On boundary values of the Cauchy-type integral for \alpha-hyperholomorphic functions in R²
by
Oleg F. Gerus
Zhitomir State Pedagogical University, Zhitomir, Ukraine
Coauthors: Michael V. Shapiro, Mexico City, Mexico (Instituto Politecnico Nacional)

We consider an analog of the Cauchy-type integral in the theory of \alpha-hyperholomorphic functions [1, 2] acting from Euclidean plane to the algebra of complex quaternions. For a domain with a closed rectifiable Jordan boundary we have established sufficient conditions for the Cauchy-type integral to be continuously extended onto the closure of the domain. We also have proved formulae for its boundary values which are similar to the N. A. Davydov formulae [3].

[1] Rocha-Chávez R., Shapiro M. V., Tovar L. M., On the Hilbert Operator for \alpha-Hyperholomorphic Function Theory in R². Complex variables. Theory and Applications. - 2000. - V. 43, no 1, p. 1-28.

[2] Kravchenko V. V., Shapiro M. V. Integral Representations for Spatial Models of Mathematical Physics. Pitman Research Notes in Mathematics Series. - 1996. - V. 351, 248 p.

[3] Davydov N. A. The continuity of the Cauchy-type integral in a closed region. Dokl. Akad. Nauk SSSR. - 1949. V. 64, no 6, p. 759-762. (In Russian).

Date received: June 20, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-29.