Hyperholomorphic functions and second order partial differential equations in

Author:
R. Z. Yeh

Journal:
Trans. Amer. Math. Soc. **325** (1991), 287-318

MSC:
Primary 35E20; Secondary 35A25

DOI:
https://doi.org/10.1090/S0002-9947-1991-1015927-9

MathSciNet review:
1015927

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Hyperholomorphic functions in with are introduced, extending the hitherto considered hyperholomorphic functions in . A Taylor formula is derived, and with it a unique representation theorem is proved for hyperholomorphic functions that are real analytic at the origin. Hyperanalyticity is seen to be generally a consequence of hyperholomorphy and real analyticity combined. Results for hyperholomorphic functions are applied to gradients of solutions of second order homogeneous partial differential equations with constant coefficients. Polynomial solutions of such a second order equation are obtained by a matrix algorithm. These polynomials are modified and combined to form polynomial bases for real analytic solutions. It is calculated that in such a basis there are homogeneous polynomials of degree .

**[1]**R. Delanghe,*On regular-analytic functions with values in a Clifford algebra*, Math. Ann.**185**(1970), 91-111. MR**0265618 (42:527)****[2]**R. P. Gilbert and G. N. Hile,*Generalized hypercomplex function theory*, Trans. Amer. Math. Soc.**195**(1974), 1-29. MR**0450577 (56:8870)****[3]**G. N. Hile,*Representations of solutions of a special class of first order systems*, J. Differential Equation**25**(1977), 410-424. MR**0499218 (58:17136)****[4]**-,*Elliptic systems in the plane with first order terms and constant coefficients*, Comm. Partial Differential Equations**3**(1978), 949-977. MR**507124 (80d:35050)****[5]**M. H. Protter and G. N. Hile,*Maximum principles for a class of first order elliptic systems*, J. Differential Equations**24**(1977), 136-151. MR**0460878 (57:869)****[6]**R. Z. Yeh,*Hyperholomorphic functions and higher order partial differential equations in the plane*, Pacific J. Math.**142**(1990), 379-399. MR**1042052 (91a:30046)****[7]**-,*Solutions of polyharmonic Dirichlet problems derived from general solutions in the plane*, J. Math. Anal. Appl.**154**(1991), 341-363. MR**1088636 (91k:35051)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35E20,
35A25

Retrieve articles in all journals with MSC: 35E20, 35A25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-1015927-9

Keywords:
Hyperholomorphic functions,
symmetric powers,
Taylor formula,
Leibniz formula,
hyperanalyticity,
real analytic solutions,
polynomial bases

Article copyright:
© Copyright 1991
American Mathematical Society