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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Analysis and applications of holomorphic functions in higher dimensions

Author: R. Z. Yeh
Journal: Trans. Amer. Math. Soc. 345 (1994), 151-177
MSC: Primary 26E05; Secondary 30G35, 31B05, 35C10, 35J99
Erratum: Trans. Amer. Math. Soc. 347 (1995), null.
MathSciNet review: 1260207
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Abstract: Holomorphic functions in $ {R^n}$ are defined to generalize those in $ {R^2}$. 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.

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  • [1] R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 2, Wiley-Interscience, New York, 1962.
  • [2] R. Delanghe, On regular analytic functions with values in Clifford algebra, Math. Ann. 185 (1970), 91-111. 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), 152-158. MR 0267114 (42:2016)
  • [5] G. N. Hile, Representations of solutions of a special class of first order systems, J. Differential Equations 25 (1977), 410-424. MR 0499218 (58:17136)
  • [6] G. N. Hile and P. Lounesto, Matrix representations of Clifford algebra, Linear Algebra Appl. 128 (1990), 51-63. MR 1049072 (91d:15056)
  • [7] 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)
  • [8] -, Solutions of polyharmonic Dirichlet problems derived from general solutions in the plane, J. Math. Anal. Appl. 154 (1991), 341-363. MR 1088636 (91k:35051)
  • [9] -, Hyperholomorphic functions and second order partial differential equations in $ {R^n}$, Trans. Amer. Math. Soc. 325 (1991), 287-318. MR 1015927 (91h:35083)

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Additional Information

Keywords: Hypercomplex numbers, holomorphic functions, Cauchy-Riemann 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

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