CUBO A Mathematical Journal
Vol.14, No¯ 02, (61–80). June 2012

Bicomplex Numbers and their Elementary Functions

M.E. Luna-Elizarrarás, M. Shapiro

Departamento de Matemáticas,

E.S.F.M del I.P.N., México.

and

D.C. Struppa1, A. Vajiac

Schmid College of Science and Technology,

Chapman University, Orange California,
1 email: struppa@chapman.edu

ABSTRACT

In this paper we introduce the algebra of bicomplex numbers as a generalization of the

field of complex numbers. We describe how to define elementary functions in such an

algebra (polynomials, exponential functions, and trigonometric functions) as well as

their inverse functions (roots, logarithms, inverse trigonometric functions). Our goal

is to show that a function theory on bicomplex numbers is, in some sense, a better

generalization of the theory of holomorphic functions of one variable, than the classical

theory of holomorphic functions in two complex variables.

RESUMEN

En este art́ıculo introducimos el álgebra de números bicomplejos como una general-

ización del campo de números complejos. Describimos cómo definir funciones elemen-

tales en tales álgebras (polinomios y funciones exponenciales y trigonométricas) aśı

como sus funciones inversas (ráıces, logaritmos, funciones trigonométricas inversas).

Nuestro objetivo es mostrar que una teoŕıa de funciones sobre números bicomplejos es,

en cierto sentido, una mejor generalización de la teoŕıa de funciones holomorfas de una

variable compleja, que la teoŕıa de funciones holomorfas en dos variables complejas.

Keywords and Phrases: Bicomplex numbers, Elementary functions

2010 AMS Mathematics Subject Classification: 30G35



62 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
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1 Introduction

Consider two C1 functions u,v from R2 = {(x1,x2) : x1 ∈ R,x2 ∈ R} to R. It is well known that if
these two functions satisfy the so-called Cauchy-Riemann system

∂u

∂x1
=
∂v

∂x2

∂u

∂x2
= −

∂v

∂x1

then the function f(x1 + ix2) := u(x1,x2) + iv(x1,x2) admits complex derivative (i.e. the limit

limh→0
f(z+h)−f(z)

h
exists), and is what we call a holomorphic function. This observation is the key

point of the theory of one complex variable, and shows that the entire theory relies on considering

pairs of differentiable functions connected by a simple system of linear, constant coefficients, first

order, partial differential equations.

There are different ways to attempt to generalize this observation to the case of more pairs

of real variables. For example, if we consider two such pairs defined on two independent sets of

variables (i.e. a map defined on R4 with values in R4), one can consider quaternion valued functions

of a quaternionic variable, and a very interesting theory of holomorphicity was developed by Fueter

[3] (though others like Moisil [4] and Mosil-Teodorescu [5] introduced similar ideas before him).

Another way to generalize this observation consists in looking at maps ~f = (f1,f2) from C
2 to C2,

and to ask that each component f1,f2 be holomorphic in both variables in C
2, without assuming

any additional relationship between them. Though both generalizations are important, and give

rise to large and interesting theories, we believe that there is another even more appropriate

generalization, which so far has not received enough attention.

To this purpose, we propose to complexify the Cauchy-Riemann system and to apply it to

pairs of holomorphic functions u,v from C2 = {(z1,z2) : z1 ∈ C,z2 ∈ C} to C, so that the pair
(u,v) can be interpreted as a map of C2 to itself. It is then natural to ask whether it makes any

sense to consider pairs (u,v) for which the following system is satisfied:

∂u

∂z1
=
∂v

∂z2

∂u

∂z2
= −

∂v

∂z1
.

Formally, we have replaced R by C, and differentiability in the real sense by holomorphicity.

Does this have any implications on the pair (u,v)? As it turns out, it is possible to give a very

interesting interpretation of this complexified Cauchy-Riemann system, if we endow the pair (z1,z2)

with a special algebraic structure. Instead of considering (z1,z2) as a point in C
2 we now consider,

in analogy with what we did in the case of R2, a new space whose elements are of the form

Z = z1 + jz2, where j is a new imaginary unit (i.e. j
2 = −1), which commutes with the original

imaginary unit i. This creates a new algebra, the algebra of bicomplex numbers, and as we will



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Bicomplex Numbers and their Elementary Functions 63

show in the next sections, such an algebra enjoys most of the properties one would expect from a

good generalization of the field of complex numbers.

There is another equally interesting way of introducing bicomplex numbers. We recall, for

example, that complex numbers are important because they allow the factorization of the positively

definite real quadratic form

x21 + x
2
2 = (x1 + ix2)(x1 − ix2),

which defines the geometry of the plane and is the symbol of the 2−dimensional Laplace operator.

One may therefore ask whether it is possible to factor the complex analog of the form, i.e. the

C−valued quadratic form z21 + z
2
2, which is the symbol of the 2−dimensional complex Laplace

operator. A trivial answer is to express such form as a product of two linear complex valued

factors as

z21 + z
2
2 = (z1 + iz2)(z1 − iz2).

These two factorizations may appear superficially similar, but in fact there is a deep difference

between them. The factorization of x21 + x
2
2 is realized through real 2−dimensional factors, while

the factorization of z21 +z
2
2 is realized through complex 1−dimensional factors. One may therefore

ask whether it is possible to factor the complex quadratic form through two factors which have

complex dimension 2. In order to do so, we consider a distributive, and associative (but not

necessarily commutative) complex algebra over C2, and we assume we have two elements a,b in

this algebra, such that

z21 + z
2
2 = (z1 + az2)(z1 + bz2) = z

2
1 + az2z1 + z1bz2 + az2bz2. (1.1)

This implies immediately that, for every z1,z2 we must have

az2z1 + z1bz2 = 0, (1.2)

which, for z1 = 1, gives

az2 + bz2 = 0,

and therefore a = −b. By substituting in (1.2) we obtain

az2z1 − z1az2 = 0,

which, for z2 = 1, gives

az1 = z1a.

This shows that a is not a complex number, but it commutes with every complex number. By

inserting these results in (1.1) we obtain

z21 + z
2
2 = z

2
1 − a

2z22,

i.e. a2 = −1. Once again, we have arrived to a new structure, which requires a to be a second

imaginary unit, which we will call j in the sequel, that commutes with the initial imaginary unit i.



64 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
14, 2 (2012)

The algebra which one obtains is the bicomplex algebra. In this paper we show how to introduce

elementary functions, such as polynomials, exponentials, trigonometric functions, in this algebra,

as well as their inverses (something that, incidentally, is not possible in the case of quaternions).

We will show how these elementary functions enjoy properties that are very similar to those enjoyed

by their complex counterparts. In addition, as we will indicate below, this algebraic structure will

allow us to show that any pair of holomorphic functions that satisfies the complexified Cauchy-

Riemann system admits derivative in the sense of bicomplex numbers.

2 The bicomplex numbers

We gave, in the introduction, a couple of justifications for the introduction of the notion of bi-

complex numbers. It is however also possible to arrive to bicomplex numbers by means of purely

algebraic considerations. For example, if in a complex number a + ib we replace the real numbers

a and b by complex numbers z1 = a1 + ia2 and z2 = b1 + ib2, then we get just another complex

number:

z1 + iz2 = (a1 + ia2) + i (b1 + ib2) = (a1 − b2) + i (a2 + b1) .

If we want to obtain a new type of number, then we must use another imaginary unit, say j, with

j2 = −1, and set

z1 + jz2 = (a1 + ia2) + j (b1 + ib2) ,

which gives a new object, outside the field of complex numbers.

If we want to be able to operate with these new numbers, we need to define the product

of the two imaginary units. This was a problem that was solved by Hamilton by requiring that

they anticommute, and his solution led to the introduction of quaternions. Hamilton’s decision

was influenced by many considerations, including the desire to obtain a field, which of course the

quaternions form (a skew field). But one could explore what happens if we assume that the two

new imaginary units commute. In this case we obtain a new, and lesser known object, the algebra

of bicomplex numbers.

The set BC of bicomplex numbers is therefore defined as follows:

BC = {z1 + jz2
∣

∣z1,z2 ∈ C},

where i and j are commuting imaginary units, i.e. ij = ji, i2 = j2 = −1, and C is the set of

complex numbers with the imaginary unit i. Thus bicomplex numbers are “complex numbers with

complex coefficients”, which explains the name of bicomplex, and in what follows we will try to

emphazise the deep similarities between the properties of complex and bicomplex numbers. We

should probably point out that bicomplex numbers were apparently first introduced in 1892 by

Segre, [12], that the origin of their function theory is due to the Italian school of Scorza-Dragoni

([13], [14], [15]), and that a first theory of differentiability in BC was developed by Price in [7].



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Bicomplex Numbers and their Elementary Functions 65

Subsequently, other authors have developed further the study of these objects, [2], [6], [8], [9] . A

key role, in this evolution, has been played by John Ryan, who was probably the first to understand

the importance of complex Clifford Algebras (of which BC is the simplest example, and the only

commutative one), and to highlight their role in analysis, [10], [11].

A bicomplex number can be written in cartesian form as Z = z1 + jz2 or, at least as long as

z2
1
+ z2

2
6= 0, in trigonometric form as

Z = z1 + jz2 =

√

z2
1
+ z2

2





z1
√

z2
1
+ z2

2

+ j
z2

√

z2
1
+ z2

2





=

√

z2
1
+ z2

2
(cosθ + j sinθ) ,

(2.1)

where the complex number θ is a solution of the system

cos(θ) =
z1

√

z2
1
+ z2

2

, sin(θ) =
z2

√

z2
1
+ z2

2

. (2.2)

Since for the particular case of real z1 and z2,
√

z2
1
+ z2

2
is the modulus of a complex number and

θ is its argument, the complex number
√

z2
1
+ z2

2
is called the complex modulus of the bicomplex

number Z, denoted by |Z|c, and θ is called the complex argument of Z, denoted by argc(Z). These

names can be given a deeper justification, but this is beyond the scope of this article.

It can be shown by elementary complex analysis that the apparent ambiguities in formula (2.1)

can always be resolved by choosing either the principal or the secondary branch of the complex

square root
√

z2
1
+ z2

2
. Either way formula (2.1) is well-defined.

Now the addition and the multiplication of bicomplex numbers are introduced in a natural

way: given Z1 = z11 + jz12 and Z2 = z21 + jz22 in BC, then

Z1 + Z2 := (z11 + z21) + j(z12 + z22). (2.3)

and

Z1 · Z2 := (z11 + jz12)(z21 + jz22) = (z11z21 − z12z22) + j(z11z22 + z21z12). (2.4)

It is a simple exercise left to the reader to verify the following

Proposition 1. (BC,+, ·) is a commutative ring, i.e.

(1) The addition is associative, commutative, with identity element 0 = 0+j0, and all bicomplex

numbers have an additive inverse. This is to say that (BC,+) is an Abelian group.

(2) The multiplication is associative, commutative, with identity element 1 = 1 + j0.



66 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
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(3) The multiplication is distributive with respect to the addition, i.e. for any Z,Z1,Z2 ∈ BC,
we have:

Z(Z1 + Z2) = ZZ1 + ZZ2. (2.5)

Remark 2. The process described above has allowed us to endow the complex linear space C2

with a structure of a commutative complex algebra. Note that quaternions form only a real, not a

complex algebra. On the other hand, quaternions are a (skew) field, while we will soon show that

not all bicomplex numbers have a multiplicative inverse.

2.1 Bicomplex Conjugation

Given a bicomplex number Z = z1 + jz2, its (bicomplex) conjugate is defined by

Z† := z1 − jz2.

We immediately notice that

Z · Z† = z21 + z22 ∈ C. (2.6)

This last equality is only apparently similar to the corresponding identity for complex numbers;

however the quadratic form in (2.6) takes complex values (rather than real non-negative values),

and this implicates significant differences with the complex situation. In particular it implies that

a bicomplex number Z = z1 + jz2 is invertible if and only if

Z · Z† = z21 + z22 6= 0. (2.7)

In this case, it is easy to verify that the inverse of Z is given by

Z−1 =
Z†

z2
1
+ z2

2

.

If both z1 and z2 are non-zero but the sum z
2
1 +z

2
2 = 0, then the corresponding bicomplex number

Z = z1 + jz2 is a zero divisor. In fact all zero divisors Z = z1 + jz2 in BC are characterized by the

equations z21 = −z
2
2, i.e. z1 = ±iz2. Thus all zero divisors are of the form:

Z = λ(1 ± ij),

for any λ ∈ C \ {0}.

The following proposition is a simple exercise.

Proposition 3. The bicomplex numbers

e :=
1 + ij

2
and e† :=

1 − ij

2



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Bicomplex Numbers and their Elementary Functions 67

are zero divisors, which are linearly independent in the C-linear space C2, and satisfy the identities:

e + e† = 1, e − e† = ij,

e · e† = 0, e2 = e, e†2 = e†.

The next property has no analog for complex numbers, and it exemplifies one of the interesting

peculiarities of the bicomplex setting. For any bicomplex number Z = z1 +jz2 ∈ BC one can write

Z = αe + βe†, (2.8)

where α = z1 −iz2 and β = z1 +iz2 are uniquely defined complex numbers. Formula (2.8) is called

the idempotent representation of Z.

This shows that the set {e,e†} is another basis for the complex space BC, and writing Z as a

pair (z1,z2) in C
2, one has the transition formula









z1

z2









=











1

2

1

2

−
1

2i

1

2i



















α

β









. (2.9)

Note that this new basis is orthogonal with respect to the Euclidean inner product in C2 which is

given for (z1,w1) and (z2,w2) by

〈(z1,w1),(z2,w2)〉C2 := z1z2 + w1w2.

Since e =

(

1

2
,
i

2

)

and e† =

(

1

2
,−

i

2

)

in C2, we have 〈e,e†〉C2 = 0, and

〈e,e〉C2 = 〈e†,e†〉C2 =
1

2
,

so that we have an orthogonal but not orthonormal basis for C2.

The following result shows the importance of the idempotent representation of bicomplex

numbers in all algebraic operations.

Proposition 4. The addition and multiplication of bicomplex numbers can be realized “term-by-

term” in the idempotent representation. Specifically, if Z1 = αe + βe
† and Z2 = γe + δe

† are

two bicomplex numbers, then

Z1 + Z2 = (α + γ) e + (β + δ) e
†,

Z1 · Z2 = αγe + βδe†,
Zn1 = α

n
e + βn e†.



68 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
14, 2 (2012)

Moreover, the inverse of an invertible bicomplex number Z = αe + βe† is given by

Z−1 = α−1e + β−1 e†,

where α−1 and β−1 are the complex multiplicative inverses of α and β, respectively.

3 Bicomplex Derivatives

3.1

Let f : U ⊂ BC → BC be a bicomplex function. There is a definition for the derivative of a
bicomplex function (see e.g. [7]) which looks quite similar to its complex counterpart.

Definition 5. The derivative of the function f at a point Z0 ∈ U is the limit, if it exists,

f′(Z0) := lim
Z→Z0

f(Z) − f(Z0)

Z − Z0
, (3.1)

for Z in the domain of f such that Z − Z0 is an invertible bicomplex number.

It is never emphasized in the literature that this limit is tacitly taken in the usual Euclidean

topology of C2 (we call this the usual Euclidean convergence in BC), which seems not to be the one

generated by the natural structure of BC (see for instance (2.7)). We will show below that there

is another, equivalent approach to the limit of a bicomplex function which employs the specific

algebraic structure of bicomplex numbers.

3.2

Let Zn = αne + βne
† for n ≥ 1, be a sequence of bicomplex numbers.

Definition 6. The sequence {Zn}n≥1 is said to converge component-wise if the sequences of complex

numbers {αn} and {βn} are convergent in the complex plane to complex numbers α0 and β0. We

then write that Zn → Z0 := α0e + β0e
†, and we say that Zn has limit Z0.

Formula (2.9) shows the equivalence between the usual Euclidean convergence and the component-

wise convergence defined above. Moreover, the Euclidean convergence is the general definition of

convergence in C2, but the component-wise convergence expresses the relation between the topol-

ogy and the algebraic structure of BC.

3.3

Component-wise convergence of sequences allows us to define the notion of component-wise limits

of bicomplex functions.



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Bicomplex Numbers and their Elementary Functions 69

Consider a bicomplex function f = g+jh, f : U ⊂ BC → BC, and its idempotent representation
f = ue +ve†. If for any sequence {Zn}n≥1 component-wise convergent to Z0, the complex number

sequences {u(Zn)} and {v(Zn)} are convergent in C to λ and µ respectively, then the function f has

a (usual) limit as Z → Z0 (with respect to the canonical topology in C
2), and we have:

lim
Z→Z0

f(Z) = λe + µe†.

A description of continuity of a bicomplex function which is compatible with the algebraic structure

of BC follows immediately.

We mentioned in the introduction that bicomplex numbers are the appropriate setting to

consider a complexification of the Cauchy-Riemann equations. That this is the case is demonstrated

by the following important result:

Theorem 7. Let U be an open set in BC, whose variable we indicate with Z = z1 + jz2 and let

f : U → BC be such that f = u+ jv ∈ C1(U). Then f admits bicomplex derivative f′ if and only if:

(1) u and v are complex holomorphic in z1 and z2

(2)
∂u

∂z1
=
∂v

∂z2
and

∂u

∂z2
= −

∂v

∂z1
on U.

4 Bicomplex polynomials

Let

p(Z) =

n∑

k=0

AkZ
k

be a bicomplex polynomial of degree n, with Z = z1 + jz2 = αe + βe
†, and bicomplex coefficients

Ak = γke + δke
†, for k = 0. . .n. Then Zk = αke + βke† and we can rewrite p(Z) as

p(Z) =

n∑

k=0

(

γkα
k
)

e +

n∑

k=0

(

δkβ
k
)

e
† =: φ(α)e + ψ(β)e†.

If we denote the set of distinct roots of φ and ψ by S1 and S2, and if we denote by S the set
of distinct roots of the polynomial p, it is easy to see that

S = S1e + S2e†,

so that the structure of the zero-set of a bicomplex polynomial p(Z) of degree n is fully described

by the following three cases:

(1) If both polynomials φ and ψ are of degree at least one, and if S1 = {α1, . . . ,αk} and S2 =
{β1, . . . ,βℓ}, then the set of distinct roots of p is given by

S = {Zs,t = αse + βte†
∣

∣s = 1, . . . ,k, t = 1, . . . , ℓ} .



70 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
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(2) If φ ≡ 0, then S1 = C and S2 = {β1, . . . ,βℓ}, with ℓ ≤ n. Then

S = {Zt = λe + βte†
∣

∣λ ∈ C, t = 1, . . . , ℓ} .

Similarly, if ψ ≡ 0, then S2 = C and S1 = {α1, . . . ,αk}, where k ≤ n. Then

S = {Zs = αse + λe†
∣

∣λ ∈ C, s = 1, . . . ,k} .

(3) If all the coefficients Ak with the exception of A0 = γ0e + δ0e
† are complex multiples of e

(respectively of e†), but A0 has δ0 6= 0 (respectively γ0 6= 0), then p has no roots.

We now discuss a few examples, to give a flavor for computations in BC. First, consider the

polynomial

p(Z) =

(

1

2
+ j

i

2

)

Z5 + (−(1 + 4i) + 2j(2 − i))Z4 + ((−11 + 6i) − j (12 + 11i))Z3

+

((

29

2
+ 13i

)

+ j

(

−13 +
47

2
i

))

Z2 +

((

13

2
− 17i

)

+ j

(

17 +
13

2
i

))

Z

−

(

11

2
+ i

)

+ j

(

1 −
11

2
i

)

.

The corresponding complex polynomials are:

φ(α) = α5 − (3 + 8i)α4 + 2(−11 + 9i)α3 + 2(19 + 13i)α2 + (13 − 34i)α − (11 + 2i) ,

ψ(β) = β4 − 6iβ3 − 9β2.

Their distinct roots are S1 = {i,1 + 2i} and S2 = {0,3i}. Then p has the following four roots:

S =
{
1

2
i −

1

2
j, 2i + j,

1 + 2i

2
+ j

−2 + i

2
,
1 + 5i

2
+ j
1 + i

2

}

.

As another example, consider the polynomial

p(Z) = (1 + ji)Z2 − (i − j) .

The associated complex polynomials are:

φ(α) = 2(α2 − i), ψ(β) ≡ 0.

The null set of p is

S =
{

±
(√

2

2
+ i

√
2

2

)

e + λe†
∣

∣λ ∈ C
}

.

Slightly adjusting the previous example, i.e. taking ψ(β) ≡ 2, we get the polynomial

p(Z) = (1 + ji)Z2 + (1 − i) + j (1 − i) ,



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Bicomplex Numbers and their Elementary Functions 71

which has no roots.

It is also important to note that a bicomplex polynomial may not have a unique factorization

into linear polynomials. For example, the polynomial p(Z) = Z3 − 1 has 9 solutions. Indeed, the

associated complex polynomials are

φ(α) = α3 − 1, φ(β) = β3 − 1.

The set of zeros of φ and ψ are, respectively:

S1 =
{

α1 = 1,α2 = −
1

2
+ i

√
3

2
,α3 = −

1

2
− i

√
3

2

}

S2 =
{

β1 = 1,β2 = −
1

2
+ i

√
3

2
,β3 = −

1

2
− i

√
3

2

}

Then the set of solutions of p is

S =
{
Zkl = αke + βℓe

†
∣

∣k,ℓ = 1. . .3
}
,

and we have at least two distinct factorizations:

Z3 − 1 = (Z − 1)

(

Z +
1

2
−

√
3

2
i

)(

Z +
1

2
+

√
3

2
i

)

and

Z3 − 1 = (Z − 1)

(

Z +
1

2
− j

√
3

2

)(

Z +
1

2
+ j

√
3

2

)

.

It is therefore clear from what we have indicated that bicomplex polynomials do not satisfy

the Fundamental Theorem of Algebra in its original form. At the same time, the following is true

and summarizes the comments above.

Theorem 8 (Analogue of the Fundamental Theorem of Algebra for bicomplex polynomials). Con-

sider a bicomplex polynomial p(Z) =
n∑

k=0

AkZ
k. If all the coefficients Ak with the exception of

the free term A0 = γ0e + δ0e
† are complex multiple of e (respectively of e†), but A0 has δ0 6= 0

(respectively γ0 6= 0), then p has no roots. In all other cases, p has at least one root.
Corollary 9. Assume that a bicomplex polynomial p of degree n ≥ 1 has at least one root. Then:

(1) If at least one of the coefficients Ak, for k = 1. . .n, is invertible, then p has at most n
2

distinct roots.

(2) If all coefficients are complex multiples of e (respectively e†) then p has infinitely many roots.

Note that zeros of bicomplex polynomials were originally investigated in [6].



72 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
14, 2 (2012)

5 The exponential function in bicomplex numbers

In this section, we are going to introduce the exponential function of a bicomplex variable. Our ap-

proach is based on the following theorem, whose proof requires the usage of minimal mathematical

tools.

Theorem 10. Let Z = z1 + jz2 be any bicomplex number. Then the sequence

Zn :=

(

1 +
Z

n

)n

is convergent.

Proof. The computation below proves that the sequence is component-wise convergent. Set as

before Z = αe + βe†. Then

(

1 +
Z

n

)n

=

(

1 +
α

n
e +

β

n
e
†

)n

=

(

e + e† +
α

n
e +

β

n
e
†

)n

=

(

(

1 +
α

n

)

e +

(

1 +
β

n

)

e
†

)n

=
(

1 +
α

n

)n

e +

(

1 +
β

n

)n

e
† .

By taking the limit as n → ∞, and relying on the fact that the corresponding sequences of

complex numbers
(

1 +
α

n

)n

and

(

1 +
β

n

)n

are convergent to the complex exponentials eα and

eβ, respectively, we get that the limit of the right-hand-side exists and

lim
n→∞

(

1 +
Z

n

)n

= lim
n→∞

(

(

1 +
α

n

)n

e +

(

1 +
β

n

)n

e
†

)

= eαe + eβe†

=
1

2
(eα + eβ) + j

i

2
(eα − eβ)

=
1

2
(ez1−iz2 + ez1+iz2) + j

i

2
(ez1−iz2 − ez1+iz2)

= ez1
(

1

2
(e−iz2 + eiz2) + j

i

2
(e−iz2 − eiz2)

)

= ez1 (cos(z2) + j sin(z2)) .

(5.1)

This concludes our proof.

Clearly, the theorem justifies the following definition.



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Bicomplex Numbers and their Elementary Functions 73

Definition 11. We set

eZ := lim
n→∞

(

1 +
Z

n

)n

= ez1 (cos(z2) + j sin(z2)) .

One observes a marvelous similarity with the definitions of the Euler number e and with the

exponential functions in real and complex numbers.

We pass now to the properties of this newly introduced bicomplex exponential function.

• First we note that the bicomplex exponential is an extension to BC of the complex exponential
function: indeed, for Z = z1 + j0 ∈ C, we have that

eZ = ez1 (cos(0) + j sin(0)) = ez1,

which is the usual complex exponential function.

• Note that ez1 is the complex modulus of the bicomplex number eZ, and z2 is the complex
argument of the same bicomplex number eZ. The reader may find it instructive to compare

this fact with what happens in the complex case.

• For Z = 0 = 0e + 0e†, we have: e0 = 1e + 1e† = 1.

• For any bicomplex number Z, the exponential eZ is invertible. This is because

eZ = ez1−iz2e + ez1+iz2e†

and the exponential terms ez1−iz2 and ez1+iz2 are complex exponential functions, so they

are never zero. The inverse multiplicative of eZ is

e−Z = e−(z1−iz2)e + e−(z1+iz2)e† = e−z1 (cos(z2) − j sin(z2)) .

Thus, the range of the bicomplex exponential function does not contain neither the zero nor

any zero divisors.

• Two curious facts. For e = 1 · e + 0 · e†, and e† = 0 · e + 1 · e†, we have:

ee = e · e + 1 · e† = e12
(

cos

(

i

2

)

+ j sin

(

i

2

))

= e
1
2

(

cosh

(

1

2

)

+ ji sinh

(

1

2

))

.

Similarly:

ee
†

= 1 · e + e · e† = e12
(

cos

(

i

2

)

− j sin

(

i

2

))

= e
1
2

(

cosh

(

1

2

)

− ji sinh

(

1

2

))

.



74 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
14, 2 (2012)

• Due to the commutativity of the multiplication in BC, we can show that for any Z1 =
z11 + jz12 and Z2 = z21 + jz22 in BC, the following formula holds:

eZ1eZ2 = eZ1+Z2. (5.2)

Indeed, we have:

eZ1eZ2 = (ez11 (cos(z12) + j sin(z12))) (e
z21 (cos(z22) + j sin(z22)))

= ez11ez21 ((cos(z12) cos(z22) − sin(z12) sin(z22))

+j(sin(z12) cos(z22) + sin(z22) cos(z12)))

= ez11+z21 (cos(z12 + z22) + j sin(z12 + z22)) = e
Z1+Z2 .

This equality means that the exponential function is a homomorphism from the additive

group of bicomplex numbers into the multiplicative group of invertible bicomplex numbers.

• In the case Z = 0 + jz2, we have:

eZ = ejz2 = cos(z2) + j sin(z2).

• The complex formula eiπ+1 = 0 remains valid for bicomplex numbers, but it is complemented
with its mirror image ejπ + 1 = 0.

• For any Z = αe+βe† ∈ BC, and any invertible bicomplex number W = γe+δe†, i.e. γδ 6= 0,
the equation eZ = W is equivalent to the system eα = γ and eβ = δ. Because γδ 6= 0, it
follows that there is always a solution.

• We leave as an exercise to the reader to verify that the bicomplex derivative of eZ is still eZ.

• Recalling that the complex exponential function and the complex trigonometric functions are
periodic, we obtain that

eZ = ez1 (cos(z2) + j sin(z2))

= ez1+2πim (cos(z2 + 2πn) + j sin(z2 + 2πn))

= eZ+2π(mi+nj),

for m and n integer numbers. Thus the bicomplex exponential function is periodic with

bicomplex periods 2π(mi + nj). One can prove that these are the only periods.

6 Trigonometric functions of a bicomplex variable

Adding and subtracting the formulas ejz2 = cos(z2) + j sin(z2) and e
−jz2 = cos(z2) − j sin(z2), for

any z2 ∈ C, we express the complex cosine and sine via the bicomplex exponential:



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Bicomplex Numbers and their Elementary Functions 75

cosz2 =
ejz2 + e−jz2

2
,

sinz2 =
ejz2 − e−jz2

2j
. (6.1)

Thus we are in a position to introduce the bicomplex sine and cosine functions which are direct

extensions of their complex antecedents.

Definition 12. Let Z = z1 + jz2 ∈ BC. We define the bicomplex cosine and sine functions of a
bicomplex variable as follows:

cosZ :=
ejZ + e−jZ

2
,

sinZ :=
ejZ − e−jZ

2j
. (6.2)

Given Z = z1 + jz2 = αe + βe
† ∈ BC, the properties of the bicomplex exponential bring us

immediately to the idempotent representation of cosZ and sinZ:

cosZ = cos(α)e + cos(β)e† ,

sinZ = sin(α)e + sin(β)e† . (6.3)

In terms of the components of the cartesian representation, one gets:

cosZ = cos(z1 − iz2)e + cos(z1 + iz2)e
† .

Since for a complex variable z the following formulas hold:

cosh(z) = cos(iz), sinh(z) = −i sin(iz),

we obtain that

cosZ = cosh(z2) cos(z1) − j sinh(z2) sin(z1).

We continue with a description of some basic properties of the trigonometric bicomplex func-

tions.

• Since the complex sine and cosine functions are periodic with principal period 2π, then taking
Z = αe +βe† and setting Zk,ℓ = (α+2kπ)e + (β+2ℓπ)e

† for arbitrary integers k,ℓ we have:

cos(Zk,ℓ) = cos(Z), sin(Zk,ℓ) = sin(Z).

Thus the real number (2π)e + (2π)e† = 2π remains the principal period of both bicomplex

sine and cosine functions.



76 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
14, 2 (2012)

• From (6.3), the equation cosZ = 0 is equivalent to the equations in complex variables α and
β:

cos(α) = 0, cos(β) = 0.

The solutions are α =
π

2
+ kπ, and β =

π

2
+ ℓπ, for k,ℓ ∈ Z. Note that α and β are never

0, so the bicomplex solutions Z to cosZ = 0 are always invertible. In the {1,j} basis, we get

the general solution to cosZ = 0 as

Z = z1 + jz2 = ((1 + k + ℓ) + j i(k − ℓ))
π

2
. (6.4)

• Similarly, the equation sinZ = 0 is equivalent to

sin(α) = 0, sin(β) = 0.

The solutions are α = kπ, and β = ℓπ, for k,l ∈ Z. Note that there are non-invertible
solutions for sinZ = 0, e.g. for α = 0, i.e. k = 0, and β 6= 0. In the {1,j} basis, we get the
general solution for sinZ = 0 as

Z = z1 + jz2 = (k + ℓ + j i(k − ℓ))
π

2
.

• The component-wise formulas (6.3) guarantee that the usual trigonometric identities are true,
e.g., the sums and differences of angle formulas, the double angle identities, etc. For example:

sin2 Z + cos2 Z = (sin2(α) + cos2(α))e + (sin2(β) + cos2(β))e† = 1.

• It turns out that both functions have the derivatives which extend directly their complex
antecedents, i.e.

(cosZ)′ = − sinZ, (6.5)

(sinZ)′ = cosZ. (6.6)

7 Bicomplex Radicals

In this and the next section we begin the study of inverse functions in BC. We start by looking

at the equation Zn = W, where Z = z1 + jz2 = αe + βe
†, and W = w1 + jw2 = ae + be

†. This

system is equivalent to the following two complex equations in variables α and β:

αn = a, βn = b.

If W is invertible, i.e. ab 6= 0, each complex equation has n distinct complex solutions, and the
equations are independent of each other. Denote these solutions by ak ∈ n

√
a and bℓ ∈ n

√
b,

respectively. Therefore the bicomplex equation Zn = W has, in general, n2 solutions given by the

bicomplex numbers

Zkℓ = ake + bℓe
† =

ak + bℓ

2
+ j
bℓ − ak

2i



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Bicomplex Numbers and their Elementary Functions 77

for all k,ℓ = 1. . .n. We define the n-th root of W to be the set of all of these solutions,
n
√
W :=

{Zkℓ}.

Note that if we start with formula (2.1) for the bicomplex number W = w1 + jw2, i.e.

W = |W|c(cosθ + j sinθ)

where |W|c =
√

w2
1
+ w2

2
is the complex modulus of W, and θ is the complex argument of W,

then the solutions Zkℓ of the equation Z
n = W have complex modulus n

√

|W|c, which is a set of n

complex numbers, and arguments
θ + 2ℓπ

n
, for ℓ = 1. . .n. In conclusion, we find again that there

are n2 bicomplex n − th roots, and more precisely

n
√
W = {

n
√

|W|c(cos
θ + 2ℓπ

n
+ j sin

θ + 2ℓπ

n
) : ℓ ∈ {0,1, . . . ,n − 1}}.

If W = ae + be† is a zero divisor then exactly one of the complex numbers a or b is zero, so

the bicomplex equation Zn = W has exactly n solutions, all of them zero divisors. Obviously if

W = 0 there is only one solution, Z = 0, to the equation Zn = 0.

8 The bicomplex logarithmic function

In this section we define the notion of the logarithm of a bicomplex number. In the complex case,

we look for the solutions of ez = w, where z and w 6= 0 are complex numbers. If ln(|w|) is the real
logarithm of the positive number |w|, and arg(w) is the principal argument of w, then the complex

logarithm is defined as the set

Log(w) := {ln |w| + i(arg(w) + 2mπ) : m ∈ Z}

and its m−th branch is defined by

logm(w) := ln |w| + i(arg(w) + 2mπ). (8.1)

We will finally denote by log(w) the principal branch of Log(w), i.e. the branch for m = 0.

Similarly, we will denote by arg(w) the principal argument, so that

Arg(w) := {arg(w) + 2mπ
∣

∣m ∈ Z}.

We note, in this respect, that our notation differs a bit from other more frequently used, but we

believe our convention will be useful in discussing the bicomplex case.

We pass now to our task to define the logarithm of a bicomplex number. Take a bicomplex

number Z = z1 + jz2, and an invertible bicomplex number W = w1 + jw2. We study the solutions

to the bicomplex equation eZ = W. Recall again from (2.1) that

W = |W|c(cosθ + j sinθ).



78 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
14, 2 (2012)

Then

ez1 = |W|c, z2 = θ + 2kπ

for any k ∈ Z. The first equation implies that z1 is in the set Log|W|c, i.e. for every m ∈
Z the complex number z1 = logm |W|c, is a solution of the first equation. In the idempotent

representation, taking γ = w1 − iw2 and δ = w1 + iw2, then w
2
1
+ w2

2
= γδ. Then the complex

modulus of W is |W|c =
√

w2
1
+ w2

2
=

√
γδ, and

logm |W|c = logm
√

γδ. (8.2)

One can show that

θ = i logℓ

√

γ

δ
(8.3)

for any ℓ ∈ Z. The (principal) complex argument argc(W) is given by the formula above for ℓ = 0.
Finally we obtain

z2 = θ + 2kπ = argc(W) + 2nπ,

where n = k + ℓ is an integer number. Set

Argc(W) := {argc(W) + 2nπ
∣

∣n ∈ Z} = iLog
√

γ

δ
.

These facts motivate the following definition:

Definition 13. The bicomplex logarithm is defined by:

Log(W) := Log|W|c + j Argc(W),

which is an infinite set of bicomplex numbers. The (m,n)-th branch of the bicomplex logarithm

of W is given by:

logm,n(W) := logm |W|c + j(argc(W) + 2nπ)

for m and n integer numbers.

In the idempotent representation W = γe + δe†, it turns out that

Log(W) = Log(γ)e + Log(δ)e†. (8.4)

As in the case of the complex logarithm, the formula (8.4) has to be interpreted in the sense that

both terms of the right-hand-side represent the infinite sets of complex numbers multiplied by e

and e†, respectively. This formula is obtained immediately by studying the equation eZ = W in its

idempotent form. In more detail, if Z = αe + βe†, then eZ = W is equivalent to the two complex

equations

eα = γ, eβ = δ,



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Bicomplex Numbers and their Elementary Functions 79

which have as solutions the complex logarithms Log(γ) and Log(δ), respectively. In the cartesian

basis, formula (8.4) agrees with definition 13:

Log(W) =
1

2
Log(γδ) +

1

2
ji Log

γ

δ
= Log

√

γδ + ji Log

√

γ

δ

= Log|W|c + j Argc(W).

We state below some properties of the bicomplex logarithm.

• The bicomplex logarithm is not defined for zero-divisors, as the bicomplex exponential W =
eZ is always invertible.

• If Z = z1 + jz2 is an invertible bicomplex number, if m,n ∈ Z, then:

elogm,n(Z) = elogm |Z|c+j argc(Z)+2nπj = elogm |Z|cej argc(Z)

= |Z|c(cos(argc(Z)) + j sin(argc(Z))) = Z

• For Z = 1 = 1 + j0, we have:

logm,n(1) = 0 + 2mπi + 2nπj

for all m,n ∈ Z.

• For Z1 and Z2 two invertible bicomplex numbers, the following formula holds

Log(Z1Z2) = Log(Z1) + Log(Z2). (8.5)

The inverses of the bicomplex trigonometric functions are defined in complete analogy with the

complex case, as we have already properly defined the notions of bicomplex exponential, logarithm,

and square root.

For example, the inverse of the bicomplex cosine function is obtained by solving the equation

cos(Z) =
ejZ + e−jZ

2
= W.

This is a quadratic equation in ejZ with roots

ejZ = W ±
√

W2 − 1.

Therefore, for m,n ∈ Z,

arccos(W) := −j logm,n(W ±
√

W2 − 1) = ±j logm,n(W +
√

W2 − 1).

Received: May 2011. Revised: October 2011.



80 M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa and A. Vajiac CUBO
14, 2 (2012)

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	Introduction
	The bicomplex numbers
	Bicomplex Conjugation

	Bicomplex Derivatives
	
	
	

	Bicomplex polynomials
	The exponential function in bicomplex numbers
	Trigonometric functions of a bicomplex variable
	Bicomplex Radicals
	The bicomplex logarithmic function