Normal Families of Bicomplex Holomorphic Functions

1.2 MB·English

by  K. S. Charak

Normal Families of Bicomplex
Holomorphic Functions
8 K.S. Charak1, D. Rochon2, N. Sharma3
0
0
2
1 Department of Mathematics, University of Jammu, Jammu-180 006, INDIA.
c
e E-mail: kscharak7@rediffmail.com
D

5
2 D´epartement de math´ematiques et d’informatique, Universit´e du Qu´ebec `a

] Trois-Rivi`eres, C.P. 500 Trois-Rivi`eres, Qu´ebec, Canada G9A 5H7.
V
E-mail: [email protected], Web: www.3dfractals.com
C
.
h
3 Department of Mathematics, University of Jammu, Jammu-180 006, INDIA.
t
a E-mail: [email protected]
m
[

4 Abstract
v
Inthisarticle,weintroducetheconceptofnormalfamiliesofbicomplex
3
holomorphicfunctionstoobtainabicomplexMonteltheorem. Moreover,
0
we give a general definition of Fatou and Julia sets for bicomplex poly-
4
nomials and we obtain a characterization of bicomplex Fatou and Julia
4
. sets in terms of Fatou set, Julia set and filled-in Julia set of one complex
6 variable. Some 3D visual examples of bicomplex Julia sets are also given
0
for the specific slice j=0.
8
0
:
v
Xi Keywords: BicomplexNumbers,ComplexCliffordAlgebras,NormalFamilies,
Montel Theorem, Bicomplex Dynamics, Julia Set, Fatou Set.
r
a
1
1 Introduction
A family F of holomorphic functions defined on a domain D ⊆C is said to be
normal in D if every sequence in F has a subsequence converging uniformly on
compactsubsetsofD toafunctionf. Thelimitfunctionf isholomorphiconD
(byWeierstrassTheorem)ortheconstantinfinity. Variousauthorswhilestudy-
ing the normality of a family of holomorphic functions take the limit function
f (cid:54)= ∞ but for studying the normal families from complex dynamics point of
view,oneneedstoincludethecasewherethelimitfunctionf ≡∞. Theformer
approach we shall call as restrictive approach while the later will be called the
general approach towards normal families. The concept of normal families was
introduced by P. Montel in 1907 [19]. For comprehensive account of normal
familiesofmeromorphicfunctionsondomainsinConecanrefertoJoelSchiff’s
text [16], C.T. Chuang’s text [20] and Zalcman’s survey article [21]. With the
renewed interest in normal families of meromorphic functions, arising largely
from the important role they play in Complex Dynamics, it seems sensible to
talkaboutnormalfamiliesofholomorphicfunctionsondifferentdomainsofdif-
ferent spaces thereby enabling one to study the dynamics of such functions. In
this article we have considered the families of bicomplex holomorphic functions
on bicomplex domains. Since this article lays the foundations of the subject
Normal Families of Bicomplex Holomorphic Functions for future inves-
tigations in various possible directions, it is necessary to adopt a dual approach
towards the study of normality of families of bicomplex holomorphic functions
on bicomplex domains. The first approach is restrictive approach which gives
risetomoreinterestingresultswhenthenormalfamiliesarestudiedintheirown
right. For example the converse of Montel Theorem holds under this approach.
The second approach is the general approach in which though the converse of
MontelTheoremfailstoholdbutisessentiallyrequiredwhenthenormalfamilies
arestudiedfromthebicomplexdynamicspointofview. Duringourdiscussions,
weshallcomeacrossthesituationswherethedifferencesleadtointerestingcon-
clusions. Besides complete discussion on Montel Theorem in various situations,
we have defined Fatou, Julia and filled-in Julia sets of bicomplex polynomials
and their representations in terms of their complex counterparts are obtained
for the specific case of non-degenerate bicomplex polynomials of degree d ≥ 2.
Also, some 3D visual examples of bicomplex Julia sets are given for the specific
slice j=0.
2 Preliminaries
2.1 Bicomplex Numbers
Bicomplex numbers are defined as
T:={z +z i | z ,z ∈C(i )} (2.1)
1 2 2 1 2 1
2
where the imaginary units i ,i and j are governed by the rules: i2 =i2 =−1,
1 2 1 2
j2 =1 and
i i = i i = j,
1 2 2 1
i j = ji = −i , (2.2)
1 1 2
i j = ji = −i .
2 2 1
Note that we define C(i ) := {x+yi | i2 = −1 and x,y ∈ R} for k = 1,2.
k k k
Hence, it is easy to see that the multiplication of two bicomplex numbers is
commutative. In fact, the bicomplex numbers
T∼=ClC(1,0)∼=ClC(0,1)
are unique among the Complex Clifford algebras (see [14]) in that they are
commutative but not division algebra. It is also convenient to write the set of
bicomplex numbers as
T:={w +w i +w i +w j | w ,w ,w ,w ∈R}. (2.3)
0 1 1 2 2 3 0 1 2 3
In particular, in equation (2.1), if we put z = x and z = yi with x,y ∈
1 2 1
R, then we obtain the following subalgebra of hyperbolic numbers, also called
duplex numbers (see, e.g. [12], [18]):
D:={x+yj | j2 =1, x,y ∈R}∼=ClR(0,1).
Complex conjugation plays an important role both for algebraic and geo-
metric properties of C. For bicomplex numbers, there are three possible conju-
gations. Let w ∈T and z ,z ∈C(i ) such that w =z +z i . Then we define
1 2 1 1 2 2
the three conjugations as:
w†1 =(z1+z2i2)†1 :=z1+z2i2, (2.4a)
w†2 =(z1+z2i2)†2 :=z1−z2i2, (2.4b)
w†3 =(z1+z2i2)†3 :=z1−z2i2, (2.4c)
where z is the standard complex conjugate of complex number z ∈ C(i ). If
k k 1
we say that the bicomplex number w =z +z i =w +w i +w i +w j has
1 2 2 0 1 1 2 2 3
the “signature” (++++), then the conjugations of type 1,2 or 3 of w have,
respectively, the signatures (+−+−), (++−−) and (+−−+). We can verify
easily that the composition of the conjugates gives the four-dimensional Klein
group:
◦ † † † †
0 1 2 3
† † † † †
0 0 1 2 3
† † † † † (2.5)
1 1 0 3 2
† † † † †
2 2 3 0 1
† † † † †
3 3 2 1 0
where w†0 :=w ∀w ∈T.
3
The three kinds of conjugation all have some of the standard properties of
conjugations, such as:
(s+t)†k = s†k +t†k, (2.6)
(cid:0)s†k(cid:1)†k = s, (2.7)
(s·t)†k = s†k ·t†k, (2.8)
for s,t∈T and k =0,1,2,3.
Weknowthattheproductofastandardcomplexnumberwithitsconjugate
givesthesquareoftheEuclideanmetricinR2. Theanalogsofthis,forbicomplex
numbers, are the following. Let z ,z ∈ C(i ) and w = z +z i ∈ T, then we
1 2 1 1 2 2
have that :
|w|2i1 :=w·w†2 =z12+z22 ∈C(i1), (2.9a)
|w|2i2 :=w·w†1 =(cid:0)|z1|2−|z2|2(cid:1)+2Re(z1z2)i2 ∈C(i2), (2.9b)
|w|2j :=w·w†3 =(cid:0)|z1|2+|z2|2(cid:1)−2Im(z1z2)j∈D, (2.9c)
where the subscript of the square modulus refers to the subalgebra C(i ),C(i )
1 2
or D of T in which w is projected. Note that for z ,z ∈ C(i ) and w =
1 2 1
z +z i ∈ T, we can define the usual (Euclidean in R4) norm of w as (cid:107)w(cid:107) =
1 2 2
(cid:112) (cid:113)
|z |2+|z |2 = Re(|w|2).
1 2 j
w†2
It is easy to verify that w· =1. Hence, the inverse of w is given by
|w|2
i1
w−1 = w†2 . (2.10)
|w|2
i1
From this, we find that the set NC of zero divisors of T, called the null-cone, is
given by {z +z i | z2+z2 =0}, which can be rewritten as
1 2 2 1 2
NC ={z(i ±i )| z ∈C(i )}. (2.11)
1 2 1
2.2 Bicomplex Holomorphic Functions
It is also possible to define differentiability of a function at a point of T:
Definition 1 Let U be an open set of T and w ∈ U. Then, f : U ⊆ T −→ T
0
is said to be T-differentiable at w with derivative equal to f(cid:48)(w )∈T if
0 0
f(w)−f(w )
lim 0 =f(cid:48)(w ).
w→w0 w−w0 0
(w−w0 inv.)
Note: The subscript of the limit is there to recall that the division is possible
only if w−w is invertible.
0
4
We also say that the function f is bicomplex holomorphic (T-holomorphic)
on an open set U if and only if f is T-differentiable at each point of U. Using
w =z +z i ,abicomplexnumberwcanbeseenasanelement(z ,z )ofC2,so
1 2 2 1 2
a function f(z +z i )=f (z ,z )+f (z ,z )i of T can be seen as a mapping
1 2 2 1 1 2 2 1 2 2
f(z ,z )=(f (z ,z ),f (z ,z ))ofC2. Herewehaveacharacterizationofsuch
1 2 1 1 2 2 1 2
mappings:
Theorem 1 Let U be an open set and f : U ⊆ T −→ T such that f ∈ C1(U),
and let f(z +z i ) =f (z ,z )+f (z ,z )i . Then f is T-holomorphic on U
1 2 2 1 1 2 2 1 2 2
if and only if
f and f are holomorphic in z and z ,
1 2 1 2
and
∂f ∂f ∂f ∂f
1 = 2 and 2 =− 1 on U.
∂z ∂z ∂z ∂z
1 2 1 2
Moreover, f(cid:48) = ∂f1 + ∂f2i and f(cid:48)(w) is invertible if and only if detJ (w)(cid:54)=0.
∂z1 ∂z1 2 f
This theorem can be obtained from results in [8] and [11]. Moreover, by the
Hartogs theorem [17], it is possible to show that “f ∈ C1(U)” can be dropped
from the hypotheses. Hence, it is natural to define the corresponding class of
mappings for C2:
Definition 2 The class of T-holomorphic mappings on a open set U ⊆ C2 is
defined as follows:
∂f ∂f ∂f ∂f
TH(U):={f:U ⊆C2 −→C2|f ∈H(U) and 1 = 2, 2 =− 1 on U}.
∂z ∂z ∂z ∂z
1 2 1 2
It is the subclass of holomorphic mappings of C2 satisfying the complexified
Cauchy-Riemann equations.
Weremarkthatf ∈TH(U)intermsofC2 ifandonlyiff isT-differentiable
on U. It is also important to know that every bicomplex number z +z i has
1 2 2
the following unique idempotent representation:
z +z i =(z −z i )e +(z +z i )e . (2.12)
1 2 2 1 2 1 1 1 2 1 2
where e = 1+j and e = 1−j. This representation is very useful because
1 2 2 2
addition, multiplication and division can be done term-by-term. It is also easy
to verify the following characterization of the non-invertible elements.
Proposition 1 An element w = z +z i is in the null-cone if and only if
1 2 2
z −z i =0 or z +z i =0.
1 2 1 1 2 1
Thenotionofholomorphicitycanalsobeseenwiththiskindofnotation. For
this we need to define the projections P ,P : T −→ C(i ) as P (z +z i ) =
1 2 1 1 1 2 2
z −z i and P (z +z i )=z +z i .
1 2 1 2 1 2 2 1 2 1
5
Definition 3 WesaythatX ⊆TisaT-cartesiansetdeterminedbyX andX
1 2
ifX =X × X :={z +z i ∈T:z +z i =w e +w e ,(w ,w )∈X ×X }.
1 e 2 1 2 2 1 2 2 1 1 2 2 1 2 1 2
In [8] it is shown that if X and X are domains (open and connected) of
1 2
C(i ) then X × X is also a domain of T. Then, a way to construct some
1 1 e 2
“discus” (of center 0) in T is to take the T-cartesian product of two discs (of
center 0) in C(i ). Hence, we define the “discus” with center a = a +a i
1 1 2 2
of radius r and r of T as follows [8]: D(a;r ,r ) := B1(a − a i ,r ) ×
1 2 1 2 1 2 1 1 e
B1(a +a i ,r ) = {z +z i : z +z i = w e +w e ,|w −(a −a i )| <
1 2 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 1
r ,|w −(a +a i )| < r } where B1(z,r) is the open ball with center z ∈
1 2 1 2 1 2
C(i ) and radius r > 0. In the particular case where r = r = r , D(a;r,r)
1 1 2
will be called the T-disc with center a and radius r. In particular, we define
D(a;r ,r ):=B1(a −a i ,r )× B1(a +a i ,r )⊂D(a;r ,r ). We remark
1 2 1 2 1 1 e 1 2 1 2 1 2
that D(0;r,r) is, in fact, the Lie Ball (see [1]) of radius r in T.
Now, it is possible to state the following striking theorems (see [8]):
Theorem 2 Let X and X be open sets in C(i ). If f : X −→ C(i )
1 2 1 e1 1 1
and f : X −→ C(i ) are holomorphic functions of C(i ) on X and X
e2 2 1 1 1 2
respectively, then the function f :X × X −→T defined as
1 e 2
f(z +z i )=f (z −z i )e +f (z +z i )e ∀ z +z i ∈X × X
1 2 2 e1 1 2 1 1 e2 1 2 1 2 1 2 2 1 e 2
is T-holomorphic on the open set X × X and
1 e 2
f(cid:48)(z +z i )=f(cid:48) (z −z i )e +f(cid:48) (z +z i )e
1 2 2 e1 1 2 1 1 e2 1 2 1 2
∀ z +z i ∈X × X .
1 2 2 1 e 2
Theorem 3 LetX beanopensetinT, andletf :X −→TbeaT-holomorphic
function on X. Then there exist holomorphic functions f : X −→ C(i ) and
e1 1 1
f :X −→C(i ) with X =P (X) and X =P (X), such that:
e2 2 1 1 1 2 2
f(z +z i )=f (z −z i )e +f (z +z i )e
1 2 2 e1 1 2 1 1 e2 1 2 1 2
∀z +z i ∈X.
1 2 2
3 Bicomplex Montel Theorem
Sincetheconceptslikeuniformboundedness,localuniformboundedness,
anduniform convergence on compact setsaredefinedforfunctionsonany
metricspaceanddonotdependonbicomplexholomorphicfunctions,weassume
these concepts in our discussion and refer the reader to any standard text on
Analysis (e.g. see [4] and [13]). We start our discussion with the following
definition of normality.
Definition 4 A family F of bicomplex holomorphic functions defined on a do-
main D ⊆ T is said to be normal in D if every sequence in F contains a
subsequence which converges locally uniformly on D. F is said to be normal
at a point z ∈D if it is normal in some neighborhood of z in D.
6
Let us consider f : D −→ T be a T-holomorphic function on D. Then by
Theorem 3, there exist holomorphic functions f : P (D) −→ C(i ) and f :
e1 1 1 e2
P (D)−→C(i ) such that
2 1
f(z +z i )=f (z −z i )e +f (z +z i )e ∀ z +z i ∈D.
1 2 2 e1 1 2 1 1 e2 1 2 1 2 1 2 2
We define the norm of f on D as
|f (z −z i )|2+|f (z +z i )|2
(cid:107)f(cid:107)=(cid:107)f(z)(cid:107)={ e1 1 2 1 2 e2 1 2 1 }12,z =z1+z2i2 ∈D.
One can easily see that
• (cid:107)f(cid:107)≥0,(cid:107)f(cid:107)=0 iff f ≡0 on D;
• (cid:107)af(cid:107)=|a|(cid:107)f(cid:107), a∈C(i );
1
• (cid:107)f +g(cid:107)≤(cid:107)f(cid:107)+(cid:107)g(cid:107);
√
• (cid:107)fg(cid:107)≤ 2(cid:107)f(cid:107)(cid:107)g(cid:107).
Thus, the linear space of bicomplex holomorphic functions on a domain D ⊆T
is a normed space under the above norm.
We start with a uniformly bounded family F of bicomplex holomorphic
functions. In this case, we can verify directly the following result.
Theorem 4 A family F of bicomplex holomorphic functions defined on a bi-
complex cartesian domain D is uniformly bounded on D if and only if F =
ei
P (F) is uniformly bounded on P (D), i=1,2.
i i
If we consider now a locally uniformly bounded family F of bicomplex
holomorphic functions, we can prove a similar result since a set K =P (K)×
1 e
P (K) is compact if and only if P (K) is compact for i=1,2.
2 i
Theorem 5 A family F of bicomplex holomorphic functions defined on a bi-
complex cartesian domain D is locally uniformly bounded on D if and only if
F =P (F) is locally uniformly bounded on P (D), i=1,2.
ei i i
Proof Let F be locally uniformly bounded on D. Then for every compact set
K ⊂D there is a constant M(K) such that
(cid:107)f(z)(cid:107)≤M, ∀f ∈F,z =z +z i ∈K.
1 2 2
Thus,
|f (z −z i )|2+|f (z +z i )|2
{ e1 1 2 1 2 e2 1 2 1 }12 ≤M, ∀fei ∈Fei, i=1,2
∀z −z i ∈P (K), z +z i ∈P (K).
1 2 1 1 1 2 1 2
7
Therefore,
√
|f (z −z i )|≤ 2M, ∀f ∈F , ∀z −z i ∈P (K) (3.1)
e1 1 2 1 e1 e1 1 2 1 1
and
√
|f (z +z i )|≤ 2M, ∀f ∈F ∀z +z i ∈P (K). (3.2)
e2 1 2 1 e2 e2 1 2 1 2
Now,letK beacompactsubsetofP (D). Thenthereis(always)acompact
1 1
subset K of P (D) (even singleton will do) such that K × K = K(cid:48) say, is
2 2 1 e 2
a compact subset of D with P (K(cid:48)) = K , i = 1,2. Thus (3.1) holds for any
i i
compact subset of P (D), and similarly for (3.2).
1
Conversely, suppose F is locally uniformly bounded on P (D), i = 1,2.
ei i
Let K be any compact subset of D. Then by continuity of P , K = P (K) is
i i i
compact subset of P (D), i = 1,2 and hence there are constants M (K ) and
i 1 1
M (K ) such that
2 2
|f (z −z i )|≤M , ∀f ∈F , ∀z −z i ∈K
e1 1 2 1 1 e1 e1 1 2 1 1
and
|f (z +z i )|≤M , ∀f ∈F ∀z +z i ∈K .
e2 1 2 1 2 e2 e2 1 2 1 2
Therefore,
M 2+M 2
(cid:107)f(z)(cid:107)≤{ 1 2 2 }21, ∀f ∈F, z =z1+z2i2 ∈P1(K)×eP2(K). (3.3)
SinceK ⊆P (K)× P (K),(3.3)holdsforK alsoandthiscompletestheproof.
1 e 2
(cid:50)
What happens if D is not a bicomplex cartesian product? In the case of
uniformly bounded family of bicomplex holomorphic functions (Theorem 4), it
is easy to verify that the result is true for any domain. In the case of locally
uniformlyboundedfamilyofbicomplexholomorphicfunctions,weneedtorecall
the following results from the bicomplex function theory.
Remark 1 A domain D ⊆ T is a domain of holomorphism for functions of a
bicomplex variable if and only if D is a T-cartesian set ([8], Theorem 15.11),
and if D is not a domain of holomorphism then D (cid:40)P (D)× P (D), and there
1 e 2
exists a holomorphic function which is a bicomplex holomorphic continuation of
the given function from D to P (D)× P (D) ([8] Corollary 15.4).
1 e 2
Theorem 6 A family F of bicomplex holomorphic functions defined on a ar-
bitrary bicomplex domain D is locally uniformly bounded on D if and only if
F =P (F) is locally uniformly bounded on P (D), i=1,2.
ei i i
Proof If F = P (F) is locally uniformly bounded on P (D) for i =1,2, from
ei i i
Remark1, wecanextendD toP (D)× P (D)andapplyTheorem5toobtain
1 e 2
thatF islocallyuniformlyboundedonP (D)× P (D). Fortheotherside, we
1 e 2
need to recall that a family F is locally uniformly bounded on D if and only if
the family F is locally bounded on D i.e. for each w ∈D there is a positive
0
8
numberM =M(w )andaneighbourhoodD(w ;r,r)⊂D suchthat||f(w)||≤
0 0
M for all w ∈ D(w ;r,r) and all f ∈ F (see [16]). Since D(w ;r,r) ⊂ D is a
0 0
bicomplex cartesian product of two discs in the plane, it is easy to verify that
√
the family F is bounded by 2M(w ) on D(P (w ),r) ⊂ P (D) for i = 1,2.
ei 0 i 0 i
As w was arbitrary, F =P (F) is locally bounded on P (D), i=1,2. (cid:50)
0 ei i i
We are now ready to prove the bicomplex version of the Montel theorem.
Lemma 1 Let F be a family of bicomplex holomorphic functions defined on a
bicomplex domain D. If F =P (F) is normal on P (D) for i=1,2 then F is
ei i i
normal on D.
Proof Suppose that F = P (F) is normal on P (D) = D , i = 1,2. We
ei i i i
want to show that F is normal in D. Let {F } be any sequence in F and
n
K be any compact subset of D. Then {P (F )} = {(f ) } is a sequence in
1 n n 1
F = P (F). Since F = P (F) is normal in P (D) then {(f ) } has a
e1 1 e1 1 1 n 1
subsequence {(f ) } which converges uniformly on P (K) to a C(i )-function.
nk 1 1 1
Now, consider {F } in F. Then {P (F )}={(f ) } is a sequence in F =
nk 2 nk nk 2 e2
P (F). SinceF =P (F)isnormalinP (D)then{(f ) }hasasubsequence
2 e2 2 1 nk 2
{(f ) }whichconvergesuniformlyonP (K)toaC(i )-function. Thisimplies
nkl 2 2 1
that{(f ) e +(f ) e }isasubsequenceof{F }whichconvergesuniformly
nkl 1 1 nkl 2 2 n
on P (K)× P (K)⊇K to a bicomplex function showing that F is normal in
1 e 2
D. (cid:50)
Theorem 7 (Montel) Every locally uniformly bounded family of bicomplex
holomorphic functions defined on a bicomplex domain is a normal family.
Proof Let F be a locally uniformly bounded family of bicomplex holomorphic
functions defined on a domain D ⊆ T. Using Theorem 6, we have that F =
ei
P (F)islocallyuniformlyboundedonP (D), i=1,2.Hence,fromtheclassical
i i
Montel Theorem, F = P (F) is normal on P (D) for i = 1,2 and by Lemma
ei i i
1 we obtain that F is normal on D.(cid:50)
Note: The converse of Bicomplex Montel Theorem is also true. Indeed,
suppose that F is normal and not locally uniformly bounded in D. Then in
some closed discus D(a;r ,r ) in the domain D, for each n ∈ N there is a
1 2
function f ∈ F and a point w ∈ D(a;r ,r ) such that (cid:107)f (w )(cid:107) > n. Since
n n 1 2 n n
F is normal, there is a subsequence {f } of {f } converging uniformly on
nk n
D(a;r ,r )toabicomplex(holomorphic)functionf. Thatis,forsomepositive
1 2
integer n , we have
0
(cid:107)f (w)−f(w)(cid:107)<1, ∀k ≥n , and w ∈D(a;r ,r ).
nk 0 1 2
Thus,ifM =max (cid:107)f(w)(cid:107),then(cid:107)f (w)(cid:107)≤1+M, ∀w ∈D(a;r ,r )
z∈D(a;r1,r2) nk 1 2
and this is a contradiction.
The above discussion permits to establish the following results.
Theorem 8 ThefamilyF ofbicomplexholomorphicfunctionsisnormalonthe
arbitrary domain D if and only if F = P (F) is normal on P (D) for i=1,2.
ei i i
9
Corollary 1 IfthefamilyF ofbicomplexholomorphicfunctionsisnormalona
arbitrary domain D (cid:54)=P (D)× P (D), then F is normal on the larger domain
1 e 2
P (D)× P (D).
1 e 2
Corollary 2 A family of bicomplex holomorphic functions F is normal on a
arbitrary domain D if and only if F is normal at each point of D.
4 Bicomplex Montel Theorem from Montel The-
orem of C2
In this section, we want to show that it is possible to see the Bicomplex Montel
Theorem (Theorem 7) as a particular case of the following Montel theorem of
several complex variables (see [15]).
Theorem 9 Let D ⊂ Cn be an open set and F ⊂ O(D,Cn) be a family of
holomorphic mappings. Then the following are equivalent:
1. The family F is locally uniformly bounded.
2. The family F is relatively compact in O(D,Cn).
Since, TH(D) ⊂ O(D,C2), we obtain directly the desired result using the
fact that a family F is relatively compact in O(D,Cn), i.e. F is compact in
O(D,Cn), if and only if F is a normal family (see [4]). Moreover, Theorem 9
will be proven for the specific class TH(D) instead of O(D,C2) if we can show
thatTH(D)isclosedinO(D,C2)withthecompactconvergencetopology. This
is a direct consequence of the following Bicomplex Weierstrass Theorem.
Lemma 2 Let {f } be a sequence of bicomplex holomorphic functions which
n
converges locally uniformly to a function f on a T-disc D(a +a i ;r,r). Then
1 2 2
f is bicomplex holomorphic in D(a +a i ;r,r).
1 2 2
Proof Since f (z +z i ) is T-holomorphic on D(a +a i ;r,r) ∀n ∈ N, we
n 1 2 2 1 2 2
have from Theorem 3 that
(f ) :P (D(a +a i ,r))−→C(i )
ei n i 1 2 2 1
is holomorphic for i = 1,2, ∀n ∈ N. Since D(a +a i ;r,r) is a bicomplex
1 2 2
cartesian product, by the Weierstrass theorem of one complex variable, the
sequence (f ) must converges locally uniformly to the holomorphic function
ei n
f on D(P (a +a i ),r) for i=1,2. Therefore, from Theorem 2, the function
ei i 1 2 2
f(z +z i )=f (z −z i )e +f (z +z i )e is T-holomorphic on D(a +
1 2 2 e1 1 2 1 1 e2 1 2 1 2 1
a i ;r,r).(cid:50)
2 2
Theorem 10 (Weierstrass)Let{f }beasequenceofbicomplexholomorphic
n
functions on a domain D which converges uniformly on compact subsets of D
to a function f. Then f is bicomplex holomorphic in D.
Proof Foranarbitraryw ∈D,chooseaT-discD(w ;r,r)⊂D. Sincef (w)→
0 0 n
f(w) locally uniformly on D, by Lemma 2, f is T-holomorphic on D. As w
0
was arbitrary, f(w) is T-holomorphic on D.(cid:50)
10