International Journal of Pure and Applied Mathematics
————————————————————————–
Volume 71 No. 3 2011, 383-389

LIOUVILLE’S THEOREM AND

POWER SERIES FOR QUATERNIONIC FUNCTIONS

J.A. Marão1, M.F. Borges2 §

1Department of Mathematics
UFMA - Federal University of Maranhão

65085-580, Maranhão, BRAZIL
2UNESP - São Paulo State University

S.J. Rio Preto Campus
15054-000, São José do Rio Preto, BRAZIL

Abstract: In recent years quaternionic functions have been an intense and
prosperous object of research, and important results were determined [1]-[6].
Some of these results are similar to well known cases in one complex variable,
op. cit. [5], [6]. In this paper the hypercomplex expansion of a function in a
power series as well as determination of a Liouville’s type theorem have been
investigated to the quaternionic functions. In this case, it is observed that the
Liouville’s type theorem is true for second order derivatives, which differs from
its classical version.

AMS Subject Classification: 30G99, 30E99
Key Words: quaternions series, hypercomplex, quaternions

1. Introduction and Motivation

The development in series of powers is a natural resource to determine how a
function might be expanded as a sum of powers. It helps the determination of
integration in some cases. To fix ideas, it is necessary to show what happens to
the case of functions of one real variable, as seen in [7]. The Taylor series for a
analytic function f(z), is such that:

Received: April 23, 2011 c© 2011 Academic Publications, Ltd.
§Correspondence author



384 J.A. Marão, M.F. Borges

f(z) = f(a) +
z − a

1!
f ′(z) +

(z − a)2

2!
f ′′(z) + . . . +

(z − a)n

n!
fn(z) + Rn(z), (1)

f(z) =
∞∑

m=0

an(z − a)n, (2)

where an = fn(a)
n! e limn→∞ Rn(z) = 0.

The series given in (1) are called the Taylor series centered at a. For the
case where a = 0, the series is called Maclaurin Series.

The determination of the Taylor series for quaternionic functions, you need
some steps on their rigorous proof. Thus, an issue raised in determining the
convergence of the rest to zero as n tends to infinity requires the area of 4-
dimensional hyper sphere. To this end, the formula generally used is [8]:

2π
n

2 rn−1

Γ(n
2 )

, (3)

where Γ(∗) is Gamma Function. This formula is necessary determine the area
of hyper surface in the case n = 4. Thus, it follows that:

AHE = 2π2r3. (4)

2. Liouville’s Theorem and Liouville’s Like Theorems

Liouville’s like Theorems have all been treated as main source of research in
few number of recent mathematical works [9], [10], [11], [12]. They have, for
instance been used for radial solutions for the bi-harmonic operator and for
studying both quasi-harmonic functions and foliations with complex leaves.
Under a quaternionic point of view, a long-time ago quaternion holomorphs
were presented and a Liouville’s Theorem helped to approach a solution of
some regular integral functions.

The Liouville’s Theorem, was first developed by French mathematician
Joseph Liouville (1809-1882) for functions of a complex variable. Your state-
ment is given below:

“If f(z) is analytic and limited in absolute terms for a complex z finite
number, then f(z) is a constant”, see [7].

To fix ideas, it will be shown now a relationship that will assist in the
statement of Liouville’s theorem to the quaternionic case. Following [9], the
derivation by Cauchy’s integral formula, is such that:



LIOUVILLE’S THEOREM AND... 385

fn(q) =
1

(i + j + 2k)π

∫
Λ

f(q′)

(q′ − q)
dq′ (5)

So, considering Λ a hyper sphere of radius r centered at q and M is still a
maximum of |f(q)| on Λ. It follows that:

|fn(q)| =
n!

6π
|

∫
Λ

f(q′)

(q′ − q)
dq′| ≤

n!

6π

K

rn+1
2π2r3 (6)

|fn(q)| ≤
n!πK

rn−2
. (7)

Using the statement above, will now be made a statement to the Liouville
theorem for quaternions.

Theorem 1. If f ′′(q) is analytic and limited in absolute terms for a

complex q finite number, then f ′′(q) is a constant.

Demonstration. By assumption, |f ′′(q)| < M for any q Thus, f ′′′(q) = M
r

.

Since this is valid for all r, can be considered r as large as you want, and conclude
that f ′′′(q) = 0.

As it is easyly observed in this theorem there is a fundamental difference
compared to classical Liouville theorem, where now the theorem is initially true
for second derivatives.

3. Power Series for the Quaternionic Case

Take a quaternionic function that has derivatives in a neighborhood of the point
q = a. Considering now a hypersphere Λ which exists in that neighborhood and
has a center at a. The Cauchy Formula, seen in [5] can be applied as follows:

f(q) =
1

(i + j + 2k)π

∫
Λ

f(q′)

q′ − q
dq′ (8)

where the point q is an arbitrary point and fixed the interior of Λ and q′ is the
variable of integration. Now follows that:

1

q′ − q
=

1

q′ − a(q − a)
=

1

(q′ − a)(1 − q−a
q′−a

)
. (9)

With data on q′ and q it follows that:

|
q − a

q∗ − a
| < 1.



386 J.A. Marão, M.F. Borges

Moreover, it follows that:

1

1 − q−a
q′−a

= 1 +
q − a

q′ − a
+ [

q − a

q′ − a
]2 + . . . + [

q − a

q′ − a
]n +

[ q−a
q′−a

]n+1

q′−q
q′−a

which was determined by geometric progression. Here, now, by replacing in (8)
that f(q) is given by:

f(q) =
1

(i + j + 2k)π

∫
Λ

f(q′)

q′ − a
dq′ +

q − a

(i + j + 2k)π

∫
Λ

f(q′)

q′ − a
dq′

+ . . . +
(q − a)n

(i + j + 2k)π

∫
Λ

f(q′)

q′ − q
dq′ + Rn(q) (10)

where

Rn(q) =
(q − a)n

(i + j + 2k)π

∫
Λ

f(q′)

(q′ − a)n+1(q′ − q)
dq′.

Let the function f(q) be developed as given below:

f(q) = f(a) +
q − a

1!
c1 +

(q − a)2

2!
c2 + . . . +

(q − a)n

n!
cn(q) + Rn(q). (11)

or

f(q) =

∞∑
n=0

cn

n!
(q − a)n. (12)

Performing successive derivations on function f(q), then we obtain:

cn =
fn(q)

n!
(q − a)n. (13)

Moreover, using the theorem given in [9], which is derived from quaternionic,
it follows that:

fn(q) =
n!

π(i + j + 2k)

∫
Λ

f(q′)

(q′ − q)n+1
dq′. (14)

It now remains to prove that the limit below:

lim
n→∞

Rn(q) = 0.

Using the facts below:

|q′ − q| > 0,



LIOUVILLE’S THEOREM AND... 387

|
f(q′)

q′ − q
| < K

and

|q′ − a| = r.

taking into account that f(q′) is analytic inside Λ and q′ ∈ Λ. Since then the
area of the sphere 4πr2, it follows that:

|Rn| =
|q − a|n+1

6π
|

∫
Λ

f(q′)

(q′ − a)n+1(q′ − q)
dq′| <

|q − a|n+1

6π
K

1

rn+1
2π2r3

=
1

3
πr3K|

q − a

r
|n+1

but as | q
q′−a

| = | q
r
| < 1 when n → ∞ the expression on the right approaches

zero.

The results are resumed in the following Theorem:

Theorem 2. If f(q) analytic in P and q = a one point in P. Exists, one

power series with center in a such that:

f(q) =
∞∑

n=0

cn(q − a)n

where

cn =
1

n!
fn(q), n = 0, 1, 2, ...

and

lim
n→∞

Rn = 0.

4. Conclusion

In (12) and (13) was seen a generalization to power series, in the case of quater-
nionic functions. Moreover, we noted that the Liouville theorem is valid from
the second derivative of the function, differently from what happens in case of
functions of one complex variable. It is worth mentioning that this work can
be extended to the determination of Power series as well as to the extension of
Liouville’s theorem for the octonionic case. These are main purpose of future
works.



388 J.A. Marão, M.F. Borges

5. Acknowledgments

The author José Antônio Pires Ferreira Marão appreciates the support and en-
couragement of Camila Marinho Penha, Framilson José Ferreira Carneiro, Pe-
dro Ribeiro de Alencar, Artur Silva Santos, Lucimá Góes de Sousa and Benedito
dos Santos Raposo.

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