We consider a new class of quaternionic mappings associated with three-dimensional partial differential equations and propose a description of all mappings from this class by using four analytic functions of complex variable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers, Wiley , Chichester (1997).
V. V. Kravchenko and M. V. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, Addison Wesley Longman, Harlow (1996).
G. C. Moisil and N. Theodoresco, “Functions holomorphes dans l’espace,” Mathematica (Cluj), 5, 142–159 (1931).
A.V. Bitsadze, Boundary-Value Problems for Elliptic Equations of the Second Order [in Russian], Nauka, Moscow (1966).
R. Fueter, “Die Funktionentheorie der Differentialgleichungen ∆u = 0 und ∆∆u = 0 mit vier reellen Variablen,” Comment. Math. Helv., 7, 307–330 (1935).
A. Sudbery, “Quaternionic analysis,” Math. Proc. Cambridge Phil. Soc., 85, 199–225 (1979).
H. Leutwiler, “Modified quaternionic analysis in ℝ3 ,” Complex Variables Theory Appl., 20, 19–51 (1992).
T. Hempfling and H. Leutwiler, “Modified quaternionic analysis in ℝ4 ,” in: Clifford Algebras and Their Applications in Mathematical Physics, Kluwer, Dordrecht (1998), pp. 227–238.
S.-L. Eriksson-Bique, “A correspondence of hyperholomorphic and monogenic functions in ℝ4 ,” Clifford Analysis and its Applications. NATO Sci. Ser., 25, 71–80 (2001).
C. G. Cullen, “An integral theorem for analytic intrinsic functions on quaternions,” Duke Math. J., 32, 139–148 (1965).
G. Gentili and D. C. Struppa, “A new approach to Cullen-regular functions of a quaternionic variable,” Compt. Rend. Math., 342, No. 10, 741–744 (2006).
F. Colombo, S. Sabadini, and D. C. Struppa, “Noncommutative functional calculus: theory and applications of slice hyperholomorphic functions,” Progr. Math., 289 (2011).
G. Gentili, C. Stoppato, and D. Struppa, Regular Functions of a Quaternionic Variable, Springer, Berlin (2013).
C. Segre, “The real representations of complex elements and extension to bicomplex systems,” Math. Ann., 40, 413–467 (1892).
B. L. van der Waerden, Algebra [Russian translation], Nauka, Moscow (1976).
S. A. Plaksa and R. P. Pukhtaevich, “Constructive description of monogenic functions in an n-dimensional semisimple algebra,” An. ¸Sti. Univ. Ovidius Constanţa, 22, No. 1, 221–235 (2014).
S. A. Plaksa and V. S. Shpakovskii, “Constructive description of monogenic functions in a harmonic algebra of the third rank,” Ukr. Math. J., 62, No. 8, 1251–1266 (2011).
G. P. Tolstov, “On curvilinear and iterated integrals,” Tr. Mat. Inst. Akad. Nauk SSSR, 35, 3–101 (1950).
O. F. Herus, “On hyperholomorphic functions of the space variable,” Ukr. Math. J., 63, No. 4, 530–537 (2011).
O. F. Gerus and M. Shapiro, “On the boundary values of a quaternionic generalization of the Cauchy-type integral in ℝ2 for rectifiable curves,” J. Natur. Geom., 24, No. 1–2, 121–136 (2003).
B. Schneider, “Some properties of a Cauchy-type integral for the Moisil–Theodoresco system of partial differential equations,” Ukr. Math. J., 58, No. 1, 105–112 (2006).
C. Flaut and V. Shpakivskyi, “Holomorphic functions in generalized Cayley–Dickson algebras,” Adv. Appl. Clifford Alg., 25, No. 1, 95–112 (2015).
P. W. Ketchum, “Analytic functions of hypercomplex variables,” Trans. Amer. Math. Soc., 30, No. 4, 641–667 (1928).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 1, pp. 117–130, January, 2015.
Rights and permissions
About this article
Cite this article
Shpakivs’kyi, V.S., Kuz’menko, T.S. On One Class of Quaternionic Mappings. Ukr Math J 68, 127–143 (2016). https://doi.org/10.1007/s11253-016-1213-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-016-1213-6