Take a bicomplex function F : Ω ⊂BC→BC on a domain Ω. We write all the bicomplex numbers involved inC(i)-idempotent form, for instance,
Z =β1e+β2e†= (1+im1)e+ (2+im2)e†, F(Z) =G1(Z)e+G2(Z)e†,
H =η1e+η2e†= (u1+iv1)e+ (u2+iv2)e†. Let us introduce the sets
Ω1:={β1β1e+β2e† ∈Ω} ⊂C(i) (7.58) and
Ω2:={β2β1e+β2e† ∈Ω} ⊂C(i). (7.59) It is easy to prove that Ω1and Ω2 are domains inC(i), see book [56].
We assume that F ∈ C1(Ω) where the real partial derivatives are taken with respect to the “idempotent real variables”:1, m1, 2, m2; how this is related with the cartesian variables x1, y1, x2, y2 will be discussed later. The condition F ∈ C1(Ω) ensures the real differentiability of F in Ω:
F(Z+H)−F(Z) = ∂F
∂1(Z)ãu1+ ∂F
∂m1(Z)ãv1 +∂F
∂2(Z)ãu2+ ∂F
∂m2(Z)ãv2+o(H) (7.60) forH →0. We are going to follow Section 7.4 so we omit many details. First of all, let us translate formula (7.60) inC(i)-complex language: since
u1=1
2(η1+η1); u2= 1
2(η2+η2);
v1= i
2(η1−η1); v2= i
2(η2−η2), then
F(Z+H)−F(Z) = ∂F
∂1(Z)ã 1
2(η1+η1) + ∂F
∂m1(Z)ã i
2(η1−η1) + ∂F
∂2(Z)ã1
2(η2+η2) + ∂F
∂m2(Z)ã i
2(η2−η2) +o(H)
=η11 2
∂F
∂1(Z)−i ∂F
∂m1(Z)
+η11 2
∂F
∂1(Z) +i∂F
∂m1(Z)
+ η21 2
∂F
∂2(Z)−i ∂F
∂m2(Z)
+η21 2
∂F
∂2(Z) +i ∂F
∂m2(Z)
+o(H)
=:η1∂F
∂β1(Z) +η1∂F
∂β1(Z) +η2∂F
∂β2(Z) +η2∂F
∂β2(Z) +o(H).
7.6. Bicomplex holomorphy: the idempotent representation 163 Remark 7.6.1. Note that the above calculations show that, as is well known, the bicomplex functionF of class C1, seen as a mapping from C2(i)→C2(i), is holo- morphic with respect to βq (q= 1,2) if and only if ∂F
∂βq(Z) = 0in Ω. Note that if F is a bicomplex function, and we express it in cartesian coordinates, it turns out that F isBC-holomorphic if and only if its components are holomorphic as func- tions of two complex variables and satisfy a Cauchy-Riemann type relation between them. As we will show later, this is definitely not the case when we express F in the idempotent representation. In this case, BC-holomorphy will be equivalent to the requirement that each component is a holomorphic function of a single complex variable and there are no relations between the components.
We emphasize that now we are considering the identification between BC andC2(i),
Z=β1e+β2e† ←→(β1, β2)∈C2(i),
where however the basis inC2(i) is not the canonical basis {1,j}, but rather the idempotent basis{e,e†}.
For the next step recall the formulas
H =η1e+η2e†, H†=η2e+η1e†, H =η2e+η1e†, H∗=η1e+η2e†, which imply that
η1=He+H†e†, η1=H∗e+He†, η2=H†e+He†, η2=He+H∗e†. The condition of real differentiability after substitutions becomes:
F(Z+H)−F(Z)
=H ∂F
∂β1(Z)e+ ∂F
∂β2(Z)e†
+H† ∂F
∂β2(Z)e+ ∂F
∂β1(Z)e†
+H ∂F
∂β2(Z)e+ ∂F
∂β1(Z)e†
+H∗ ∂F
∂β1(Z)e+ ∂F
∂β2(Z)e†
+o(H). (7.61)
Note that the expressions in the parentheses are not, yet, the idempotent forms of anything, since the coefficients of e and e† are bicomplex numbers, notC(i)- complex numbers. Thus we are required to make one more step. Using the formula F =G1e+G2e† we arrive at
F(Z+H)−F(Z)
=H ∂G1
∂β1(Z)e+∂G2
∂β2(Z)e†
+H† ∂G1
∂β2(Z)e+∂G2
∂β1(Z)e†
+H ∂G1
∂β2(Z)e+∂G2
∂β1(Z)e†
+H∗ ∂G1
∂β1(Z)e+∂G2
∂β2(Z)e†
+o(H). (7.62)
This formula is valid for any F in C1(Ω), so let us analyze how BC-holomorphic functions are singled out among those of classC1.
Theorem 7.6.2. The C1-function F is BC-holomorphic if and only if the three bicomplex coefficients of H†, H andH∗ in (7.62)are all zero for any Z inΩ.
Proof. The if direction follows as in Theorem 7.4.3. Specifically, since F is BC- holomorphic, formula (7.9) holds for allH /∈S0. But F is a C1-function, hence (7.62) holds as well for anyH = 0, thus both formulas hold for non-zero-divisors.
Then the result follows directly by recalling that both (7.9) and (7.62) are unique representations for a given functionF, and by comparing them.
In order to prove theonly if, it is helpful to write explicitly the meaning of the vanishing of these coefficients, namely:
∂G1
∂β2(Z)e+∂G2
∂β1(Z)e†= 0,
∂G1
∂β2(Z)e+∂G2
∂β1(Z)e†= 0,
∂G1
∂β1(Z)e+∂G2
∂β2(Z)e†= 0. (7.63)
Now note that the second and the third equations, because of the independence ofeande†, impose thatG1andG2areC(i)-valued holomorphic functions of the complex variablesβ1,β2and thus they have authentic complex partial derivatives.
What is more, the first equation in (7.63) says that one of the partial derivatives of eachG1 and G2 is identically zero: ∂G1
∂β2(Z) = 0, ∂G2
∂β1(Z) = 0 for any Z ∈Ω.
Hence, using (7.58) and (7.59),G1is a holomorphic function of the single variable β1∈Ω1 andG2 is a holomorphic function of the single variableβ2∈Ω2. We now want to show that these equations imply thatF isBC-holomorphic. But in fact, because of these equations, we have that for any invertibleH there holds:
F(Z+H)−F(Z)
H =G1(β1+η1)−G1(β1)
η1 e+G2(β2+η2)−G2(β2) η2 e†, whereZ is an arbitrary point in Ω.
Now, by the properties ofG1andG2we deduce that the right-hand side has, for S0 H → 0, the limit G1(β1)e+G2(β2)e†, which concludes the proof: the limit in the left-hand side exists also for any Z ∈ Ω with S0 H → 0 and it coincides with the derivativeF(Z) makingF BC-holomorphic in Ω.
As a matter of fact, the proof allows us to make a more precise characteri- zation ofC1-functions which areBC-holomorphic.
Theorem 7.6.3. A bicomplex function F =G1e+G2e† : Ω⊂BC→BC of class C1 isBC-holomorphic if and only if the following two conditions hold:
7.6. Bicomplex holomorphy: the idempotent representation 165 (I) The componentG1, seen as a C(i)-valued function of two complex variables (β1, β2)is holomorphic; moreover, it does not depend on the variableβ2 and thusG1 is a holomorphic function of the variableβ1.
(II) The componentG2, seen as a C(i)-valued function of two complex variables (β1, β2)is holomorphic; moreover, it does not depend on the variableβ1 and thusG2 is a holomorphic function of the variableβ2.
Remark 7.6.4. The functions G1 andG2 are independent in the sense that there are no Cauchy-Riemann type conditions relating them.
We are in a position now to prove that the converse to Theorem 7.4.3 is true as well.
Theorem 7.6.5. Given F ∈ C1(Ω,BC), then condition (7.53) implies that F is BC-holomorphic.
Proof. If (7.53) holds, then a direct computation shows that all the three formulas in (7.63) are true, and by Theorem 7.6.2F isBC-holomorphic.
The direct computation mentioned above is quite useful and instructive and we will perform it later.
Corollary 7.6.6. Let F be aBC-holomorphic function inΩ, thenF is of the form F(Z) =G1(β1)e+G2(β2)e† with Z =β1e+β2e† ∈Ω and its derivative is given by
F(Z) =G1(β1)e+G2(β2)e†.
Taking into account the relations between β1, β2 and the cartesian compo- nentsz1, z2, we have also that
F(z1+jz2) =G1(z1−iz2)e+G2(z1+iz2)e†; F(Z) =G1(Ze+Z†e†)e+G2(Z†e+Ze†)e†.
This implies that aBC-holomorphic function has derivatives of any order and F(n)(Z) =G(1n)(β1)e+G(2n)(β2)e†
=G(1n)(Ze+Z†e†)e + G(2n)(Z†e+Ze†)e†.
Remark 7.6.7. Although formula (7.62) is quite similar to formula (7.48)its con- sequences for the function F are paradoxically different: while formula (7.48) has allowed us to conclude that the cartesian componentsf1, f2 are holomorphic func- tions of two complex variables which are not independent, formula (7.62)explains to us that the idempotent components G1, G2 are usual holomorphic functions of one complex variable which are, besides, independent.
Remark 7.6.8. We have proved that if F is BC-holomorphic, then for any Z = β1e+β2e† ∈Ωit is of the form
F(Z) =G1(β1)e+G2(β2)e†.
But the right-hand side of the latter is well-defined on the wider setΩ := Ω" 1ãe+ Ω2ãe† ⊃Ω (in general, this inclusion is proper), with the notations as in (7.58) and (7.59). Moreover, by Theorem7.6.3 the functionF" defined by
"
F(Z) :=G1(β1)e+G2(β2)e†, Z∈Ω", is BC-holomorphic in Ω. Since" F"
Ω ≡ F we see that, unlike what happens in the complex case, not every domain inBC is a domain of BC-holomorphy: every function which is BC-holomorphic in a domainΩextendsBC-holomorphically up to the minimal set of the formX1ãe+X2ãe† containingΩ. One can compare this with[81].
Remark 7.6.9. We recall that ifΩ⊂C(i)is an open set bounded by a simple closed curve, then there exists a holomorphic functionf on Ωwith the following proper- ty: if Ω" is any open set which strictly contains Ω, then there is no holomorphic functionf"onΩ" such thatf"restricted to Ωequalsf. UsuallyΩis called adomain of holomorphy forf.
Consider now a bicomplex holomorphic functionF on a domain Ω. Suppose that Ω1 and Ω2 are domains of holomorphy for G1 andG2, respectively, i.e.,G1 andG2cannot be holomorphically extended to any bigger open set inC(i). Then the bicomplex functionF =G1e+G2e†, defined onΩ := Ω" 1ãe+ Ω2ãe† cannot be BC-holomorphically extended to any open set containingΩ. Combining this fact"
with the comments of the previous Remark 7.6.8, we can say thatΩ is a" domain of bicomplex holomorphyforF.
Remark 7.6.10. The same analysis can be done for the idempotent representation withC(j)coefficients.
Remark 7.6.11. Recall that at the beginning of this chapter we worked with the cartesian representation of bicomplex numbers and we investigated many proper- ties of derivable bicomplex functions, in particular, such functions proved to have complex partial derivatives with respect toz1 andz2. This approach fails immedi- ately when one tries to apply it to the case of the idempotent representation: this is because the definition of the derivative excludes precisely the values ofH which are necessary for the complex partial derivatives with respect toβ1, β2. But in the proof of Theorem7.4.3we have shown, as a matter of fact, that such partial derivatives of a BC-holomorphic functions do exist and, moreover, ∂F
∂β1(Z) =G1(β1)ãe and
∂F
∂β2(Z) =G2(β2)ãe†.