Non-constant bounded holomorphic functions of hyperbolic
numbers – Candidates for hyperbolic activation functions

arXiv:1306.1653v1 [] 7 Jun 2013

* Eckhard Hitzer (University of Fukui)
Abstract– The Liouville theorem states that bounded holomorphic complex functions
are necessarily constant. Holomorphic functions fulfill the socalled Cauchy-Riemann
(CR) conditions. The CR conditions mean that a complex z-derivative is independent
of the direction. Holomorphic functions are ideal for activation functions of complex
neural networks, but the Liouville theorem makes them useless. Yet recently the use
of hyperbolic numbers, lead to the construction of hyperbolic number neural networks.
We will describe the Cauchy-Riemann conditions for hyperbolic numbers and show that
there exists a new interesting type of bounded holomorphic functions of hyperbolic
numbers, which are not constant. We give examples of such functions. They therefore
substantially expand the available candidates for holomorphic activation functions for
hyperbolic number neural networks.
Keywords: Hyperbolic numbers, Liouville theorem, Cauchy-Riemann conditions,
bounded holomorphic functions

1

Introduction

For the sake of mathematical clarity, we first carefully
review the notion of holomorphic functions in the two
number systems of complex and hyperbolic numbers.
The Liouville theorem states that bounded holomorphic complex functions f : C → C are necessarily constant [1]. Holomorphic functions are functions
that fulfill the socalled Cauchy-Riemann (CR) conditions. The CR conditions mean that a complex
z-derivative
df (z)
, z = x + iy ∈ C, x, y ∈ R, ii = −1,
dz

Contrary to the complex case, the hyperbolic logistic function is bounded. This is
due to the absence of singularities. Thus,
in general terms, this seems to be a suitable
activation function. Concretely, the following facts, however, might be of disadvantage. The real and imaginary part have different squashing values. Both component
functions do only significantly differ from
zero around the lines1 x = y (x > 0) and
−x = y (x < 0).

(1)

is independent of the direction with respect to which
the incremental ratio, that defines the derivative, is
taken [5]. Holomorphic functions would be ideal for
activation functions of complex neural networks, but
the Liouville theorem means that careful measures
need to be taken in order to avoid poles (where the
function becomes infinite).
Yet recently the use of hyperbolic numbers
z = x + h y, h2 = 1, x, y ∈ R, h ∈
/ R.

boundaries (hyperplanes) of the real and the unipotent h-part of the output. But Buchholz argued in
[4], p. 114, that

(2)

lead to the construction of hyperbolic number neural
networks. We will describe the generalized CauchyRiemann conditions for hyperbolic numbers and show
that there exist bounded holomorphic functions of
hyperbolic numbers, which are not constant. We give
a new example of such a function. They are therefore
excellent candidates for holomorphic activation functions for hyperbolic number neural networks [2, 3].
In [3] it was shown, that hyperbolic number neural
networks allow to control the angle of the decision

Complex numbers are isomorphic to the Clifford
geometric algebra Cl0,1 which is generated by a single
vector e1 of negative square e1 = −1, with algebraic
basis {1, e1 }. The isomorphism C ∼
= Cl0,1 is realized
by mapping i 7→ e1 .
Hyperbolic numbers are isomorphic to the Clifford
geometric algebra Cl1,0 which is generated by a single
vector e1 of positive square e1 = +1, with algebraic
basis {1, e1 }. The isomorphism between hyperoblic
numbers and Cl1,0 is realized by mapping h 7→ e1 .

2

Complex variable functions

We follow the treatment given in [5]. We assume
a complex function given by an absolute convergent
1 Note that we slightly correct the two formulas of Buchholz,
because we think it necessary to delete e1 in Buchholz’ original
x = ye1 (x > 0), etc.

power series.
w = f (z) = f (x + iy) = u(x, y) + iv(x, y),

(3)

where u, v : R2 → R are real functions of the real
variables x, y. Since u, v are obtained in an algebraic way from the complex number z = x + iy, they
cannot be arbitrary functions but must satisfy certain conditions. There are several equivalent ways
to obtain these conditions. Following Riemann, we
state that a function w = f (z) = u(x, y) + iv(x, y)
is a function of the complex variable z if its derivative is independent of the direction (in the complex
plane) with respect to which the incremental ratio is
taken. This requirement leads to two partial differential equations, named after Cauchy and Riemann
(CR), which relate u and v.
One method for obtaining these equations is the
following. We consider the expression w = u(x, y) +
iv(x, y) only as a function of z, but not of z̄, i.e. the
derivative with respect to z̄ shall be zero. First we
perform the bijective substitution
x=

1
(z + z̄),
2

1
y = −i (z − z̄),
2

(4)

based on z = x + iy, z̄ = x − iy. For computing the
derivative w,z̄ = dw
dz̄ with the help of the chain rule
we need the derivatives of x and y of (4)
x,z̄ =

1
,
2

y,z̄ =

1
i.
2

(5)

3

Hyperbolic numbers

Hyperbolic numbers are also known as split-complex
numbers. They form a two-dimensional commutative
algebra. The canonical hyperbolic system of numbers
is defined [5] by
(11)

The hyperbolic conjugate is defined as
z̄ = x − h y.
(6)

Requiring that both the real and the imaginary part
of (6) vanish we obtain the Cauchy-Riemann conditions
u,x = v,y ,
u,y = −v,x .
(7)
Functions of a complex variable that fulfill the CR
conditions are functions of x and y, but they are only
functions of z, not of z̄.
It follows from (7), that both u and v fulfill the
Laplace equation
u,xx = v,yx = v,xy = −u,yy ⇔ u,xx + u,yy = 0, (8)
and similarly
v,xx + v,yy = 0.

g(u(x, y)+iv(x, y)) = gr (u(x, y))+igi (v(x, y)). (10)

z = x + h y, h2 = 1, x, y ∈ R, h ∈
/ R.

Using the chain rule we obtain
w,z̄ = u,x x,z̄ + u,y y,z̄ + i(v,x x,z̄ + v,y y,z̄ )
1
1
1
1
= u,x + iu,y + i( v,x + iv,y )
2
2
2
2
1
!
= [u,x − v,y + i(v,x + u,y )] = 0.
2

are called harmonic functions and are important in
many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because
they can be used to accurately describe the behavior
of electric, gravitational, and fluid potentials. In the
study of heat conduction, the Laplace equation is the
steady-state heat equation [6].
Liouville’s theorem [1] states, that any bounded
holomorphic function f : C → C, which fulfills the
CR conditions is constant. Therefore for complex
neural networks it is not very meaningful to use holomorphic functions as activation functions. If they
are used, special measures need to be taken to avoid
poles in the complex plane. Instead separate componentwise (split) real scalar functions for the real
part gr : R → R, u(x, y) 7→ gr (u(x, y)), and for the
imaginary part gi : R → R, v(x, y) 7→ gi (v(x, y)), are
usually adopted. Therefore a standard split activation function in the complex domain is given by

(9)

The Laplace equation is a simple example of an elliptic partial differential equation. The general theory
of solutions to the Laplace equation is known as potential theory. The solutions of the Laplace equation

(12)

Taking the hyperbolic conjugate corresponds in the
isomorphic algebra Cl1,0 to taking the main involution (grade involution), which maps 1 7→ 1, e1 7→
−e1 .
The hyperbolic invariant (corresponding to the
Lorentz invariant in physics for y = ct), or modulus,
is defined as
z z̄ = (x + h y)(x − h y) = x2 − y 2 ,

(13)

which is not positive definite.
Hyperbolic numbers are fundamentally different
from complex numbers. Complex numbers and
quaternions are division algebras, every non-zero element has a unique inverse. Hyperbolic numbers do
not always have an inverse, but instead there are
idempotents and divisors of zero.
We can define the following idempotent basis
n1 =

1
(1 + h),
2

n2 =

1
(1 − h),
2

(14)

which fulfills
1
1
(1 + h)(1 + h) = (2 + 2h) = n1 ,
4
4
n22 = n2 ,
n1 + n2 = 1,
1
1
n1 n2 = (1 + h)(1 − h) = (1 − 1) = 0,
4
4
n̄1 = n2 , n̄2 = n1 .
(15)
n21 =

The inverse basis transformation is simply
1 = n1 + n2 ,

h = n1 − n2 .

(16)

Setting
z = x + hy = ξn1 + ηn2 ,

(17)

we get the corresponding coordinate transformation
x=

1
(ξ + η),
2

y=

1
(ξ − η),
2

(18)

as well as the inverse coordinate transformation
ξ = x + y ∈ R,

η = x − y ∈ R.

(19)

Figure 1: The hyperbolic number plane [9] with horizontal x-axis and vertical yh-axis, showing: (a) Hyperbolas with modulus z z̄ = −1 (green). (b) Straight
lines with modulus z z̄ = 0 ⇔ x2 = y 2 (red), i.e. divisors of zero. (c) Hyperbolas with modulus z z̄ = 1
(blue).

The hyperbolic conjugate becomes, due to (15), in
the idempotent basis
z̄ = ξn̄1 + ηn̄2 = ηn1 + ξn2 .

(20)

In the idempotent basis, using (20) and (15), the hyperbolic invariant becomes multiplicative
z z̄ = (ξn1 + ηn2 )(ηn1 + ξn2 )
= ξη(n1 + n2 ) = ξη = x2 − y 2 .

(21)

In the following we consider the product and quotient
of two hyperbolic numbers z, z 0 both expressed in the
idempotent basis {n1 , n2 }
zz 0 = (ξn1 +ηn2 )(ξ 0 n1 +η 0 n2 ) = ξξ 0 n1 +ηη 0 n2 , (22)

due to (15). We repeat that in (24) the product is
zero, even though the factors are non-zero. The numbers ξn1 , ηn2 along the n1 , n2 axis are therefore called
divisors of zero. The divisors of zero have no inverse.
The hyperbolic plane with the diagonal lines of divisors of zero (b), and the pairs of hyperbolas with
constant modulus z z̄ = 1 (c), and z z̄ = −1 (a) is
shown in Fig. 1.

4

Hyperbolic number functions

We assume a hyperbolic number function given by
an absolute convergent power series

and

w = f (z) = f (x + hy) = u(x, y) + hv(x, y),
z
ξn1 + ηn2
z z̄ 0
= 0
= 0 0
0
0
z
ξ n1 + η n2
z z̄
(ξn1 + ηn2 )(η 0 n1 + ξ 0 n2 )
= 0
(ξ n1 + η 0 n2 )(η 0 n1 + ξ 0 n2 )
(ξη 0 n1 + ηξ 0 n2 )(η 0 n1 + ξ 0 n2 )
=
ξ0 η0
ξ
η
= 0 n1 + 0 n2 .
ξ
η

h2 = 1,

h∈
/ R.

(25)

where u, v : R2 → R are real functions of the real
variables x, y. An example of a hyperbolic number
function is the exponential function
ez = ex+hy = ex ehy = ex (cosh y + h sinh y)
= u(x, y) + hv(x, y),

(23)

(26)

with
Because of (23) it is not possible to divide by z 0 if
ξ 0 = 0, or if η 0 = 0. Moreover, the product of a
hyperbolic number with ξ = 0 (on the n2 axis) times
a hyperbolic number with η = 0 (on the n1 axis) is
(ξn1 + 0n2 )(0n1 + ηn2 ) = ξηn1 n2 = 0,

(24)

u(x, y) = ex cosh y,

v(x, y) = ex sinh y.

(27)

Since u, v are obtained in an algebraic way from the
hyperbolic number z = x + hy, they cannot be arbitrary functions but must satisfy certain conditions.

There are several equivalent ways to obtain these conditions. A function w = f (z) = u(x, y) + hv(x, y) is
a function of the hyperbolic variable z, if its derivative is independent of the direction (in the hyperbolic
plane) with respect to which the incremental ratio is
taken. This requirement leads to two partial differential equations, so called generalized Cauchy-Riemann
(GCR) conditions, which relate u and v.
To obtain the GCR conditions we consider the expression w = u(x, y) + hv(x, y) only as a function of
z, but not of z̄ = x − hy, i.e. the derivative with
respect to z̄ shall be zero. First we perform the bijective substitution
x=

1
(z + z̄),
2

1
y = h (z − z̄),
2

(28)

based on z = x + hy, z̄ = x − hy. For computing the
derivative w,z̄ = dw
dz̄ with the help of the chain rule
we need the derivatives of x and y of (28)
x,z̄ =

1
,
2

1
y,z̄ = − h.
2

w,z̄ = u,x x,z̄ + u,y y,z̄ + h(v,x x,z̄ + v,y y,z̄ )
1
1
1
1
= u,x − hu,y + h( v,x − hv,y )
2
2
2
2
1
!
= [u,x − v,y + h(v,x − u,y )] = 0.
(30)
2
Requiring that both the real and the h-part of (30)
vanish we obtain the GCR conditions
u,y = v,x .

f (z) = u(x, y) + h v(x, y),
1
.
u(x, y) = v(x, y) =
1 + e−x e−y

(31)

Functions of a hyperbolic variable that fulfill the
GCR conditions are functions of x and y, but they
are only functions of z, not of z̄. Such functions are
called (hyperbolic) holomorphic functions.
It follows from (31), that u and v fulfill the wave
equation

−1
(−e−x e−y )
(1 + e−x e−y )2
e−x e−y
=
,
(1 + e−x e−y )2

u,x =

u,y = v,x = v,y =

0<

1
1+

u,x = ex cosh y,

e−x e−y

lim

v,y = ex cosh y = u,x . (34)

< 1.

1
= 0,
1 + e−x e−y

and
lim

x,y→∞

1
1+

(38)

e−x e−y

= 1.

(39)

(40)

The function (35) is representative for how to turn
any real neural node activation function r(x) into
holomorphic hyperbolic activation function via
f (x) = r(x + y) (1 + h).

(41)

We note that in [3, 4] another holomorphic hyperbolic activation function was studied, namely

u,y = ex sinh y,

v,x = ex sinh y = u,y ,

(37)

We especially have

and similarly
The wave equation is an important second-order
linear partial differential equation for the description
of waves – as they occur in physics – such as sound
waves, light waves and water waves. It arises in fields
like acoustics, electromagnetics, and fluid dynamics.
The wave equation is the prototype of a hyperbolic
partial differential equation [7].
Let us compute the partial derivatives u,x , u,y ,
v,x , v,y for the exponential function ez of (26):

e−x e−y
.
(1 + e−x e−y )2

The GCR conditions (31) are therefore clearly fulfilled, which means that the hyperbolic function f (z)
of (35) is holomorphic. Since the exponential function e−x has a range of (0, ∞), the product e−x e−y
also has values in the range of (0, ∞). Therefore the
function 1 + e−x e−y has values in (1, ∞), and the
components of the function f (z) of (35) have values

x,y→−∞

(33)

(36)

where we repeatedly applied the chain rule for differentiation. Similarly we obtain

u,xx = v,yx = v,xy = u,yy ⇔ u,xx − u,yy = 0, (32)
v,xx − v,yy = 0.

(35)

The function u(x, y) is pictured in Fig. 2.
Let us verify that the function f of (35) fulfills the
GCR conditions

(29)

Using the chain rule we obtain

u,x = v,y ,

We clearly see that the partial derivatives (34) fulfill
the GCR conditions (31) for the exponential function
ez , as expected by its definition (26). The exponential function ez is therefore a manifestly holomorphic
hyperpolic function, but it is not bounded.
In the case of holomorphic hyperbolic functions the
GCR conditions do not imply a Liouville type theorem like for holomorphic complex functions. This can
most easily be demonstrated with a counter example

f 0 (z) =

1
,
1 + e−z

(42)

2. x2 > y 2 , x < 0:
θ = artanh (y/x),

z = −ρehθ ,

i.e. the quadrant in Fig. 1 including the negative
x-axis (to the left).
3. x2 < y 2 , y > 0:
θ = artanh (x/y),

z = hρehθ ,

i.e. the quadrant in Fig. 1 including the positive
y-axis (top).
4. x2 < y 2 , y < 0:
θ = artanh (x/y),
Figure 2: Function u(x, y) = 1/(1 + e−x e−y ). Horizontal axis −3 ≤ x ≤ 3, from left corner into paper
plane −3 ≤ y ≤ 3. Vertical axis 0 ≤ u ≤ 1. (Figure
produced with [8].)
but compare the quote from [4], p. 114, given in the
introduction. The split activation function used in
[2]
1
1
+h
,
(43)
f 00 (x, y) =
−x
1+e
1 + e−y
is clearly not holomorphic, because the real part u =
1/(1 + e−x ) depends only on x and not on y, and the
h-part v = 1/(1 + e−y ) depends only on y and not on
x, thus the GCR conditions (31) can not be fulfilled.

5

Geometric interpretation of
multiplication of hyperbolic
numbers

In order to geometrically interpret the product of two
complex numbers, it proves useful to introduce polar
coordinates in the complex plane. Similarly, for the
geometric interpretation of the product of two hyperbolic numbers, we first introduce hyperbolic polar
coordinates for z = x + hy with radial coordinate
p
p
ρ = |z z̄| = |x2 − y 2 | .
(44)
The hyperbolic polar coordinate transformation [5] is
then given as
1. x2 > y 2 , x > 0:
θ = artanh (y/x),

z = ρehθ ,

i.e. the quadrant in the hyperbolic plane of Fig.
1 limitted by the diagonal idempotent lines, and
including the positive x-axis (to the right).

z = −hρehθ ,

i.e. the quadrant in Fig. 1 including the negative
y-axis (bottom).
The product of a constant hyperbolic number (assuming a2x > a2y , ax > 0)
a = ax + hay = ρa ehθa ,
q
θa = artanh (ay /ax ),
ρa = a2x − a2y ,

(45)

with a hyperbolic number z (assuming x2 > y 2 , x >
0) in hyperbolic polar coordinates is
az = ρa ehθa ρ ehθ = ρa ρ eh(θ+θa ) .

(46)

The geometric interpretation is a scaling of the modulus ρ → ρa ρ and a hyperbolic rotation (movement
along a hyperbola) θ → θ + θa .
In the physics of Einstein’s special relativistic
space-time [11, 12], the hyperbolic rotation θ → θ+θa
corresponds to a Lorentz transformation from one
inertial frame with constant velocity tanh θ to another inertial frame with constant velocity tanh(θ +
θa ). Neural networks based on hyperbolic numbers
(dimensionally extended to four-dimensional spacetime) should therefore be ideal to compute with electromagnetic signals, including satellite transmission.

6

Conclusion

We have compared complex numbers and hyperbolic
numbers, as well as complex functions and hyperbolic
functions. We saw that according to Liouville’s theorem bounded complex holomorphic functions are necessarily constant, but non-constant bounded hyperbolic holomorphic functions exist. One such function
has already beeng studied in [3, 4]. We have studied a promising example of a hyperbolic holomorphic
function
1+h
,
(47)
f (z) =
1 + e−x−y

in some detail. The distinct notions of idempotents
and divisors of zero, special to hyperbolic numbers,
were introduced. After further introducing hyperbolic polar coordinates, a geometric interpretation of
the hyperbolic number multiplication was given.
Hyerbolic neural networks offer, compared to complex neural networks, therefore the advantage of suitable bounded non-constant hyperbolic holomorphic
activation functions. It would certainly be of interest
to study convergence, accuracy and decision boundaries of hyperbolic neural networks with the activation function (35), similar to [3, 4].

Acknowledgment
I want to acknowledge God [13]:
In the beginning was the Word2 , and the
Word was with God, and the Word was
God. He was with God in the beginning.
Through him all things were made; without
him nothing was made that has been made.
In him was life, and that life was the light
of all mankind.
I want to thank my dear family, as well as T. Nitta
and Y. Kuroe.

References
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the Plane and n-dimensional Space, Birkhauser,
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Como, Italy, vol. 2, 129/133 (2000).
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the International Joint Conference on Neural Networks, IJCNN’08-HongKong, June 1-6,
2973/2979(2008).
[4] S. Buchholz, PhD Thesis, A Theory of Neural
Computation with Clifford Algebras, University
of Kiel, 2005.
[5] F. Catoni et al, The Mathematics of Minkowski
Space-Time, Birkhauser, 2008.
[6] Laplace’s equation, Wikipedia, accessed 24 August 2011, http://en.wikipedia.org/wiki/
Laplace’s_equation
2 Greek term: logos. Note: A Greek philosopher named
Heraclitus first used the term Logos around 600 B.C. to designate the divine reason or plan which coordinates a changing
universe. [14]

[7] Wave equation, Wikipedia, accessed 24 August
2011, http://en.wikipedia.org/wiki/Wave_
equation
[8] Online
3D
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grapher,
//www.livephysics.com/ptools/
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http:

[9] Split-complex number, Wikipedia, accessed
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[10] Notes of collaboration with H. Ishi, Feb. 2011,
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[13] The Bible, New International Version (NIV),
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http://www.blueletterbible.org/lang/
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