Odessa State University. Petr Velikiy St. 2. Odessa, 270000. UKRAINE. ABSTRACT. A class of the hyperholomorphic quaternion-valued functions is introduced.
HELMHOLTZ OPERATOR FUNCTION TIlEORY
tHTII· A QUATERNIONIC WAVE NUMBER AND ASSOCIATED
V.V. KRAVCHENKO, M.V. SHAPIRO· Department of Hathematics Odessa State University Petr Velikiy St. 2 Odessa, 270000 UKRAINE ABSTRACT. A class of the hyperholomorphic quaternion-valued functions is introduced. It is related to the set of the metaharmonic functions (i.e., elements of the kernel of the Helmholtz operator with a quaternionic parameter (=quaternionic wave number)) just in the same way as the usual quaternionic analysis is related to the set of the harmonic functions. Theorems about the connection between these metaharmonic functions and hyperholomorphic functions were proved. There were constructed fundamental solutions of analogon of the Cauchy-Riemann operator and of the Helmholtz operator with a complex-quaternionic parameter. It is shown that there exists a bijection of a set of monochromatic electromagnetic fields in a homogenious medium, in absence of currents and charges, onto a set of pairs of functions belonging to the conjugate classes of a hyperholomorphy. The last allowed to obtain some assertions for such electro-magnetic fields as the consequences of a theory of the hyperholomorphic functions. A part of these assertions is known, but a part is possibly new. Introduction
It is known quite well that a theory of the holomorphic functions of one complex variable is closely connected with a theory of the harmonic functions, 1. e., of the solutions of the (two-dimensional) Laplace equation. So any essential progress in each of these theories results progress in the other one. There is no such connection for the n-dimensional complex analysis (n>1). and this is one of the possible explanations of the paradoxical differences existing between the cases n=1 and n>1. These paradoxes disappear i f we consider the so-called Clifford (and in particular - quaternionic) analysis (see [BOS] for the Clifford analysis and for example [GS]. [SV] for the quaternionic analysis) which allows to connect with a theory of the harmonic functions of an arbi trary number of variables the exact structural analogue of the one-dimensional complex analysis. But side by side with the Laplace operator the Helmholtz operator • The second author was a Visiting Professor at the Department of Hathematlcs, CINVESTAV del I.P.N .• Hexlco City. HEXICO, during the final part of the preparation of this work. 101 J. Lawrynowicz (ed.), Deformations ofMathematical Structures lI, 101-128. © 1994 Kluwer Academic Publishers.
V. V. KRAVCHENKO AND M. V. SHAPIRO
102
!J."A.: =!J.+"A.,
(O.ll
plays an important role and is often met in the applications. It can be considered as a most simple and natural enough generalization of the Laplace operator. That is why it seems as much natural to have a wish to connect with a theory of the metaharmonic functions (i.e., functions from a kernel of the Helmholtz operator) a corresponding generalization of the Clifford (and in particular - of the quaternionic) analysis. But there exist a few works realizing this wish, and they appeared not long ago. A quaternion function theory associated with the operator (0.1) for "A.e~+:=(O,+oo) has been developed in a series of the papers of Sprossig and his disciples. Corresponding results one can find in the book of Gurlebeck, Sprossig [GS). Some of them were developed for the functions with values in a Clifford algebra in the papers of Xu Zhenyuan [XZ), Sommen and Xu Zhenyuan [SX], Kravchenko and Shapiro [KS]. In the papers of Obolashvili lOb) and Huang Liede [HLl], [HL2] an analogous theory has been developed for "A.e~ . Some special questions have been touched up in the short communications of Shapiro [51), [52]. Underline that in the mentioned above papers of Obolashvili and Huang Liede not a technique of the quaternions was used but a technique of the matrices connected with the regular representations of the quaternions. Al though both theories ("A.e~ + and "A.e~_ ) were developed in the similar ways they were formally different and had no intersections. A wish to construct a theory joining them and also including an important for the applications case "A.eiC - led us to an understanding of a necessity to consider a situation when "A. is a complex quaternion in (0.1). Now let us describe shortly a content of the separate parts of the paper. Part 1 is quite technical and auxiliary. Mainly it is devoted to the properties of the quaternions (real and complex). These properties are well-known of course although a content of a section 1. 3 containing several descriptions of the zero-divisors in the non-commutative algebra H(iC) of the complex quaternions has a certain interest. In Part 2 the Helmholtz operator ~ with "A.eH(iC) is introduced on a set of H(iC)-valued functions of three real variables: (0.2) where !J. is the three-dimensional Laplace operator. The operators I/Jn « and I/Jn (~4
-
(1. 2) E
C,
usual scalar product. Note that
2 aa"l a I ~8
:
3 2 (1) 2 (2) 2 = L I a k I = Ia I + Ia I. k=a
Denote by Ei the set of the zero divisors of the ring H(C), 1. e. , 6:={aeH(C)!a"O; 3 boOO: ab=O}, Denote by GH(C) the group of invertible elements from H(C). If aEEi, then (see details in the next section) a- 1 :=a/(aa) is an inverse to a complex quaternion a. Therefore, obviously, GH(C)=H(C)\Ei, 3
1.3. PROPOSITION. Let O"a= L akik=aa+~ e H(C). The following assertions k=O are equivalent:
lOS
HELMHOLTZ OPERATOR WITH A QUATERNIONIC WAVE NUMBER
1. a e 6
2. aa=O 3. a~=~2, i.e., aoe C is one of the square roots of the complex number
~2 (among complex quaternions there are other square roots of this number, for example, ±~). 2 4. a =2a a o
5. a2=2~a 3
6.
L k=O
7.
2
a =0 k
lall) 1=la(2) I
and
O. It is well known (see, for example, [VIl) that for ~+~2 fundamental solutions are given by the formula
