The corresponding Cauchy - Riemann system for dual quaternion-valued functions Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 7(25), No. 1, 2018, pp. 59-69

DOI: 10.15393/j3.art.2018.4830

59

UDC 517.548, 517.552

J. E. KIM

THE CORRESPONDING CAUCHY- RIEMANN SYSTEM FOR DUAL QUATERNION-VALUED FUNCTIONS

Abstract. This paper provides differential operators in dual quaternions and represents the regularity of dual quaternion-valued functions using the dual Cauchy-Riemann system in dual quaternions. Also, we give the corresponding Cauchy theorem of the dual quaternion-valued function in Clifford analysis.

Key words: quaternion, dual number, Cauchy-Riemann system, Cauchy theorem, Clifford analysis

2010 Mathematical Subject Classification: 32A99, 32W50, 30G35, 11E88

1. Introduction. A dual quaternion can be represented in the form p + eq, where p and q are ordinary quaternions and e is the so-called dual unit, an element that commutes with every element of the algebra and is such that e2 = 0. Unlike quaternions, not every dual quaternion has an inverse. The set of dual quaternions is the following Clifford algebra:

Dq : = {Z = P! + ep2 | Pi,P2 € H},

where H is the set of quaternions which are combined by the basis elements 1, i, j, k. It has the product rule for i, j and k given by

i2 = j2 = k2 = ijk = —1

and

ij = —ji = k, jk = —kj = i, ki = —ik = j.

For two quaternions p = z1 + z2j and q = w1 + w2j, where z1 = x0 + ix!, z2 = x2 + ix3, w1 = y0 + iy1 and w2 = y2 + iy3, the rule of addition is:

p + q = (z1 + w1) + (z2 + w2) j

©Petrozavodsk State University, 2018

and multiplication is:

pq = (ziwi - Z2w) + (ziW2 + Z2W[)j-From the above rules, we give a norm for a quaternion as follows:

\p\2 := pp* = ziZi + Z2Z2

and the inverse of p as follows:

1 p *

P-1 = (P = 0)

Hamilton [7, 8] introduced quaternions in 1843, and in 1873 Clifford [4, 5] obtained a broad generalization of these numbers, which is now called the Clifford algebra [14]. At the turn of the 20th century, Kotelnikov [12] and Study [15] developed dual vectors and dual quaternions. In 1891 Study realized that this associative algebra was ideal for describing the group of motions of three-dimensional spaces. He further developed the idea in [15]. Kajiwara et al. [9] gave a basic estimate for inhomogeneous Cauchy - Riemann system and applied the theory to a closed densely defined operator in a Hilbert space. Kim et al. [10] obtained a corresponding inverse of functions and their properties and a regularity of functions on the form of multidual complex variables in Clifford analysis. Also, we [11] researched corresponding Cauchy - Riemann systems and properties of functions with values in special quaternions and split quaternions by using a regular function with values in dual split quaternions. Mathematicians has appeared null solutions of the Douglis operator, called hyperanalytic functions theory. Blaya et al. [1] presented the definition of conjugate hyperharmonic Douglis algebra-valued functions which proposed generalization of the classical conjugate harmonic functions in the Complex analysis case. They established an upper bound for the norm of a fractal Hilbert transform in the space of Holder analytic functions and characterized the monogenicity of functions and generalizations of certain two-sided monogenic extension results in the sense of Douglis operator (see [2, 3]).

This paper investigates the expression of differential operators in dual quaternions. The paper also represents a corresponding Cauchy theorem of dual quaternion-valued functions by using a dual Cauchy - Riemann system in dual quaternions.

2. Preliminaries. We consider the following set

Dq = {Z = pi + ep2 \ Pr e H, e2 =0, r = 1, 2},

which is isomorphic to H2 and R8. For Z = p1 + ep2 and W = q1 + eq2, we have the following rules of addition and multiplication on Dq:

Z + W = (pi + qi) + e(p2 + q2)

and

ZW = piqi + e(pi q2 + p2qi), respectively. We give a complex conjugate element of Dq as follows:

Z * = p* + ep*

and then, the norm of Z, denoted by \ Z\ , is described by \Z\2 = 1(ZZ * + Z * Z) = pip* + 2eS(pip*), where S(pip*) is the scalar part of pip* such that

S(pi p*) = Xoyo + Xiyi + X2V2 + X3 y3.

The elements of the set {ep \ p e H} are not invertible; for a dual quaternion Z = pi + ep2 outside this set, the inverse is given by

Z-i = ^ (pi =0), pip*

where

pi - e'pi ^p2p'i, called the dual conjugate of Z with ZZ^ = ZtZ = pip*.

Zt = p* - ™-i~ -*

3. Hyperholomorphic function in dual quaternions. Let Q be

a bounded open set in Dq. A function is given by

F :Q ^ Dq; F(Z) = fi(pi,p2) + e/2(pi,p2),

where

/i = 9i (Zi,Z2,Wi,W2) + jg2(Zi,Z2,Wi,W2)

and

f2 = hi (zi ,Z2 ,Wi,W2 ) + jh2 (zi ,Z2 ,Wi,W2 )

are quaternion-valued functions and gi = u0 + iui, g22 = u2 — iu3, hi = = v0 + ivi and h2 = v2 — iv3 are complex-valued functions with real-valued functions ur and vr (r = 0,1, 2, 3).

We consider the corresponding differential operators:

D ( d ■ d d k d \ + id .a . d k ® \ Vdx0 dxi j dx2 dx3) V dy0 dyi j dy2 0y3)

and

( d . d . d , d \ ( d . d . d

D = U—+ ^ + + + eU— + + + k^- .

Vdxo dxi dx2 0x3/ \dyo dyi dy2 oy3J

Using the properties of the basis elements 1, i, j, and k, we also write as follows:

( d . d \ + ( d .8 \

V dzi j dz2) V dwi j dw2)

D :=

and

' d + - + ( d + , d ' , dzi j dz2) V dwi j dw2.

where -J^, , 75= and (r = 1, 2) are the usual differential operators

OZr ' OWrl azr awr \ ' ' 1

in complex analysis. We let

d d d d d d

— -77- := ~--j-

and

Then

dpi dzi dz2 dp2 dwi dw2

d d d d d d = ^= + 7TT = ^= +

dpi dzi dz2 dp** dwi dw2

t^ d d n __ d d

D : = ---+ e^ and D* = —- + e-

dpi dp2 dp* dp2 Let Ci(Q, Dq) be the set of continuous functions from Q to Dq. The

corresponding Laplace operator on Ci (Q, Dq) is

A := 1(DD* + D* D) = -(D*D + DD*) 22

5 d _ ( d d

+e2S(;

dpi dp* Vdpi dp2

d d + 2S( d d dpi dpi \dp* dp2

Consider the following calculations with the differential operators defined above:

Remark 1. On a bounded open set Q in H2, we have

DF = / + e( / + / )■

dpi Vdpi dp2J

D* F = / + e( f + f ) = dp* Vdp* dp2/

= (dgi _ + j(+ d^i)) +

V dZi dZ2 V dZi dZ2J J

+e{( dhi _ dh2 + dgi _ ) +

IV dZi dZ2 dWi dW^J

+j(dhi + dhi + + dgi

V dZi dZ2 dWi dW2

Definition 1. Let Q be a bounded open set in H2. A function F is said to be left (resp. right) regular on Q if the components /i and /2 of F are both continuously differentiable functions and F satisfies the following equation

D*F = 0 (resp. FD* = 0). (1)

From Remark 1, the left equation of (1) is equivalent to

d/i =0 and f d/i

dp* dp* dp*

In detail, it is also equivalent to the following system:

dgi dg2

dz1 dz2 '

dg2 dg1

dz1 dz2 '

dh1 + dg1 = dw1 dh2 + dz2 dg2

dz1 dw"'

dh2 + dg2 = dw1 dh1 dg1

dz1 dz2 dw2

(2)

Clearly, the properties and progresses of the theory of left regular functions are equivalent to that of right regular functions. For the sake of the convenience we consider left regular functions, which are called just regular.

Let U be a compact oriented C^-manifold with boundary dU contained in a domain Q of H2. For r (0 < r < 3), let

dxT := dx0 A • • • A dxr-1 A dxr+1 A • • • A dx3 A dy0 A dy1 A dy2 A dy3,

dyT := dx0 A dx1 A dx2 A dx3 A dy0 A • • • A dyr-1 A dyr+1 A • • • A dy3 and

da = da(p1 ,p2) := da1 + eda2,

where and

da1 : = dx0 — idx1 + jdx2 — kdx3

a2 := dyo — idy1 + jdy2 — kdy3. Hence, for F = F(p1 ,p2) = g(z,w) + eh(z,w) in C^(Q,Dq), where

g(z, w) = u0 + u1i + u2 j + u3k

and

with

h(z, w) = vo + V1 i + V2j + V3k (z,w) = (xo ,x1,x2,x3,yo ,y1 ,y2,y3),

the corresponding dual quaternion-valued 7-form is

u := da(pi,p2)F (pi,p2) = (doq + eda2)(g(z,w) + eh(z,w)) = = daig(z, w) + edaih(z, w) + eda2g(z, w)

and the exterior derivative is

dF ^ dF ^ dF ^

du := —— dx0 A dx0 — i ——dxi A dxi + j ——dx2 A dx2 — dx0 dx± dx2

i dF ( do , dg

— k——dx3 A dx3 + e ——dy0 A dy0 — i——dy\ A dy\ + dx3 \dy0 y y dyi

dg dg \

—dy2 A dy2 — k—dy3 A dys) = dy2 dys J

= (D*F)dV(pi ,p2),

where dV(pl,p2) = dx0 A dxl A dx2 A dx3 A dy0 A dyl A dy2 A dy3.

Remark 2. For each Z G dU, let n = n(pl,p2) = N + eM, where N = n0 + nli + n2j + n3k and M = m0 + mli + m2j + m3k be the outward unit normal to dU at Z. Then we have

da(pi,p2) = n(pi, p2) dS (pi, p2)

and

u = n(pi,p2)F(pi,p2)dS(pi,p2) = Ng(z, w) + e(Nh(z, w) + Mg(z, w)),

where dS(pi,p2) is the scalar element of a surface area on dU. Following [13], let

Z * - W *

*(Z — W ) = Vfz—W'

where

Z * = Zo — Zii — Z2j — Zsk and W * = no — Vi i — V2 j — ns k

with Zr = xr + eyr and nr = K + e/ir (r = 0,1, 2, 3 and xr, yr, Xr, fxr G R), and v is the surface area of the unit sphere in H2, it is called the Cauchy kernel on Dq.

Remark 3. For Z = Z0 + Ci' + Z2 j + Z3k, the norm of Z is

zz * = z *z = z2 ■

r=0

Remark 4. Let $(Z — W) = ZZ-W|8, where v is the surface of the unit sphere in R4, is so-called the Cauchy kernel on Dg. The function $(Z — W) is left and right regular in fi. Indeed,

Z * - W *

Z* W*

D*t(Z — W) = Di VAZ—W18)

X

v\Z — W \8 ( d d d , d \ (do + 'del + jdZ2 + %3) x

(Co — no) — (Ci — ni)i — (z2 — n2)j — (z3 — n3)k )

v (E 3=o(Zr — nr )2 )4

-

where

and

Z * — W * v\Z — W\8

d d

^)D* = 0,

d(r dxr

+ e

dVr

d

Zr = 1, (r = 0,1, 2, 3).

dCr

Lemma 1. [6] Let fi be a bounded open set in Dg. Let u and v be smooth scalar-valued functions on fi. Then for all r and t (0 < r,t < 3),

u

dv + du dxr dxr

U

v^jdV = J uv nrdS

dU

and

' / dv du \ f

(—--+ —— v)dV = uv mtdS,

\dvt dyt / I

U dU

where nr and mt are defined in Remark 2.

Lemma 2. [13] Let fi be a bounded open set in Dg. Let F and ^ be smooth dual quaternion-valued functions on fi, where F = g(z,w) +

+eh(z, w) and ^ = w) + £^(z, w) with $(z, w) = a0 + a1i + a2j + a3k and p(z, w) = + i + fl2j + fl3k, where ar and flr (r = 0,1, 2, 3j are real-valued functions. Then we have

J{F(D» + (FD*)^}dV = J F(n^)dS,

U dU

where n is defined in Remark 2.

Theorem 1. Let ft be a bounded open set in Dg and U be a subset of ft. For Z e ft, if D*F = 0 and W e int(U), then we have

F(Z) = J $(Z - W){da(pi,p2)F(p1}p2)},

dU

where int(U) is the interior of U and $(Z — W) is a regular function expressed in Remark 4. Also, if D*F = 0 and W e ft \ U, then the above integral is zero.

Proof. For W e int(U) and given e > 0, let UB be U except the open ball of radius e centered at W. Then from Remark 4, the function $(Z — W) is regular in Ub and from Lemma 2, we have

J {$(Z — W )(D* F) + ($(Z — W)D* )F }dV =

ub

= J{$(Z — W)(D*F)dV = J §(Z — W)(n F)dS =

ub ub

= J $(Z — W)(n F)dS — J $(Z — W)(n F)dS,

dU Be

where Be is the sphere of radius e centered at W. Since

Z* W* Z W F

$(Z - W)(n F) = —----Z-— =---,

^ yv>Vir> v\Z — W|8 v\Z — W\ v\Z — W\7'

we have

/$(Z — w )(n f )dS = f VZW dS

Be ub

Since W E int(U) and the integral is taken over Be, as e ^ 0, we get:

f F

v\Z - W|'

Be

:dS-F

< e.

Also, for W E ü \ U, we have

J $(Z - W)(n F)dS = 0.

Ub

Therefore, we obtain

J $(Z - W)(D*F)dV = J $(Z - W)(n F)dS =

Ub Ub

= J $(Z - W )(n F )dS -J $(Z - W )(n F )dS =

9U Be

= J $(Z - W)da(pi,p2)F(Pl,P2) - F(Pl,P2).

dU

From the hypothesis D*F = 0, we obtain

0 = y$(Z - W)da(pi,p2)F(pi,P2) - F(pi,p2). □

dU

Acknowledgment. This work was supported by the Dongguk University

Research Fund of 2017.

References

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Received January 9, 2018.

In revised form, April 9, 2018.

Accepted April 11, 2018.

Published online June 19, 2018.

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