Hyperholomorphic functions with values in a modified form of quaternions Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 9 (27), No 1, 2020, pp. 83-95 83

DOI: 10.15393/j3.art.2020.6330

UDC 517.548, 517.552, 517.518.126

Ji Eun Kim

HYPERHOLOMORPHIC FUNCTIONS WITH VALUES IN A MODIFIED FORM OF QUATERNIONS

Abstract. We give the definition of hyperholomorphic pseudo-complex functions, i.e., functions with values in a special form of quaternions, and propose the necessary variables, functions, and Dirac operators to describe the Cauchy integral theorem and the generalized Cauchy-Riemman system. We investigate the properties and corollaries corresponding to the Cauchy integral theorem for the pseudo-complex number system discussed in this paper.

Key words: hyperholomorphic, quaternion, dirac operator, Cauchy-Riemman system

2010 Mathematical Subject Classification: 32A30, 30G35, 32W50, 26B05

1. Introduction. The non-commutative four-dimensional space R4 of hypercomplex numbers, which are called quaternions with four real numbers, was studied by Hamilton [5]. Since quaternions involve non-commutative multiplication, quaternions have different algebraic properties compared to the complex number system. In 1935, Fueter [2] defined regular quaternionic functions in R4. Later Deavours [1] and Subdery [12] developed quaternionic analysis, based on complex analysis.

Many formulas in R4 are simpler and more convenient to apply in physics when written in terms of C2 . In [11], Nono represented quaternions in the complex-number form. In [6], Kajiwara et al. gave an integra-bility condition for any hyperholomorphic function f\ + f2j composed of harmonic complex-valued functions f\ and f2 in a pseudoconvex domain of C4. In [7], [8], Kim et al. presented a ternary representation of real quaternions and also introduced the pseudo-complex number form with the modified basis i. The regularity of a function defined in R3 relative to the commonly known properties of regular functions was defined.

© Petrozavodsk State University, 2020

Hamilton tried to generalize complex numbers to the form a + ib + jc, where a,b,c E R and i2 = j2 = -1. However, since the set {a + ib + + jc | a,b,c E R} is not closed under multiplication (which was proved by Kenneth in 1966), this set cannot be generalized as an algebra. Later, Hamilton found a closed multiplication for complex numbers, denoted by q = ix+jy + kz, where i2 = j2 = k2 = ijk = -1. Some interesting investigations were carried out on the set {a + ib + jc | a,b,c E R}. Leutwiler [9] studied the interplay between the solutions f = u + iv + jw of the generalized Cauchy-Riemann system and functions of the reduced quaternionic variable z = x + iy + jt. Leutwiler showed that every solution f of that system defined in some neighborhood of the origin admits a series expansion in terms of the elementary polynomial solutions. In [3], [4], Gurlebeck and Sprofiig studied quaternion-valued functions that are defined in open subsets of Rra (n = 3, 4) and are solutions of generalized Cauchy-Riemann or Dirac systems. Their research is related to boundary-value problems and partial differential equations.

This paper recalls the properties resulting from the applications of the defined differential operators and the regularity of modified ternary functions. Using the properties of a modified ternary function, we present integration over the boundary of a domain in the modified ternary numbers. In addition, the present paper presents and verifies the Cauchy integral theorem for modified ternary functions. We also expose corollaries to the Cauchy integral theorem. The paper introduces the definitions of hyperholomorphic functions on the real ternary numbers and represents pseudo-complex numbers as a special form of quaternions, defined as a+bi. In section 2, we provide the necessary variables, functions, and operators used in the paper. In section 3, we refer to Naser [10] and Nono [11] in order to propose Dirac operators and Cauchy integral theorems. And then we introduce the properties and corollaries corresponding to the Cauchy integral theorem for the pseudo-complex number system.

2. Preliminaries. Let T be the set of all ternary numbers:

T = {zlz = x0 + X\e\ + x2e2, x0, X\,x2 E R},

where = e2 = —1 and e\e2 = V —1. An element z of T can be written

as

z = xo + x\ei + X2e2 =

aei + be2 ( Va2 + b2 Va2 + b2 = xo + / 2 , -xiei +-x2e2

\Ja2 + b2 \ aei + be2 aei + be2

)

ael + be2 ( axl + bx2 bxl — ax2 \

x0 T--, —, T--, ele2 ,

0 Va2 + b2\ V^TÛ2 Va2 + b2 1 )

where a and b are real non-zero numbers. Let i be the modified basis in T, denoted by

- aei + be2 -2

% = , r and % = -1. yja2 + b2

Then, an element z of T can also be written as

z = xq + izo;

such a number is called a pseudo-complex number; here

axi + bx2 bxi — ax2 z0 = —, +--, e-^e 2.

Va^+P 1

The set of pseudo-complex numbers, denoted by P, is isomorphic to R x C; that is, P = R x C. The addition and multiplication for pseudo-complex numbers are given by

z±w = (xo + izo) ± (yo + iwo) = = (xo ± yo) + i(Zo ± Wo)

and

ZW = (xo + i Zo)( yo + i Wo) =

= (xoyo — Zo Wo) + i(xoWo + Zoyo),

respectively. From the multiplication over P, we can obtain z0i = iz0. Hence, the multiplication over P is closed and associative but not commutative.

Let z be the conjugate of z, denoted by z = x0 — iz0 with zz = zz. Also, the norm | • | is written by

|z| := \[zz = \Jxl + zozo = \jx2 + x2 + x2,.

The inverse element z 1 of P is denoted by

-i = * Z = kl2 ■

Now, consider the definition of hyperholomorphy for pseudo-complex functions. First, the differential operators are given by

D

d , d — , -

d

dx0 d z0 dx0

ei

d dx1

e2

dx2

and

- d , d D = ---+ i

dx0

d

d z0 dx0

d d + &2

dx1

dx2

where dzo

1 d

(

b

2 dxi\ Va2 + b2 Va2 + b2

&1&2 +

+

d

(

b

+

dx2\ Va2 + b2 Va2 + b2

Feie2

1

d

+

b

d

2\ Va2 + b2 dxi Va2 + b2 dx2 b d

d

Va2 + b2 dxi Va2 + b2 8x2

)

and

d

1 (

2 dxi V

+

b

dzo 2 dxi\ Va2 + b2 Va2 + b2

+

+

d

b

dx2\ Va2 + b2 Va2 + b2

1

d

+

b

&1&2

d

2\ Va2 + b2 dxi Va2 + b2 0x2

+

+

b

d

d

Va2 + b2 dxi Va2 + b2 0x2

)

eie2.

Then, the Laplacian operator is given by

A := DD = DD

(

d . d — i

d2

+

d2

dx0 d z0 J \ dx, d2 d2 d2

d . d

dzo

)

dxi dz0dz0 dx0 dx2 dx2

a

a

a

a

a

a

o

3. Properties of hyperholomorphic functions. Let Q be a domain in R3. Consider a function f defined on Q and with values in Pi such that f : Q ^ P is defined by

f = u° + uiei + u2e2 = u° + if°.

That is, f satisfies

z = (x0,xl,x2) E Q M- f(z) = u0(x0,xl,x2) + ifo(xo, xl,x2) E P, where ur (r = 0,1, 2) are real-valued functions and

aul + bu2 bul — au2 = + eie2

is a complex-valued function. The function f is called a pseudo-complex function. Let the differential operators defined in Section 2 be applied to a function f : Q ^ P. Then we have the following equalities:

n , f 9 , 9 \ , fduo dfo\ *.(dfo duo\

and

Tit ( 9 , • 9 \ < , (9u° 9fA ^-(9fo . 9uo\ D = 19x0 + zWo) (u° + Wo — 97°) +l Wo + WJ .

Since the set P has non-commutative multiplication, we also apply operators to the function f from the right. We have

/ 9 • 9 \ (9u° . 9 fo\ , ,(9fo 9u°\

fD = (uo + lfo){— ld¥o) = (,+ d¥°) Wo — )

and

tn i 9 , • 9 \ (9u° 9fo\ {9fo 9uo\

fD = (u + zfo){+ zd¥o) = V— d¥o) +z[dx0 + W0) .

Hence, the equality Df = 0 implies that f satisfies the following equations:

9 u 9 u

— -— and — '1N

dx0 dz0 dx0 d z0

called the (left-)pseudo-complex Cauchy-Riemann equations. Similarly, if f satisfies fD = 0 then we obain the equations:

duo dfo dfo duo

and

дх0 д zо дх0 д z0'

called the (right-)pseudo-complex Cauchy-Riemann equations. Basing on the definition of the Laplacian, we also obtain

&-f _ {DT,)f _ (й + öSto Ь+'*) =

_ f сРщ + д 2u0 \ + , / d% + д 2fo \ V dx2 dz0dz0J V dxi dz0dz0J'

Since multiplication over P is associative,

(DD)f _ D(Df)

Therefore,

nmt\ ( 9 • 9 \ ! (du0 9fo\ ,--(dio . 9ио D(Df) _ - гд¥о){ - d7j +г + д¥о

д 2и0 д 2fo ,fd2f,о d2ut _ ------I- г -— +--

Io + /у Jo + и uo дх0 dxodzo V dxo dxod zo

bo om,o^O 4 ^o _д2uo d2fo \ + f d2fo + д2u0 \d zo dxo д zod zo) \d zodxo д zodzo, io d2uo

>+t{ %

Definition 1. Let П be an open set in R3. A function f : П ^ P,

fd2uo + d2uo \ + j/cpfo + д2fo \

\ dxo dzod zj \ dxi д zodzoJ

f(z) = Uo{Xo,Xi,X2) + ifo(Xo,Xi,X2),

is said to be left-hyperholomorphic on Q if f satisfies the following two conditions:

1) u0 is a real-analytic function and f0 is a holomorphic function,

2) f satisfies the equation Df = 0 on Q.

Involving the non-commutativity of multiplication, comparing (1) and (2), we also give the following

Definition 2. Let Q be an open set in R3. A function f : Q ^ P,

f(z) = Uo(Xo,Xi ,x2) + ifo(Xo,Xi,X2)

is said to be right hyperholomorphic on Q if f satisfies the following two conditions:

1) u0 is a real-analytic function and f0 is a holomorphic function.

2) f satisfies the equation fD = 0 on Q,

Since a right hyperholomorphic function is dealt with in a similar manner as a left hyperholomorphic function, we only consider left hyperholomor-phic functions and simply call them hyperholomorphic.

Proposition 1. Let Q be an open set in R3 and f be a hyperholomorphic function on Q. Then

Df f df ^f

DJ = J = 7T" = — 7^=-ox0 OZo

Proof. Since f is a hyperholomorphic function on Q, (1) yields

Uo o O o Uo O Uo O o O

Df = VOX° + wj +l Wo - W0) = dïo + ldX0 = W0f-

Moreover, by (1), for Df we also have

dfo ,du0 ,2dfo ,du0 , { d , d \ ,d

DJ = »--lTFr = — »--Z7)zr = — "kF7"+ = J'

dzo d z0 dzo d z0 \d z0 d z0 J d z0

□

Let us now consider the properties of hyperholomorphic functions in pseudo-complex numbers.

Proposition 2. Let Q be an open set in R3 and f and g be hyperholo-morphic functions on Q. Then

1) af is hyperholomorphic on Q if a is any real constant,

2) fa is hyperholomorphic on Q if a is any ternary constant,

3) f ± g is hyperholomorphic on Q.

Proof. The condition that f and g are both hyperholomorphic functions means that they satisfy (1). For proving items 1) - 3), it suffices to satisfy the second condition of Definition 1.

1) When a is any real constant, it is obvious that D(af) = 0.

2) Let a be a pseudo-complex constant, a = a0 + ia0, where a0 is real and

Cidi + C2CL2 C2di — CiO,2

ao = —, 0 0 +--, 0 0 eie2

\J cf + c2

2 2

-2 Vci +

with cr and ar (r = 1, 2) being real numbers. By (1), we infer

D(fa) =

(

d * d \ — *

+ i^T I {(Uoao — foao) + i(uoao + focio)}

dx{

dzot

duo dfo duo dfo . , T^cio — T^ao — -r^ao — ^ao ) + dxo dxo dzo dzo

o

+

duo d fo duo dfo ■7;—ao + t;—ao + t^^^o — dxo dxo dzo a zo

)

3) Since f and g are hyperholomorphic functions on Q, we have

D(f ± g)

(

d * d \

+ 1 ^- ) {(uo ± Wo) + 1 (fo ± go)}

dx<

(

d zo,

duo ± dvo — dfo ^ dgo dxo dxo dzo dzo

+

duo . dvo , d fo , dgo ± ---+ -—ao ±

d zo d zo dxo

dxi

)

□

Example. Let Q be an open set in R3 and f and g be hyperholomorphic functions on Q. Then is not always hyperholomorphic on Q. Since and are hyperholomorphic functions on Q, we obtain

D(fg)

(

d * d \ - *

+ 1 {(uoVo — fogo) + 1 (Uo9o + foVo)}

dx,

dzot

(duo_ dfo\v +u i^o _ \ V&Co dzo) o \dxo dzo)

dfo + duo

dxo Td 9o

+

dxo duo

dzr

\ f-FU yo . f wo jyo — ( Jo— + fo —

dfo . + r, — T^T go + Uo dxo azo

dxo dg o

+

dgo dzo

d zo dv o

+

dxo dzo

+

0

0

, • / (9 fo , d*>\ . ( , 9vo T д go

+ ,{+ Wo) Vo +{ЛэГо - foШ0

_ T®9o fdvo.-.f dvo dgo

— — Joö--Jo^--H г jo^--г jowzr

dxo о zo dxo о zo

_Tdvo , dvo *.f dgo *.T dgo _T dg dg

d o o o d o d o d o

— dg dg

If f0^z--f0^— = 0 then the function fg is hyperholomorphic on Q. For

d zo dzo

example, if f is a real-valued function then fg is hyperholomorphic on Q. — dg dg

However, when fo^r = foт.—, fg is not hyperholomorphic on Q. d Zo dZo

Put

ш = dzo Л d zo — idxo Л dzo.

Theorem 1. Let Q be a domain in R3 and U be any domain in Q with smooth boundary bU such that U С Q. If f is a hyperholomorphic on Q then

f ш f = 0.

bU

Proof. We have

wf =(dzo Лdzo — idxo Л dzo)(uo + ifo) =

= (uodzo Л dzo + fodxo Л dzo) + i(fodzo Л dzo — uodxo Л dzo).

Let d and d be the following operators:

о 1 d 7 d 1 1 - 1 d d

d = --— dxo + ^—dzo and d = --— dxo + -wzr dzo. 2 dxo dzo 2 dxo d zo

Then

- id d d d(wf) — (d + d)(wf) — [—dxo + zo + zo) (wf)

\dxo dzo о Zo J

io д Zo

( duo d fo\ fo duo\

---—— dxo Л d zo Ad zo + г —--+ 7—- dxo Л d zo Ad zo — 0

\dxo dzoj \dxo d Zo

in U. It now suffices to apply the Stokes theorem. □

Theorem 2. Let Q be an open set in R3. If f is hyperholomorphic on Q then uo and fo are harmonic functions on Q. Moreover, f is harmonic on Q.

Proof. It suffices to show that Au0 = 0 and Af0 = 0. Indeed, we have

a fnn\ d2uo d2uo d dfo d dfo

Auo = (DD)uo = -t^t + t.—ttzt = --= 0,

ox0 oz0o z0 ox0 oz0 az0 ox0

A f = (DD) f = 92 fo + 92fo = 9 9uo + 9 9uo =0 A f0 = (DD) f0 = + = - dX0 d¥o + d¥o dX0 = 0'

By these equalities and (1), we can obtain the equality A f = 0 as follows:

^ ( ) ^ d-"2 dx0dz0 + dz2^x0 + dz0dz^) + + iid%+ d2uo - d2uo + d2fo N = ydx" dx0d z0 d z0dx$ dz0dz0) d f du0 df0\ + /dfo + dW +

dx0 \dx0 dz0 J dz0 \dx0 d z0

+ / d fo + 5uo \ +1 9 f 9uo + ® fo\ =0 dx0 \dx0 d z0J d z0 \ dx0 dz0 J

Thus, f is a harmonic function on Q. □

Consider the following example related to the statement presented in Theorem 2.

Example. Let u0 be a real-valued harmonic function such that

, \ x0 uo(z) =

in a domain D C P. Then the hyper-conjugate harmonic function f0 of u0 can be found in D as

f _ Zo

fo = - M4'

Moreover, u0 + if0 is hyperholomorphic in D.

The following theorem is the Cauchy integral formula for a hyperholomor-phic function in P.

Theorem 3. Let Q be a bounded domain in P and f = u0 + if0 be hyperholomorphic on Q. Then, for every z = x0 + iz0 G Q, f can be

expressed as

f(z)

2n2

(C - z) j 1

K—T^cf(0

2n2

bQ

« - z) \C-z\4

d"cf (0,

where ( = y0 + i(0 and u = d(0 A d(0 — idy0 A d(0.

Proof. In order to conveniently find the formula of Theorem 3, we put , z) = (( — z) and , z) = — zj4. Let R be the distance between bQ and z. Let B = B(z, p) be the open ball of radius p with center z G Q, where 0 < p < R. Suppose Q(z, p) hyperholomorphic, by the Stokes theorem, we infer

Q — B. Since is

<t>(C,z)

Q(z,p)

№, z) M, z)

d"cf (0

fd{

4«, z)

n(z,p)

di-m ^ (c)l

bn(z,p)

H<, z) M, z)

"cf (0

'(0 -i-M»<f(0.

bn

4>((, z)

№, z)

Moreover, we obtain

H(z) M, z)

u,

'cf (0

B

B

H<, z)

d<- W {-tf} d^>

B

I f(C)dxo Ad Zo Ad Zo - / 0(C, z)ducf «) ^ .

\B B

Since f is hyperholomorphic in Q, we have 1

lim — f(()dx0 Adz0 Adz0 = -2n f(z),

p^o p

B

1

^ J №, z)ducf (0 = 0.

B

1

Hence,

/ - tzW™ = / - KB)(^

n bn

Thus, the function f(z) can be expressed as

™ = * I - ^^<0 - i / - tf «)•

n bn

□

Corollary 1. Let Q be a bounded domain in T and f = u0 + if0 be hyperholomorphic in a bounded domain Q C T. Then, for every z G Q, the function f can be expressed as

■f(z) = ^ J if-f -no =

bn

= 1 [ (yo - Xo) - i(f0 - zo)

(|yo -Xo\2 + |fo - Zo\2)2-Cl<f).

bn

Proof. Since f is a hyperholomorphic function on Q, we have

duc f (f) = 0

and the corollary follows by Theorem 3. □

Acknowledgment. The author was supported by the Dongguk University Research Fund 2018 and the National Research Foundation of Korea (NRF) (2017R1C1B5073944).

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DOI: https://doi .org/10.1017/S0305004100055638.

Received May 27, 2019. In revised form,, December 15, 2019. Accepted December 18, 2019. Published online January 10, 2020.

Department of Mathematics, Dongguk University, Gyeongju-si 38066, Republic of Korea E-mail: jeunkim@pusan.ac.kr