Analysis and applications of holomorphic functions in higher dimensions
Author:
R. Z. Yeh
Journal:
Trans. Amer. Math. Soc. 345 (1994), 151177
MSC:
Primary 26E05; Secondary 30G35, 31B05, 35C10, 35J99
Erratum:
Trans. Amer. Math. Soc. 347 (1995), null.
MathSciNet review:
1260207
Fulltext PDF Free Access
Abstract 
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Abstract: Holomorphic functions in are defined to generalize those in . A Taylor formula and a Cauchy integral formula are presented. An application of the Taylor formula to the kernel of the Cauchy integral formula results in Taylor series expansions of holomorphic functions. Real harmonic functions are expanded in series of homogeneous harmonic polynomials.
 [1]
R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 2, WileyInterscience, New York, 1962.
 [2]
Richard
Delanghe, On regularanalytic functions with values in a Clifford
algebra, Math. Ann. 185 (1970), 91–111. MR 0265618
(42 #527)
 [3]
William
Feller, An introduction to probability theory and its applications.
Vol. I, Third edition, John Wiley & Sons Inc., New York, 1968. MR 0228020
(37 #3604)
 [4]
W.
K. Hayman, Power series expansions for harmonic functions,
Bull. London Math. Soc. 2 (1970), 152–158. MR 0267114
(42 #2016)
 [5]
Gerald
N. Hile, Representations of solutions of a special class of first
order systems, J. Differential Equations 25 (1977),
no. 3, 410–424. MR 0499218
(58 #17136)
 [6]
Gerald
N. Hile and Pertti
Lounesto, Matrix representations of Clifford algebras, Linear
Algebra Appl. 128 (1990), 51–63. MR 1049072
(91d:15056), http://dx.doi.org/10.1016/00243795(90)90282H
 [7]
R.
Z. Yeh, Hyperholomorphic functions and higher order partial
differential equations in the plane, Pacific J. Math.
142 (1990), no. 2, 379–399. MR 1042052
(91a:30046)
 [8]
R.
Z. Yeh, Solutions of polyharmonic Dirichlet problems derived from
general solutions in the plane, J. Math. Anal. Appl.
154 (1991), no. 2, 341–363. MR 1088636
(91k:35051), http://dx.doi.org/10.1016/0022247X(91)90042X
 [9]
R.
Z. Yeh, Hyperholomorphic functions and second
order partial differential equations in 𝑅ⁿ, Trans. Amer. Math. Soc. 325 (1991), no. 1, 287–318. MR 1015927
(91h:35083), http://dx.doi.org/10.1090/S00029947199110159279
 [1]
 R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 2, WileyInterscience, New York, 1962.
 [2]
 R. Delanghe, On regular analytic functions with values in Clifford algebra, Math. Ann. 185 (1970), 91111. MR 0265618 (42:527)
 [3]
 W. Feller, An introduction to probability theory and its applications, Vol. 1, Wiley, New York, 1968. MR 0228020 (37:3604)
 [4]
 W. K. Hayman, Power series expansions for harmonic functions, Bull. London Math. Soc. 2 (1970), 152158. MR 0267114 (42:2016)
 [5]
 G. N. Hile, Representations of solutions of a special class of first order systems, J. Differential Equations 25 (1977), 410424. MR 0499218 (58:17136)
 [6]
 G. N. Hile and P. Lounesto, Matrix representations of Clifford algebra, Linear Algebra Appl. 128 (1990), 5163. MR 1049072 (91d:15056)
 [7]
 R. Z. Yeh, Hyperholomorphic functions and higher order partial differential equations in the plane, Pacific J. Math. 142 (1990), 379399. MR 1042052 (91a:30046)
 [8]
 , Solutions of polyharmonic Dirichlet problems derived from general solutions in the plane, J. Math. Anal. Appl. 154 (1991), 341363. MR 1088636 (91k:35051)
 [9]
 , Hyperholomorphic functions and second order partial differential equations in , Trans. Amer. Math. Soc. 325 (1991), 287318. MR 1015927 (91h:35083)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412602072
PII:
S 00029947(1994)12602072
Keywords:
Hypercomplex numbers,
holomorphic functions,
CauchyRiemann equations,
symmetric powers,
Stieltjes line integrals,
Taylor formula,
Cauchy integral formula,
divergence theorem,
Leibniz's rule,
power series,
harmonic functions,
Poisson equation,
Laplace equation,
polynomial expansions
Article copyright:
© Copyright 1994 American Mathematical Society
