Monogenic functions taking values in generalized Clifford algebras

  • Doan Cong Dinh School Appl. Math. and Informatics, Hanoi Univ. Sci. and Technology, Vietnam
Keywords: Generalized Clifford algebras, ; Generalized Clifford analysis, Monogenic functions, Cauchy integral representation formula


UDC 512.579

Generalized Clifford algebras are constructed by various methods and have some applications in mathematics and physics.
In this paper we introduce a new type of generalized Clifford algebra such that all components of a monogenic function
are solutions of an elliptic partial differential equation. One of our aims is to cover more partial differential equations in
framework of Clifford analysis. We shall prove some Cauchy integral representation formulae for monogenic functions in
those cases.


F. Brackx, R. Delanghe, F. Sommen, Clifford analysis, Res. Notes Math., vol. 76, Pitman, Boston, MA (1982).

A. O. Morris, On a generalized Clifford algebra, Quart. J. Math., Ser.(2)18, № 1, 7 – 12 (1967); DOI:

W. Tutschke, An elementary approach to Clifford analysis, 1st ed., World Sci., 402 – 408 (1995).

W. Tutschke, C. J. Vanegas, Clifford algebras depending on parameters and their applications to partial differential equations, Some Topics on Value Distribution and Differentiability in Complex and $p$-Adic Analysis, Sci. Press (2008).

W. Tutschke, C. J. Vanegas, A boundary value problem for monogenic functions in parameter-depending Clifford algebras, Complex Var. and Elliptic Equat., 56, № 1-4, 113 – 118 (2011); DOI:

A. R. M. Granik, On a new basis for a generalized Clifford algebra and its application to quantum mechanics, Clifford Algebras with Numeric and Symbolic Computations, Birkhauser, Boston, MA, 101 – 110 (1996). DOI:

S. Barry, Generalized Clifford algebras and their representations, Clifford Algebras and their Applications in Mathematical Physics, Springer, 133 – 141 (2011).

R. Jagannathan, On generalized Clifford algebras and their physical applications, Springer, New York, 465 – 489 (2010), DOI:

E. Obolashvili, Higher order partial differential equations in Clifford analysis, Progr. Math. Phys., vol. 28, Birkhauser, ¨Basel (2002), DOI:

M. N. Rosculet, Functii monogene pe algebre comutative, Acad. Rep. Soc. Romania, Bucuresti (1975).

S. A. Plaksa, Commutative algebras associated with classic equations of mathematical physics, Adv. Appl. Anal., Springer, Basel, 177 – 223 (2012), DOI:

V. S. Shpakivskyi, Constructive description of monogenic functions in a finite-dimensional commutative associative algebra, Adv. Pure and Appl. Math., 7, № 1, 63 – 75 (2016), DOI:

V. S. Shpakivskyi, Curvilinear integral theorems for monogenic functions in commutative associative algebras, Adv. Appl. Clifford Algebras, 26, № 1, 417 – 434 (2016); DOI:

D. Alpay, A. Vajiac, M. B. Vajiac, Gleason’s problem associated to a real ternary algebra and applications, Adv. Appl. Clifford Algebras, 28, № 2 (2018); DOI:

M. N. Rosculet, O teorie a functiilor de o variabila hipercomplexa in spa ?tiul cu trei dimen- siuni, Stud. Cerc. Mat. 5, № 3-4, 361 – 401 (1954).

K. Gu¨rlebeck, U. Kähler, On a boundary value problem of the biharmonic equation, Math. Meth. Appl. Sci., 20, № 10, 867 – 883 (1997),<867::AID-MMA888>3.3.CO;2-N DOI:<867::AID-MMA888>3.0.CO;2-W

L. Sobrero, Nuovo metodo per lo studio dei problemi di elasticita, con applicazione al problema della piastra forata, Ric. Ingegn., 13, № 2, 255 – 264 (1934).

V. F. Kovalev, I. P. Melnichenko, Biharmonic functions on biharmonic plane, Dopov. Akad. Nauk Ukr. Ser A, № 8, 25 – 27 (1981).

S. V. Gryshchuk, S. A. Plaksa, Monogenic functions in the biharmonic boundary value problem, Math. Methods Appl. Sci., 38, № 11, 2939 – 2952 (2016), DOI:

How to Cite
Dinh, D. C. “Monogenic Functions Taking Values in Generalized Clifford Algebras”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 11, Nov. 2021, pp. 1483 -91, doi:10.37863/umzh.v73i11.1033.
Research articles