C2 extensions of analytic functions defined in the complex plane
John Ryan2001, Advances in Applied Clifford Algebras
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Abstract
Analytic continuations to C '~ of solutions to elliptic differential equations defined over domains in R ~ have been studied in a general context by Kiselman [5]. For the ease where the elliptic operator is the Laplacian a detailed analysis of these analytic continuations have been given by Lelong [6], Siciak [10] and Jarnicki [4]. Related results have also been described by Hayman [2]. In this paper we consider some of the properties of analytic continuations to C 2 of analytic functions defined over domains in the complex plane. The main result proved is: Theorem 1. Suppose that ~ is a domain lying in the complex plane. Then there exist two unbounded domains ~1 and ~2 lying in C 2 and containing ~, such that for each analytic function f (z) defined on ~ there are two holomorphic functions
Key takeaways
- This enables us to extend the classical Canchy integral formula to an integral formula in C 2, and to factorize the Cauchy kernal into two separate kernals.
- Notation: We shall denote the set of matrices
- Under matrix multiplication and addition the set R (2) is an algebra over the field of real numbers.
- Under matrix multiplication and addition the set C (2) is an algebra over the field of complex numbers.
- By the arguments given in [6] it may be observed that for each point zoI + On placing fs" = f, for each j, we obtain the result.
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