

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6041
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

The Complete List of Solutions to Möbius’s
Exponential Equation

Rasimate Maungchang1, Watchareepan Atiponrat2, Tarid Suwansri1,
Jaturon Wattanapan2,3, Teerapong Suksumran2,∗

1 School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand
2 Department of Mathematics, Faculty of Science,
Chiang Mai University, Chiang Mai 50200, Thailand
3 Office of Research Administration, Chiang Mai University, Chiang Mai 50200, Thailand

Abstract. Möbius addition is a non-associative binary operation defined on the complex open

unit disk D = {z ∈ C : |z| < 1} by a ⊕M b =
a+ b

1 + ab
, and Möbius’s Exponential Equation is a

non-linear functional equation of the form L(a ⊕M b) = L(a)L(b), where L is a complex-valued
function defined on D. In [Aequat. Math. 91 (2017), 491–503], the authors address the problem
of determining the solutions to Möbius’s Exponential Equation. In this paper, we determine the
complete list of solutions to Möbius’s Exponential Equation using an algebraic approach.

2020 Mathematics Subject Classifications: 39B32, 30D05, 20N05
Key Words and Phrases: Möbius addition, Möbius’s Exponential Equation, associator function,
gyrogroup, non-linear equation

1. Introduction

Möbius addition arises naturally in the study of hyperbolic geometry, special relativity,
and non-Euclidean structures [1, 2]. In particular, Möbius addition captures key aspects of
gyrogroup structures, which have applications in complex analysis and differential geom-
etry, for instance, and serves as a primary motivation for the development of gyrogroup
theory. Furthermore, in optimization theory, Möbius transformations and Möbius ad-
dition on the complex open unit disk provide tools to formulate and solve problems in
non-Euclidean domains, model non-associative dynamics, enforce multiplicative structure
constraints, and operate within compact curved spaces. These frameworks are particu-
larly useful for modern applications in machine learning, signal processing, and geometric
optimization, where classical Euclidean assumptions are insufficient.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6041

Email addresses: mate105@gmail.com (R. Maungchang), watchareepan.a@cmu.ac.th (W. Atiponrat),
tarid.su@mail.wu.ac.th (T. Suwansri), jaturon.w@cmu.ac.th (J. Wattanapan),
teerapong.suksumran@cmu.ac.th (T. Suksumran)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)


R. Maungchang et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6041 2 of 8

Recall that Möbius addition is a non-associative binary operation defined on the com-
plex open unit disk D = {z ∈ C : |z| < 1} by the formula

a⊕M b =
a+ b

1 + ab
(1)

for all a, b ∈ D. It turns out that (D,⊕M ) forms a non-associative group-like structure
that shares several common properties with groups. In particular, a⊕M b belongs to D for
all a, b ∈ D. In [3], the authors introduce a functional equation in connection with Schur’s
Lemma of the form

L(a⊕M b) = L(a)L(b) (ME)

called Möbius’s Exponential Equation, which is in some sense an extension of Cauchy’s
Exponential Equation, where L is a complex-valued function defined on D. They also give
partial solutions on the open interval (−1, 1) and then address the problem of determining
the complete solutions to Möbius’s Exponential Equation on the whole disk (see Problem
1.3 of [3]). Actually, Möbius’s Exponential Equation arises when one attempts to deter-
mine the irreducible linear representations of (D,⊕M ), as shown in Section 3.2 of [3]. In
this paper, we give the complete list of solutions to Möbius’s Exponential Equation using
the recent notion of associators formulated in [4].

2. The solutions to Möbius’s Exponential Equation

Suppose that L is a complex-valued function defined on the disk D. As noted on page
492 of [3], if L satisfies Möbius’s Exponential Equation, then L is either everywhere or
nowhere zero. Furthermore, if L is assumed to be a Borel measurable function, then either
L is the zero function or there exists a complex constant λ for which

L(r) = eλ tanh−1 r

for all r ∈ (−1, 1) (see Theorem 1.2 of [3]). The authors of [3] establish that Möbius’s
Exponential Equation does not have any non-trivial holomorphic solutions using the Open
Mapping Theorem in complex analysis. Here, we will apply an algebraic tool, together
with the well-known Intermediate Value Theorem, to prove the following theorem.

Theorem 1. The solutions to Möbius’s Exponential Equation are only the functions L :
z 7→ 0 and L : z 7→ 1, z ∈ D.

Theorem 1 states that Möbius’s Exponential Equation does not have any non-trivial
solutions, which refines the result in [3] as the condition of being a holomorphic function is
removed. In particular, this implies that the disk D, endowed with Möbius addition ⊕M ,
has no non-trivial representations (see the remark at the end of [3] for more details). The
proof of Theorem 1 is presented in the next section for convenience.


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3. The complex Möbius gyrogroup and its associators

It is known that (D,⊕M ) forms a non-associative group-like structure. This system
is called the (complex) Möbius gyrogroup due to the fact that the function τa, defined
as τa(b) = a ⊕M b, b ∈ D, is a familiar Möbius transformation on D for all a ∈ D. As
noted in Section 2, if L satisfies Möbius’s Exponential Equation and L(w) ̸= 0 for some
w ∈ D, then L(z) ̸= 0 for all z ∈ D. From this point of view, any non-zero solution to
Möbius’s Exponential Equation is indeed a homomorphism from the Möbius gyrogroup to
the multiplicative group of non-zero complex numbers. It is clear that the functions defined
by L(z) = 0 for all z ∈ D and L(z) = 1 for all z ∈ D are solutions to Möbius’s Exponential
Equation. In this section, we prove that these functions are the only solutions to Möbius’s
Exponential Equation using an algebraic approach. In fact, we prove a stronger result: any
homomorphism from the Möbius gyrogroup to a group (which is viewed as a degenerate
gyrogroup) is necessarily trivial.

For basic definitions and relevant notations, we refer the reader to [2, 4, 5]. As intro-
duced in [4], for all a, b, c ∈ D, the associator of the triple (a, b, c) in (D,⊕M ) is denoted
by [a, b, c] and is defined by the formula

[a, b, c] = ⊖M (a⊕M (b⊕M c))⊕M ((a⊕M b)⊕M c), (2)

where ⊖M is defined as ⊖Ma = −a. For convenience, we define the associator function of
the Möbius gyrogroup, denoted by AM , by the formula

AM (a, b, c) = [a, b, c] (3)

for all a, b, c ∈ D. Note that (a⊕M b)⊕M c = a⊕M (b⊕M c) if and only if [a, b, c] = 0. We
present an alternative useful form of formula (3) in the following proposition.

Proposition 1. The associator function of the Möbius gyrogroup is given by the formula

AM (a, b, c) =
c(ab− ab)(1 + ab+ ac+ bc)

(1 + ab− |c|2(1 + ab))(1 + ab+ ac+ bc)
(4)

for all a, b, c ∈ D.

Proof. The proof of the proposition can be done by algebraic manipulation using
formulas (1), (2), and (3).

To see some aspects of the associator function of the Möbius gyrogroup, we plot a few
images under AM with specification of a and b in Figures 1 and 2. In light of Proposition
1, we obtain a sufficient and necessary condition for three elements in the disk to be
associative with respect to Möbius addition, as shown in Corollary 1. First, let us prove
the following theorem, which gives a sufficient and necessary condition for an associator
to be zero. Here, C is viewed as a real vector space in the usual way.

Theorem 2. Let a, b, c ∈ D. Then [a, b, c] = 0 if and only if c = 0 or a and b are linearly
dependent in C.


R. Maungchang et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6041 4 of 8

Figure 1: The image of the interval [0, 1) under AM , where a = 0.5 and b = 0.5i.

Proof. Suppose that [a, b, c] = 0. In view of (4), c(ab − ab)(1 + ab + ac + bc) = 0.
We claim that 1 + ab + ac + bc ̸= 0. By the closure property of ⊕M , b ⊕M c ∈ D. This
implies that 1 + a(b⊕M c) ̸= 0 since otherwise |a||b⊕M c| = 1, an impossibility. A direct
computation shows that

1 + a(b⊕M c) = 1 +
ab+ ac

1 + bc
=

1 + ab+ ac+ bc

1 + bc
,

and so 1 + ab + ac + bc ̸= 0. This implies that 1 + ab + ac + bc = 1 + ab+ ac+ bc ̸= 0,
which proves the claim. Hence, c = 0 or ab − ab = 0. In the latter case, we show that a
and b are linearly dependent. The case when a and b are both zero is clear. Therefore,
we assume that a ̸= 0 or b ̸= 0. Now, suppose that ab − ab = 0. Set a = a1 + a2i and
b = b1 + b2i, where a1, a2, b1, b2 ∈ R. By assumption, a1b2 − a2b1 = 0. This implies that
b2a − a2b = 0. If b2 = 0 and a2 = 0, then a = a1 and b = b1, and furthermore a1 and b1
are not simultaneously zero. Hence, −b1a+ a1b = 0. This shows that a and b are linearly
dependent.

Suppose conversely that c = 0 or a and b are linearly dependent. The case when c = 0
is clear. Hence, we assume that a and b are linearly dependent. Thus, there are real
numbers r and s not simultaneously zero such that ra+ sb = 0. In the case when r = 0,
we obtain that s ̸= 0, and so sb = 0 implies b = 0, which in turn implies ab−ab = 0. Now,
suppose that r ̸= 0. Set a = a1+a2i and b = b1+b2i, where a1, a2, b1, b2 ∈ R. Then solving
the system of equations induced by the equality ra+sb = 0 shows that r(a1b2−a2b1) = 0,
which implies a1b2 − a2b1 = 0. It follows that ab− ab = 0, and so [a, b, c] = 0.

Corollary 1. Let a, b, c ∈ D. Then a ⊕M (b ⊕M c) = (a ⊕M b) ⊕M c if and only if c = 0
or a and b are linearly dependent in C.

Proof. The corollary follows from the fact that

(a⊕M b)⊕M c = (a⊕M (b⊕M c))⊕M [a, b, c]


R. Maungchang et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6041 5 of 8

Figure 2: The image of the rectangular grids under AM , where a = 0.5 and b = 0.5i.

for all a, b, c ∈ D.

Next, we prove that any element of the Möbius gyrogroup can be expressed as an
associator so that the set of all associators is D itself. To do so, we need the following
lemma, which is important in its own right.

Lemma 1. If ω is a unimodular complex number (that is, |ω| = 1), then

AM (ωa, ωb, ωc) = ωAM (a, b, c)

for all a, b, c ∈ D.

Proof. Suppose that |ω| = 1. Note that if x, y ∈ C, then

ωxωy = ωxωy = |ω|2xy = xy.

Furthermore, |ωx| = |x| for all x ∈ C. Hence, the lemma follows directly from making
substitutions in formula (4).

We are now in a position to prove that the associator function of the Möbius gyro-
group is surjective. This implies that every element of the Möbius gyrogroup is indeed an
associator.

Theorem 3. The associator function of the Möbius gyrogroup is a surjective function
from D3 onto D.

Proof. Suppose that a, b, c ∈ D. First, we show that 1 + ab − |c|2(1 + ab) ̸= 0. Note
that 1+ab ̸= 0 since otherwise −1 = ab implies |a||b| = 1, an impossibility. Note also that

1 + ab− |c|2(1 + ab) = (1 + ab)

(
1 + ab

1 + ab
− |c|2

)
.


R. Maungchang et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6041 6 of 8

Hence, 1 + ab − |c|2(1 + ab) = 0 would imply
1 + ab

1 + ab
− |c|2 = 0, which would imply

|c|2 = 1 + ab

1 + ab
, and so |c| = 1, a contradiction. Next, 1 + ab+ ac+ bc ̸= 0, as proved in the

proof of Theorem 2.
Fix a, b ∈ D, and suppose that ab− ab ̸= 0. Now, let us define a function fa,b from the

close interval [0, 1] to the close disk D = {z ∈ C : |z| ≤ 1} by the formula

fa,b(λ) =
λ(ab− ab)(1 + ab+ aλ+ bλ)

(1 + ab− |λ|2(1 + ab))(1 + ab+ aλ+ bλ)
(5)

for all λ ∈ [0, 1]. Note that if λ ̸= 1, then λ ∈ D, and so

(1 + ab− |λ|2(1 + ab))(1 + ab+ aλ+ bλ) ̸= 0,

as shown above. In the case when λ = 1, we obtain that

(1 + ab− |λ|2(1 + ab))(1 + ab+ aλ+ bλ) = (ab− ab)(1 + ab+ a+ b),

which is not zero since

1 + ab+ a+ b = (1 + ab)

(
1 + ab

1 + ab
+

a+ b

1 + ab

)
= (1 + ab)

(
1 + ab

1 + ab
+ (a⊕M b)

)
.

This shows that fa,b is well defined. Since fa,b is defined using only addition, multiplication,
subtraction, division, and conjugation (which are all continuous functions), it follows that
fa,b is continuous.

Fix a, b ∈ D, and suppose that ab−ab ̸= 0. Define a function ga,b by ga,b(λ) = |fa,b(λ)|
for all λ ∈ [0, 1]. Since the complex-modulus function is continuous, it follows that ga,b is
a continuous function from [0, 1] to [0, 1]. Note that ga,b(0) = 0 and that ga,b(1) = 1 since

fa,b(1) =
(ab− ab)(1 + ab+ a+ b)

(ab− ab)(1 + ab+ a+ b)
= −1 + ab+ a+ b

1 + ab+ a+ b

so that |fa,b(1)| = 1. By the Intermediate Value Theorem (see, for instance, page 26 of
[6]), if 0 < r < 1, then ga,b(λ0) = r for some λ0 ∈ (0, 1).

We are now in a position to prove that AM is surjective. Let w ∈ D. In the case when
w = 0, we obtain that AM (0, 0, 0) = 0. Now, suppose that w ̸= 0. Using the polar form,
we can write w = |w|(cosα+ i sinα) for some α ∈ R. Note that |w| ̸= 0. Choose a, b ∈ D
such that ab − ab ̸= 0 (for example, a = 0.5 and b = 0.5i). As above, ga,b(λ0) = |w| for
some λ0 ∈ (0, 1). Note that |w| = ga,b(λ0) = |fa,b(λ0)|. Hence, fa,b(λ0) ̸= 0. Assume that
fa,b(λ0) has a polar form as fa,b(λ0) = r(cosβ + i sinβ), where 0 < r < 1 and β ∈ R.
Thus, fa,b(λ0) = |w|(cosβ + i sinβ). Set ω = cos (α− β) + i sin (α− β). Then |ω| = 1.
From Lemma 1, it follows that

AM (ωa, ωb, ωλ0) = ωAM (a, b, λ0)


R. Maungchang et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6041 7 of 8

= ωfa,b(λ0)

= (cos (α− β) + i sin (α− β))|w|(cosβ + i sinβ)

= |w|(cosα+ i sinα)

= w.

Note that ωa, ωb, and ωλ0 are in D. This shows that AM is surjective, as claimed.

The notion of associators can be used to measure the deviation from associativity of
Möbius addition. Following [4], the normal closure of the set of all associators in the
Möbius gyrogroup is called the associator normal subgyrogroup, denoted by Da, which is
the smallest normal subgyrogroup of D containing all the associators in D. According
to Proposition 3.4 of [4], Da is the unique normal subgyrogroup of D such that D/Da is
a group and if φ is a homomorphism from D to a group, then Da lies in the kernel of
φ. The quotient D/Da is referred to as the associativization of the Möbius gyrogroup.
From Theorem 3, we know that every element of the Möbius gyrogroup is an associator.
Therefore, we obtain the following corollary immediately.

Corollary 2. The associator normal subgyrogroup of the Möbius gyrogroup is the Möbius
gyrogroup itself.

We gain a better understanding of the algebraic structure of the Möbius gyrogroup,
as stated in the next theorem, which is an application of the previous corollary.

Theorem 4. There is no non-trivial homomorphism from the Möbius gyrogroup to a
group.

Proof. Suppose that Γ is a group, and suppose that φ is a homomorphism from D to
Γ. By Proposition 3.4 of [4], Da ⊆ kerφ, and so kerφ = D because Da = D. It follows
that φ is trivial, which completes the proof.

As a consequence of Theorem 4, we conclude that a non-trivial homomorphism from
the Möbius gyrogroup to the multiplicative group of non-zero complex numbers does not
exist. This proves Theorem 1, which fulfills the goal of this paper and completely solves
Problem 1.3 of [3]. Moreover, it follows that every representation of the Möbius gyrogroup
is trivial. Note that the associativization of the Möbius gyrogroup is trivial, which indi-
cates that the Möbius gyrogroup is, in some sense, most far from being a group. We
remark that extending the obtained results to higher dimensions is quite challenging and
not straightforward, due to the complexity of the Möbius addition formula. Finally, for
the future work, we remark that Möbius addition on the complex open unit disk enables
optimization in hyperbolic geometry, where geodesic-based methods replace Euclidean
approaches. Its non-associative nature models systems with order-sensitive updates, use-
ful in multi-agent and sequential decision-making. Möbius-based algorithms are suited
for learning in non-Euclidean spaces like hyperbolic spaces. As Möbius transformations
preserve the disk, they support projection-free optimization in compact domains. Appli-
cations may span machine learning, signal processing, and complex domain optimization,
where Möbius transformations maintain structural properties vital for constrained and
geometric optimization tasks.


R. Maungchang et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6041 8 of 8

Acknowledgements

This research was supported by Chiang Mai University.

Conflict of interest

The authors declare no conflict of interest.

Data availability

Data sharing not applicable to this paper as no datasets were generated or analyzed during
the current study.

Author contributions

R. Maungchang: writing—review and editing, validation, investigation, visualization; W.
Atiponrat: writing—review and editing, validation, investigation; T. Suwansri: validation,
investigation, visualization; J. Wattanapan: validation, investigation; T. Suksumran: con-
ceptualization, methodology, validation, investigation, writing—original draft preparation,
supervision, project administration. All authors have read and agreed to the published
version of the manuscript.

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