REMARKS IN QUATERNIONIC INTEGRATION:
CAUCHY'S THEOREMS AND FORMULAS
G.M. dos, Reis1, J. Marão1,2 1 Mathematics and Informatics Department
Maranhão State University
São Luís - MA, 65055-310 - BRAZIL 2B. Sc. Interdisciplinary Science and Technology
Federal University of Maranhão
São Luís - MA, 65085-580 - BRAZIL
The study of quaternions has been developed in the last decades, and some results show like a generalization of the Classical Complex Analysis Theory. The main purpose of this letter is of a presenting new results besides showing new current trends, as for instance, to deal with the integral theorem for Quaternionic Functions. Another proposal to be implemented in this work is the determination of a “closed" formula for the Cauchy Integral. A preliminary formula have been already determined [5].
You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.
References
[1]Baez, J., The octonions, Bull. Amer. Math. Soc., 39, No 2 (2001), 145-205.
[2]Borges, M. F.; Coelho, J. ; Marão, J. A., Geometrical logarithmic and
trigonometric hypercomplex functions of quaternionic type, Far East Journal of Mathematical Sciences, 50, No 1 (2011), 45-53.
[3]Borges, M.F.; Figueiredo, A. D.; Marão, J. A. Hypercomplex geometric
derivate from a Cauchy-like integral formula, International Journal of Pure
and Applied Mathematics, 68, No 1 (2011), 55-69.
[4]Borges, M. F.; Machado, J. M., New remarks on the differentiability of
hypercomplex functions, International Journal of Applied Mathematics, 8,
No 1 (2002), 85-101.
[5]Borges, M. F.; Marão, J. A.; Barreiro, R. C., A Cauchy-like theorem for
hypercomplex functions, Jornal of Geometry and Topology, 9, No 3 (2009),
263-271.
[6]Borges, M. F.; Marão, J. A.; Machado, J.M., Geometrical octonions II:
Hyper regularity and hyper periodicity of the exponential function, International Journal of Pure and Applied Math., 48, (2008), No 4 (2008),
495-500.
[7]Borges, M. F; Marão, J. A. P. F., The Laurent series for the quaternionic
case, International Journal of Pure and Applied Mathematics, 90, No 3
(2014), 281-285; doi: 10.12732/ijpam.v90i3.2.
[8]Buchmann, A., A Brief History of Quaternions and the Theory of Holomorphic Functions of Quaternionic Variables, Chapman University, p. 11.
[9]Conway, J. H., On Quaternions and Octonions: Their Geometry, Arithmetic and Symmetry, A.K.Peters, Ltd, Batiek, MA 20, (2003) p. 159.
[10] Fueter, R., Die Funktionentheorie der Differentialgleichungen ∆u = 0 und
∆∆u = 0 mit vier reelen Variablen. Comment. Math. Helv., 7, No 1 (1934),
307-330.
[11] Li, H. B., Some applications of Clifford algebra to geometries, In: Lecture
Notes on Artificial Inteligence, 1669 (1999), 156-179.
[12] Kodaira, K., Complex Analysis, Cambridge Studies in Advanced Mathematics, Cambridge University Press Cambrigde (2007), 406 pp.
[13] Sinegre, L., Quaternions and motion of a solid body about a fixed point
according to Hamilton, Rev.-Historie-Math., 1, No 1 (1995), 83-109.
[14] Marão, J. A.; Borges, M. F., A note on the hypercomplex Riemann-Cauchy
like relations for quaternions and Laplace equations, International Journal of Pure and Applied Mathematics, 90, No 4 (2014), 407-411; doi:
10.12732/ijpam.v90i4.2
[15] Marão, J. A.; Borges, M. F., Geometrical hypercomplex coupling between
electric and gravitacional fields, International Journal of Pure and Applied
Mathematics, 88, No 4 (2013), 475-482; doi: 10.12732/ijpam.v88i4.3.
[16] Marão, J. A.; Borges, M. F., Liouville’s theorem and power series for
quaternionic functions, International Journal of Pure and Applied Mathematic, 71, No 3 (2011), 383-389.
[17] Marão, J. A.; Borges, M. F., Geometrical coupling fields of a hypercomplex
type, International Journal of Pure and Applied Mathematics, 89, No 2
(2013), 215-224; doi: 10.12732/ijpam.v89i2.7.