16,17], which, although is almost a field, loses the commutativity property.This fact makes it difficult to extend the theory of holomorphic functions.Another attempt was made in 1848 by Cockle following Hamilton's ideas.He studied an algebra called tessarines [13] where the role of the imaginary units is different from the bicomplex case.Thus, an attempt has been made to consider four-dimensional algebras which preserve commutativity and contain C as a subalgebra, and in which it is possible to extend the theory of holomorphic functions.Not surprisingly, this can be done by considering, for example, two imaginary units i and j such that i 2 = j 2 = −1 and introducing ij = ji = k.This makes k an imaginary hyperbolic unit, i.e., an element such that k 2 = 1.It was not until 1892 that the mathematician Corrado Segre, also inspired by the work of Hamilton and Clifford, introduced what he called bicomplex numbers [31].Segre observed that (1 − ij)/2 and (1 + ij)/2 are idempotent and play a central role in the theory of bicomplex number.After Segre, other mathematicians, in particular Spampinato [33,34] and Scorza Dragoni [30], developed the first rudiments of a theory of functions over bicomplex numbers.The next great impulse in the study of bicomplex analysis was the work of Riley, in 1953 with his doctoral thesis [28], in which the theory of bicomplex functions is deepened.But the most important contribution was undoubtedly the work of Price [25], where the theory of bicomplex (as well as multicomplex) holomorphic functions is extensively developed.However, in recent years, there has been a resurgence of interest in the study of bicomplex holomorphic functions in one and several variables [2,7,10,14,20,21,24], as well as bicomplex meromorphic functions [5,11] and more recently there has been a remarkable activity in this field, dealing with the study of the bicomplex Bergman and Bloch [26] spaces, the bicomplex Stolz condition [27], the Möbius transformations in the bicomplex space [12] and the Cousin problems for the bicomplex case [8].
In this paper, we study several properties of the components of a bicomplex holomorphic function.Proposition 7.5.1 of [2] says that F = F 1 + j F 2 is a bicomplex holomorphic function on a domain of BC, if and only if, their components F 1 and F 2 are related to each other by a complex Cauchy-Riemann type of conditions, very similar to the complex case.If F is expressed by its idempotent form F = G 1 e + G 2 e † , Sect. 3 shows how the Cauchy-Riemann type of conditions of F 1 and F 2 determine some relations between G 1 and G 2 .
In classical complex analysis, it is well known that if u is a real harmonic function on a domain D in the complex plane, there exists a harmonic conjugate function v on the same domain such that f = u + iv is holomorphic in D, and except for a constant this function f is unique and one has a classical integral formula to obtain v in terms of u.Sect. 4 shows that a similar result is true for the bicomplex case.In addition, two different ways of obtaining its complex harmonic conjugate F 2 are presented and illustrated with examples.
The orthogonality between the contour lines of the components of an analytic function and their corresponding gradients are well known.Section 5 shows that similar results exist for the different level sets of the components of a holomorphic bicomplex function and illustrate this with an example.
In complex analysis if f = u + iv, there exists a relationship between the magnitude of the square of the derivative of f and the norm of the square of the gradient of u and v. Section 6 shows what happens in the bicomplex holomorphic case.That is, if F = F 1 + j F 2 , we exhibit various relations between the norm of the bicomplex derivative of F and the complex gradients of F 1 and F 2 , even if F is expressed in its idempotent form.
In complex analysis, the conformal transplants are used to obtain various relations on the gradient, Laplacian, and integral from the conformal transplantation of an analytic bijective mapping.These results are very useful to obtain important classical applications in Physics ( [18], Chapter 5).Section 7 shows that also in the bicomplex case, it is possible to generalize such relationships.
We expect that the generalizations to the bicomplex case presented in this article of classical results in complex analysis will lead to applications in mathematics and physics in the immediate future.

Preliminaries
This section presents several common facts about bicomplex numbers and bicomplex holomorphic functions.We will use freely results and notation of [2].
The set of bicomplex numbers BC is defined as The sum and product of bicomplex numbers are made in the expected way.A bicomplex number Z = z 1 + jz 2 admits several other forms of writing; however, in this work we use only two representations: their normal form Z = z 1 + jz 2 , with z l = x l + iy l ∈ C(i), l = 1, 2, and their idempotent form, that is, where Observe that e e † = 0; 1 = e + e † or more general λ = λ( e + e † ) with λ ∈ C(i).Moreover, In the special case that Z = β 1 e + β 2 e † and W = γ 1 e + γ 2 e † , with β l , γ l real numbers, we consider the partial order There are several conjugations of bicomplex numbers; however, we consider here only two conjugations Z † = z 1 − jz 2 and Z * = z 1 − jz 2 = β 1 e + β 2 e † , where z 1 and z 2 are the usual complex conjugates of z 1 , z 2 ∈ C(i).With these conjugations, | 2 e † and the inverse The i-norm and k-norm of Z are defined as (see Sections 1.4 and 2.7 in [20]) Let be a domain in BC and F : → BC a bicomplex function.The complex partial derivatives of F at Z 0 are defined by the following limits (if they exist): where h 1 and h 2 are complex increments and The bicomplex derivative F (Z 0 ) of the function F at a point Z 0 ∈ is the following limit, if it exists that • The C(i)−complex partial derivatives F z l exist, for l = 1, 2.
• The complex partial derivatives verify the identity which is the C(i)−complex Cauchy-Riemann system for F, that is, We would also like to mention the next comment and definition ([20], Definition 7.4.1,p. 153) The C(i)−complex partial derivatives are denoted by the symbols F z l for l = 1, 2, instead of the symbols ∂ F ∂z 1 (Z ) and ∂ F ∂z 2 (Z ), because the first ones define the limits of suitable difference quotients, while the latter ones are operators acting on C 1 functions.The relationship between these two notations is justified by the following definition: a bicomplex  Consider the following bicomplex differential operators: In bicomplex analysis, there are different ways of characterizing the bicomplex holomorphic functions as shown in the following result.Its derivatives of any order are given by The rules of derivability are the usual ones.

Relations between the idempotent components of a bicomplex holomorphic function
If we express the BC-holomorphic function F in its idempotent form (2.5), what kind of relationships must satisfy G 1 and G 2 between them?
As we can obtain a BC-holomorphic function F = G 1 e+G 2 e † : 1 e+ 2 e † → BC by just taking any pair of holomorphic functions G 1 : 1 → C and G 2 : 2 → C and setting F = G 1 e+G 2 e † : 1 e+ 2 e † → BC, then we cannot expect some relation like (2.4).However, we will see that the complex Cauchy-Riemann type of condition of F 1 and F 2 determines some relationship between G 1 and G 2 .

Then, F is a bicomplex holomorphic function on if and only if
In particular, Then, Similarly, If F is a bicomplex holomorphic function, from (2.4), we have and from here we get (3.7).Reciprocally, if (3.7) holds, then (2.4) is satisfied and F is a bicomplex holomorphic function.If we add the equalities in (3.7) and associate them, we get therefore, (3.8) follows.

The conjugate of a C(i) complex harmonic function
This section defines the concepts of C(i)−complex harmonic and bicomplex harmonic function.We will show that there is a connection between them.Let D ⊂ C 2 be an open subset and f : D → C a holomorphic function.A function f is said to be a C(i)−complex harmonic function if it satisfies a Laplace-type equation Let be an open subset of BC.A function → BC is bicomplex harmonic if F has continuous second partial derivatives and This equation is a bicomplex C(i)−Laplace-type equation in two complex variables z 1 and z 2 .The operator With this notation, Moreover, if F is a bicomplex holomorphic function, then it satisfies the Cauchy-Riemann type of system given by (2.4) and so it is clear that each component is a C(i)−complex harmonic function, that is, Given a C(i)−complex harmonic function on a domain of C 2 , we want to determine its bicomplex conjugate harmonic function.Then the following results appear to be natural generalizations of the complex cases.Recall that a closed form is locally exact.Theorem 4.1 Let ⊂ C 2 be a domain and F i : → C, i = 1, 2 be two holomorphic functions.If then the differential form Proof Let F i = u i + iv i be the real and imaginary parts of , the Cauchy-Riemann relations are written as and (4.10) rewritten as Now, the differential form (4.11) is To prove that the real part of ω is closed, it is necessary to verify These equalities are a straightforward consequence from the previous Cauchy-Riemann relations and the hypothesis.In the same way, one can prove that the imaginary part of ω is closed.
Then the differential form The curves involved in the following formulas (4.13) and (4.15) are subsets of .
Proof By Corollary 4.2 and the hypothesis, the function F 2 is well defined, that is, it does not depend on the path joining Z 0 with Z .Then we can suppose that Z , Z 0 ∈ B r (z 1 , z 0 ) ⊂ , and if h ∈ C is small enough, the differential quotient can be written as (omiting evaluations) where the last integral is calculated using the path γ : [0, 1] → defined by γ (t) = (z 1 + th, z 2 ).Since γ (t) = (h, 0), we have Thus, since F 1 is uniformly continuous on the segment.Using the path η : As in the first integral dζ 1 = 0, while in the second integral dζ 2 = 0, we have Now, we are going to present a second way to calculate the conjugate of a complex harmonic function.For that, we need the following equality: 123 Proposition 4.5 Let ⊂ C 2 be a domain such that the first homotopy group of is trivial.Let F 1 : → C be a C(i)−complex harmonic function.Then there exists an infinite number of conjugate complex harmonic functions of F 1 defined in , given by where is a family of bicomplex holomorphic functions.
Proof Let (z 0 1 , z 0 2 ), (z 1 , z 2 ) ∈ , using (4.14) define where ϕ is a holomorphic function that must be determined.Since the first homotopy group of is trivial, we have that the integral in the definition of the function F 2 is well defined and by uniform continuity on the linear path In particular, F 2 is holomorphic on z 1 and since ∂ F 2 ∂z 1 is holomorphic on z 2 , then F 2 is also holomorphic on z 2 .Since ϕ is a holomorphic function on z 2 only, its derivative dϕ dz 2 must also be independent of z 1 .Since we need that since F 1 is a C(i)−complex harmonic function.Thus, Substituting ϕ in (4.16), we have Thus, F 2 is a harmonic complex conjugate of F 1 .
The previous result gives us a classical way to calculate the harmonic conjugate of one function.

Bicomplex level sets of bicomplex harmonic conjugate
In complex analysis, one of the consequences of holomorphicity is that the level curves of the harmonic components result in orthogonals among them, see [37].We will see what happens in the bicomplex case.
be a bicomplex function in some domain ⊂ BC, where Z = x 1 + iy + j(x 2 + iy 2 ).Suppose that F l = u l + iv l where u l , v l are real functions for l = 1, 2. The equations: where a l , b l are arbitrary real constants, which define the level sets of u l and v l , l = 1, 2, respectively.
However, ∇u 1 and ∇v 2 are not orthogonal in general, because We can see also that ∇u 2 and ∇v 1 are not orthogonal in general, because In the same way, it is possible to see that ∇u 2 and ∇v 1 are not orthogonal in general.
The following result permits us to describe the intersection of the pairs of hypersurfaces u 1 = a 1 and It is well known that we can get an n-surface as the non-empty subset of R n+k obtained by the intersection are smooth functions such that {∇ f 1 ( p), . . ., ∇ f k ( p)} is linearly independent for each p ∈ S.This n− surface in R n+k results in the intersection of k, (n + k − 1)−surfaces which meet "clearly" in the sense that the normal directions are linearly independent at each point of the intersection, see [35].Since BC ≈ R 4 , we have the next corollary.
Example 5.6 Consider the bicomplex holomorphic function To simplify our analysis, we suppose y 1 > 0 and so = 0, and they are orthogonal.We consider F(1, 0, 0, = 1 and we have the orthogonal 3-surfaces: Their intersection is the 2-surface:

Bicomplex gradients
In complex analysis, we have that the following relationship exists between the modulus of derivatives and the gradients of the real and imaginary parts of an analytic function: . Now, we show what happens in the bicomplex case.
Let F be a bicomplex function in some open subset of BC.We define the bicomplex gradient of F denoted by gradF : → BC as Let F = F 1 + jF 2 be a bicomplex holomorphic function in some domain ⊂ BC.By Theorem 2.1, F z l exists for l = 1, 2 and verify the identity Since F 1 and F 2 satisfy the Cauchy-Riemann type of equations (2.4), F can be represented as where the partial derivatives are all evaluated at Z = z 1 + jz 2 .Thus, we obtain and and where | • | is the norm in C. Thus, If we write F 1 = u 1 + iv 1 where u 1 and v 1 are real functions, by Cauchy-Riemann equations ∂ F 1 ∂z 1 can be written as which implies that and On the other hand, we know that , where • 2 R 4 is the norm in R 4 .By (6.26), (6.28) and (6.29), we have In the same way, if F 2 = u 2 + jv 2 and using (6.27), . Now, if we consider the i−norm, we have On the other hand, Then, and Suppose that F = G 1 e + G 2 e † is a bicomplex holomorphic function in ⊂ BC.By (2.6), If we consider the k−norm, we get Since G 1 and G 2 are complex holomorphic functions, G l has the form G l = μ l + iν l and satisfies the following relations for l = 1, 2 : and 2 e † .

Bicomplex conformal transplants
In this section, we will give the generalization to the bicomplex case of the complex conformal transplants, which are very useful in Physics, to facilitate the calculation of potentials (temperature, electricity, stress, etc.).For more information, see [18].Let D, E be domains in BC and F, a bijective bicomplex holomorphic function from D to E, with inverse F −1 also bijective and F (Z ) = 0 for all Z ∈ D.
Let φ be a C(i)−valued function of class C 2 (D).We define in E a function ψ as follows: for any W ∈ E, let The function ψ will be called the bicomplex conformal transplant of φ under the mapping F and its process of construction is given by the bicomplex conformal transplantation.See the diagram.
Then we have the following results: Proof Applying the chain rule and the Cauchy-Riemann system given by (2.4), we obtain since by Theorem 2.1 F (Z ) = F z 1 (Z ), which completes the proof.
We will see what is the relationship between the C(i)−Laplacian of φ and the C(i)− Laplacian of ψ.By definition of C(i)−Laplacian, we have Proof Applying the chain rule again, we obtain .
As  Important applications of bicomplex analysis in physics have appeared in several works during the present century, for example, to Maxwell's equations in bicomplex analysis [1], in applications of bicomplex algebra to electromagnetism [3], as well as works on bicomplex quantum mechanics [29] and on the bicomplex quantum Coulomb potential problem [22], among others.We expect that the results presented in this work will be applied in the near future to physics problems in higher dimensions.
In a forthcoming paper, we will present some consequences of the results of this paper related to Cauchy's bicomplex integral theorem and Morera's bicomplex theorem.

Theorem 2 . 2 (
[20], Theorem 7.4.2) A C 1 −bicomplex function F is C(i)−bicomplex differentiable if and only if both the components F 1 , F 2 are holomorphic functions in the sense of two complex variables.

Theorem 4. 3
Let ⊂ C 2 be a domain and F 1 : → C a C(i)−complex harmonic function.If the first homotopy group of is trivial, then all the conjugate complex harmonic functions of F 1 are given by

Theorem 5 . 2 Theorem 5 . 3
Let U be an open set in R n and let f : U → R in C ∞ (U ).Let p ∈ U such that ∇ f ( p) = 0, and let c = f ( p).Then the set of all vectors tangent to the level set f −1 (c) is equal to [∇ f ( p)] ⊥ , where ∇ f ( p) denotes the usual gradient.In particular if ∇ f ( p) = 0 for all p ∈ U , f −1 (c) is a hypersurface.Two hypersurfaces M and N are orthogonal if for each p ∈ M ∩ N , their respective tangent spaces T p (M) = T p (N ) are orthogonal, ([6], formula (3.2.4)).Theorems 5.3 and 5.4 may be summarized by saying that the level sets of the real parts of F 1 and F 2 are two orthogonal families as well as the level sets of the imaginary parts of them.Corollary 5.5describes the intersection of these orthogonal families in BC.Let F(Z ) = F 1 (Z ) + jF 2 (Z ) be a bicomplex holomorphic function in some domain ⊂ BC.Suppose that F l = u l + iv l for l = 1, 2 and ∇u 1 ( p) = 0, p ∈ .Then ∇u 1 ( p) ⊥ ∇u 2 ( p) and ∇v 1 ( p) ⊥ ∇v 2 ( p).

. 25 )
where | • | BC is the Euclidean norm in BC and | • | is the Euclidean norm in C. On the other hand, by definition of bicomplex gradient, we have

2 2•Theorem 7 . 3
|F (z)| 2 i , which proves the theorem.The next result gives the relationship between the integral of φ over D and the integral of ψ over E. Under the hypotheses of Theorem (7.1), D φ(Z )dz 1 dz 2 = E ψ(W )|(F −1 ) (W )| The component G 1 , seen as a C(i)-valued function of the complex variables (β 1 , β 2 ), is holomorphic; moreover it does not depend on the variable β 2 and thus G 1 is a holomorphic function of the variable β 1 .(b) The component G 2 , seen as a C(i)-valued function of the complex variables (β 1 , β 2 ) is holomorhic; moreover, it does not depend on the variable β 1 and thus G 2 is a holomorphic function of the variable β 2 .
Since w 1 and w 2 represent the components F 1 and F 2 of F, it follows that C(i) 2 (w 1 ) = 0 and C(i) 2 (w 2 ) = 0.