It is known that in the study of Euclidean spaces Rn, the cases of n = 2 and n= 4 are peculiar for many reasons but in particular because the corresponding unitary spheresS1 inR2 andS3 inR4are multiplicative groups; this is thanks to the complex numbers multiplication inR2 and the quaternionic multiplication in R4.
Let us consider some analogues of the above facts related to the bicomplex multiplication and bicomplex spheres with a hyperbolic radius:
Sλ:= Z=β1e+β2e† | |Z|k=λ
whereλ∈D+. The multiplicative property of the hyperbolic modulus
|ZãW|k=|Z|kã |W|k (4.37) will be crucial for the reasoning below.
Obviously,S0={0}. Consider the bicomplex unitary sphereS1; clearly
1) 1∈S1;
2) if Z∈S1, then it is invertible andZ−1∈S1; moreover, formula (4.37) says that
3) if Z,W ∈S1, then ZãW ∈S1.
Thus, the bicomplex unitary sphere S1 is a multiplicative group, but this is an exceptional case for bicomplex spheres having a non-zero-divisor as its radius, and for other non-zero-divisor values of the parameter λthe sphere Sλ is not a multiplicative group; the same as inR2.
Recall what happens when λis a positive zero-divisor, i.e., when λ=λ1e∈ D+e, withλ1 >0, or whenλ=λ2e† ∈D+e†, withλ2>0. Ifλ∈D+e, then for the pointsZ=β1e+β2e† ofSλ we have that
|Z|k=|β1| ãe+|β2| ãe† =λ1e, that is,
|β1|=λ1, |β2|= 0, thus, the bicomplex sphereSλis
Sλ={Z |Z=β1e, |β1|=λ1>0},
and it can be seen as the circumference of radiusλ1centered at the origin which is located in the real two-dimensional plane, spanned byeandie(which is a complex line inBC).
Takeλ1= 1 here, thenSλ becomes
Se={Z |Z=β1e, |β1|= 1}.
We are now in a situation very similar to that of Section 2.7.1, and we will use a similar approach in order to makeSea multiplicative group. The setSeis endowed with the multiplicationwhich is the restriction of the bicomplex multiplication and which acts invariantly onSe; then the elementeis the neutral element for, every elementβ1einSe is-invertible and its-inverse is 1
β1e∈Se.
Thus, we conclude that (Se,) is a multiplicative group which is not a sub- group of the multiplicative group (BC\S0,ã). Of course, (Se,) is isomorphic to the multiplicative group of complex numbers of modulus one.
Note that ifλ1= 1, thenSλ1e is not a multiplicative group. In other words, among all the spheresSλ1e withλ1>0, only one,Se, is a multiplicative group.
The situation withλ=λ2e† is similar.
In what follows we assume that
λ∈D+inv, (4.38)
104 Chapter 4. Lines and curves inBC that is,λis a positive hyperbolic number; recall thatD+invis a multiplicative group.
For any such λthe sphere Sλ enjoys the property that anyZ in Sλ is invertible and
Z ∈Sλ if and only if Z−1∈Sλ−1. Notice also that ifZ∈Sλ andW ∈Sμ, then by (4.37)
|ZãW|k=λãμ∈D+inv, hence
ZãW ∈Sλμ.
These properties hint that we should consider all the spheres “together” and we set:
SD:= #
λ∈D+inv
Sλ
(i.e.,SDis a disjoint union of spheres). The setSD is a multiplicative group with respect to the bicomplex multiplication.
We discussed already some specific features of bicomplex spheres with real radii, so consider the subset ofSDdefined by
SR:= #
λ∈R+
Sλ.
A sphereSλ withλ >0 is characterized by the condition: ifZ =β1e+β2e†∈Sλ, then
|β1|=|β2|=λ=|Z|=|Z|k.
It is easily seen thatSR is a group also, and thusSR is a subgroup of SD. Since S1⊂SR and sinceS1is a group, thenS1is a subgroup of bothSRandSD.
The bicomplex spheres generate another set which can be endowed with the structure of a multiplicative group. Indeed, introduce
"
SD:={Sλ}λ∈D+
inv
and define the multiplication “◦” of bicomplex spheres by the formula Sλ◦Sμ:=Sλμ.
The unitary sphere S1 will serve as the identity in "SD (i.e., the multiplicative neutral element), and the sphereSλ−1 is the inverse of the sphereSλ.
The subset
"
SR:={Sλ}λ∈R+
ofS"D is also a group and, thus, a subgroup of the groupS"D.
Observe that the mapping
ϕ:λ∈D+inv−→Sλ∈S"D
is a group isomorphism, and its restrictionϕ|R+ is a group isomorphism as well:
ϕ|R+:λ∈R+−→Sλ∈S"R.
The setsSDandS"Dhave proved to be multiplicative groups since the zero-divisors were forbidden to be radii of the spheres. Let us return to these exceptions and let us consider the sets
Se,D:= #
λ∈D+e
Sλ; Se†,D:= #
λ∈D+e†
Sλ.
Note that the setsD+e andD+e† have become multiplicative groups when a specific multiplication was introduced in each of them. This allows us, again, to makeSe,D and Se†,D multiplicative groups following the pattern of Section 2.7.1. If Z ∈Sλ
andW ∈Sμ, i.e.,λ=λ1eandμ=μ1e,Z=β1ewith|β1|=λ1>0 andW =γ1e with|γ1|=μ1 >0; thenZW :=β1γ1e∈Sλ1μ1e=Sλμ. Hence, with this new multiplication the setSe,Dbecomes a multiplicative group. The same forSe†,D.
The analogues of"SD are the sets
"
Se,D:={Sλ}λ∈D+e and S"e†,D:={Sλ}λ∈D+
e†.
The multiplicationof bicomplex spheres inS"e,Dis defined by the formula SλSμ:=Sλμ.
Similarly forS"e†,D. The reader is invited to complete the proofs.
Most of the material covered in this Chapter is original and does not have counterparts in the existing literature. Our detailed study of complex and hyper- bolic lines and curves in BC is of great importance, as mentioned at the end of Chapter 3, e.g. for applications in the mathematics of Special Relativity, [11].
Bicomplex spheres and balls have also been studied in [49, 56] from an Eu- clidean point of view, and, from a manifold theory point of view, in [7]. We stress once more that our approach to this subject involves the hyperbolic norm| ã |k, a notion that was first developed in [2]. We strongly believe that a comprehensive study of the applications of this norm in the context of Special Relativity is well worth pursuing in the future, a realm of study that is beyond the scope of this book.
Chapter 5
Limits and Continuity
The notion of limit for complex functions is well known and we will not redis- cuss it here. Note that the formal proofs of its properties depend strongly on the properties of the modulus of a complex number;
|ab|=|a| ã |b|, |a+b| ≤ |a|+|b|, 1
a = 1
|a| for a= 0. (5.1) In particular, the existence of the limit of a function is equivalent to the existence of the limits of its real and imaginary parts; moreover, the limit of a function exists if and only if the limit of its conjugate function exists.
Our aim in this chapter is to extend the above to bicomplex functions show- ing that there exist many similarities but differences as well. Indeed, instead of properties (5.1) for the Euclidean modulus one has their analogs of the form
|ZãW| ≤√
2|Z| ã |W|, |Z+W| ≤ |Z|+|W|, (5.2) and it turns out that they allow us to repeat most of the proofs almost literally, although one needs to take into account the fact that|Z−1|is not always equal to
1
|Z|.