Abstract
A theory of quaternion-valued hyperholomorphic functions (h.h.f.) is being developed which is closely related to the Maxwell equations for monochromatic electromagnetic fields. The main integral formulas are established, and some boundary-value properties are studied.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brackx, F., Delanghe, R. and Sommen, F.:Clifford Analysis, Pitman, Boston, London, Melbourne (1982).
Gurlebeck, K. and Sprossig, W.:Quaternionic Analysis and Elliptic Boundary Value Problems, Math. Res.56, Akademie-Verlag, Berlin, 1989.
Huang Liede: The modified Heimholte equations and their boundary value problems, Selected papers of Tongji University (1989), pp. 37–54.
Huang Liede: The existence and uniqueness theorems of the linear and nonlinear Riemann-Hilbert problems for the generalized holomorphic vector of the second kind,Acta Math. Sci. (Engl. Edn) 10(2) (1990), 185–199.
Kravchenko, V. V: On the generalized holomorphic vectors, Abstracts of Republican Research Conference devoted to 200 birth anniversary of N. I. Lobachevskij, Odessa, Ukraine (1992), Part 1, p. 35 (Russian).
Kravchenko, V. G. and Shapiro, M. V.: Hypercomplex factorization of the multidimensional Hetmholte operator and some it's applications,Dokl. Sem. Inst. I. N. Vekua, Tbilisi 5(1) (1990), 106–109 (Russian).
Kravchenko, V. V. and Shapiro, M. V.: Heimholte operator with a quaternionic wave number and associated function theory, Dept. de Mat, CINVESTAV del IPN, Mexico, D. F. Reporte Interno No. 96, Abril 1992. Also in J. Lawrynowicz (ed.),Deformations of the Mathematical Structures, Kluwer Academic Publishers, Dordrecht, 1993, pp. 101–128.
Obolashvili, E. I.: Spatial generalized holomorphic vectors,Differentsial'nye Uravnenlya 11(1) (1975), 108–115 (Russian).
Shapiro, M. V: On the properties of a class of singular integral equations connected with the three-dimensional Heimholte equation, Abstracts of lectures at the 14 school on Operator Theory in Functional Spaces, Novgorod, U.S.S.R. (1989), p. 88 (Russian).
Shapiro, M. V.: On analog of the Toeplitz operators, inLinear Operators in Functional Spaces, Grozniy, U.S.S.R., 1989, pp. 178–179 (Russian).
Shapiro, M. V.: On analogues of the Riemann boundary value problem for a class of hyperholomor-phic functions, inIntegral Equations and Boundary Value Problems, World Scientific, Singapore (1991), pp. 184–188.
Shapiro, M. V. and Vasilevski, N. L.: Quaternionicψ-monogenic functions, singular operators with the quaternionic Cauchy kernel and the analogs of the Riemann problem, Deposited in Ukrn. NIINTI 06.02.87, No. 629-Uk-87, 68 pp. (Russian).
Shapiro, M. V. and Vasilevski, N. L.: Quaternionicψ-hyperholomorphic functions, singular operators with the quaternionic Cauchy kernel and analogs of the Riemann boundary value problem, Dep. de Mat., CINVESTAV del IPN, Mexico, D. F. Reporte Interno No. 102, Julio 1992, to appear as a two-part article inComplex Variables Theory Appl.
Sommen, F. and Xu Zhenyuan: Fundamental solutions for operators which are polynomials in the Dirac operator, in A. Micaliet al. (eds.),Cliford Algebras and Their Applications in Mathematical Physics, Kluwer Academic Publishers, Dordrecht, 1992, pp. 313–326.
Xu Zhenyuan: Boundary value problems and functions theory for spin-invariant differential oper-ators, PhD Thesis, Ghent University, Belgium, 1989–1990.
Xu Zhenyuan: A function theory for the operatorD -λ, Complex Variables Theory Appl.16 (1) (1991), 27–42.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kravchenko, V.V., Shapiro, M.V. Helmholtz operator with a quaternionic wave number and associated function theory. II. Integral representations. Acta Appl Math 32, 243–265 (1993). https://doi.org/10.1007/BF01082451
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01082451