6.3.1 Complex Trigonometric Functions
The complex trigonometric functions cos(z) and sin(z) are defined in terms of the complex exponential as
cos(z) :=eiz+e−iz
2 , sin(z) := eiz−e−iz
2i . (6.11)
The hyperbolic functions of a complex variable, cosh(z) and sinh(z) are de- fined in terms of the complex exponential as follows:
cosh(z) := ez+e−z
2 , sinh(z) :=ez−e−z
2 . (6.12)
All four of them are extensions of the respective functions of a real variable.
The inverse of the complex cosine function is obtained by solving the equation cos(z) = eiz+e−iz
2 =w.
This is a quadratic equation ineizwith roots eiz=w±
w2−1.
Because of the periodicity of the complex exponential this equation, with unknownz, has a countable family of solutions and the formulas
arccos(w) :=z=−ilog(w±
w2−1) =±ilog(w+
w2−1) (6.13) show how to treat precisely the arccosine function of a complex variable. In a similar fashion, the other inverse functions are dealt with.
We will now follow the same process to define bicomplex trigonometric func- tions.
124 Chapter 6. Elementary Bicomplex Functions
6.3.2 Bicomplex Trigonometric Functions
Adding and subtracting the formulasejz2= cos(z2)+jsin(z2) ande−jz2 = cos(z2)− jsin(z2), for anyz2∈C(i), we express the complex cosine and sine via the bicom- plex exponential:
cosz2= ejz2+e−jz2
2 ,
sinz2= ejz2−e−jz2 2j .
Thus we are in a position to introduce the bicomplex sine and cosine functions which are direct extensions of their complex antecedents.
Definition 6.3.2.1. LetZ ∈BC. We define the bicomplex cosine and sine functions of a bicomplex variable as follows:
cosZ :=ejZ+e−jZ
2 = eiZ+e−iZ
2 ,
sinZ :=ejZ−e−jZ
2j = eiZ−e−iZ 2i .
(6.14)
Both are well defined since a direct computation gives that, indeed, ejZ+e−jZ
2 = eiZ+e−iZ
2 and ejZ−e−jZ
2j = eiZ−e−iZ 2i . Note that ekZ+e−kZ
2 does not give the same cosZ but it gives the hy- perbolic cosine of a bicomplex number. See Section 6.3.3 below.
Given Z = z1+jz2 = β1e+β2e† ∈ BC, the properties of the bicomplex exponential bring us immediately to the idempotent representation of cosZ and sinZ:
cosZ= cos(β1)e+ cos(β2)e†,
sinZ= sin(β1)e+ sin(β)e†. (6.15) In terms of the components of the cartesian representation, one gets:
cosZ = cos(z1−iz2)e+ cos(z1+iz2)e†. Since for a complex variablezthe following formulas hold:
cosh(z) = cos(iz), sinh(z) =−isin(iz), we obtain that
cosZ= cosh(z2) cos(z1)−jsinh(z2) sin(z1).
One may writeZin different forms to observe how the above formulas change.
We continue with a description of some basic properties of the bicomplex trigonometric functions.
• Since the complex sine and cosine functions are periodic with principal period 2π, then takingZ =β1e+β2e†and settingZk, = (β1+ 2kπ)e+ (β2+ 2π)e† for arbitrary integersk, we have:
cos(Zk, ) = cos(Z), sin(Zk, ) = sin(Z).
Thus the real number (2π)e+ (2π)e† = 2πremains the principal period of both bicomplex sine and cosine functions.
• From (6.15), the equation cosZ= 0 is equivalent to the equations in complex variablesβ1 andβ2:
cos(β1) = 0, cos(β2) = 0.
The solutions areβ1=π
2 +kπ, andβ2= π
2 +π, for k, ∈Z. Note thatβ1 andβ2are never zero, so the bicomplex solutions Z to cosZ= 0 are always hyperbolic invertible numbers. In the{1,j}basis, we get the general solution to cosZ= 0 as
Z =z1+jz2= ((1 +k+) +j i(k−))π
2, (6.16)
a set of hyperbolic numbers.
• Similarly, the equation sinZ= 0 is equivalent to sin(β1) = 0, sin(β2) = 0.
The solutions are β1 = kπ, and β2 = π, for k, l ∈ Z. Note that there are non-invertible solutions for sinZ= 0, e.g., forβ1= 0, i.e.,k= 0, andβ2= 0.
In the{1,j} basis, we get the general solution for sinZ= 0 as Z=z1+jz2= (k++j i(k−))π
2, again a set of hyperbolic numbers.
• Formulas (6.15) guarantee that the usual trigonometric identities are true, e.g., the sums and differences of angle formulas, the double angle identities, etc. For example:
sin2Z+ cos2Z= (sin2(β1) + cos2(β1))e+ (sin2(β2) + cos2(β2))e† = 1.
126 Chapter 6. Elementary Bicomplex Functions
• Taking the argument equal to the idempotents e and e† gives again funny formulas. Indeed, ifZ =e, i.e., β1= 1 andβ2= 0, then
cose= cos(1)e+e†=
cos(1) + 1 2
−j
icos(1)−1 2
, sine= sin(1)e= sin(1)
2 −j
isin(1) 2
. Similarly, ifZ =e†, i.e., β1= 0 andβ2= 1, then
cose†= 1e+ cos(1)e†=
cos(1) + 1 2
+j
icos(1)−1 2
, sine†= sin(1)e† =sin(1)
2 +j
isin(1) 2
. Notice that all four are hyperbolic numbers.
As in the complex case, the other bicomplex trigonometric functions are defined in terms of the bicomplex sine and cosine functions. For example, the tangent function is the following.
Definition 6.3.2.2. Let Z be inBC. We define the bicomplex tangent function of a bicomplex variable:
tanZ:= sinZ
cosZ (6.17)
whenever cosZ is invertible, i.e., both complex numbers β1 and β2 arenot equal to π
2 plus integer multiples ofπ.
A direct computation yields:
tanZ =−jejZ−e−jZ
ejZ+e−jZ = tan(β1)e+ tan(β2)e†. In a similar fashion we have
Definition 6.3.2.3. Let Z be in BC. We define the bicomplex cotangent function of a bicomplex variable:
cotZ:= cosZ
sinZ (6.18)
wheneversinZ is invertible, i.e., both complex numbersβ1andβ2are notinteger multiples ofπ.
A direct computation yields:
cotZ=jejZ+e−jZ
ejZ−e−jZ = cot(β1)e+ cot(β2)e†.
We have described the fundamentals of the theory of trigonometric bicomplex functions. Of course, now the theory can be continued in many directions.
We mention now a curious fact. The real tangent function takes any real value. It is easy to show that the complex tangent does not contain ±i in its range. For the bicomplex tangent the excluded values are{±i,±j}.
6.3.3 Hyperbolic functions of a bicomplex variable
We want to extend directly the notion of a hyperbolic function of a complex variable onto a bicomplex variable. A direct way is clear.
Definition 6.3.3.1. LetZ inBC. Then we define the bicomplex hyperbolic sine and cosine functions as
coshZ :=eZ+e−Z
2 = ekZ+e−kZ
2 ,
sinhZ :=eZ−e−Z
2 = ekZ−e−kZ 2k .
(6.19)
Again, both are well defined since a direct computation gives now that eZ+e−Z
2 = ekZ+e−kZ
2 and eZ−e−Z
2 = ekZ−e−kZ 2k . As above, in the idempotent representationZ =β1e+β2e† we get that
coshZ= cosh(β1)e+ cosh(β2)e†,
sinhZ= sinh(β1)e+ sinh(β2)e†. (6.20) As in the previous section, these formulas would yield the usual properties analo- gous to hyperbolic complex functions. For example,
cosh2Z−sinh2Z=
cosh2(β1)−sinh2(β1) e+
cosh2(β2)−sinh2(β2)
e† = 1. The addition formulas are:
cosh(Z1+Z2) = coshZ1coshZ2+ sinhZ1sinhZ2, sinh(Z1+Z2) = sinhZ1coshZ2+ coshZ1sinhZ2. Moreover, forZ=jz2, we have:
cosh(jz2) = cos(z2), sinh(jz2) =jsin(z2).
Then forZ=z1+jz2 we have:
sinhZ= sinh(z1+jz2) = sinh(z1) cos(z2) +jcosh(z1) sin(z2).
A similar formula holds for coshZ.
128 Chapter 6. Elementary Bicomplex Functions