The Mathematical Institute, University of Oxford, Eprints Archive

Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations

Shaw, William T. (2004) Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations. SIAM Review, 46 (4). pp. 717-728.

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Abstract

Students of elementary complex analysis usually begin by seeing the derivation of the Cauchy--Riemann equations. A topic of interest to both the development of the theory and its applications is the reconstruction of a holomorphic function from its real part, or the extraction of the imaginary part from the real part, or vice versa. Usually this takes place by solving the partial differential system embodied by the Cauchy-Riemann equations. Here I show in general how this may be accomplished by purely algebraic means. Several examples are given, for functions with increasing levels of complexity. The development of these ideas within the Mathematica software system is also presented. This approach could easily serve as an alternative in the early development of complex variable theory.

Item Type:Article
Additional Information:First published in SIAM Review in Volume 46, Number 4, published by the Society for Industrial and Applied Mathematics
Uncontrolled Keywords:complex analysis, Cauchy-Riemann equations, computer algebra
Subjects:O - Z > Potential theory
H - N > Mathematics education
A - C > Computer science
D - G > Functions of a complex variable
O - Z > Several complex variables and analytic spaces
Research Groups:Oxford Centre for Industrial and Applied Mathematics
Mathematical and Computational Finance Group
ID Code:135
Deposited By:William Shaw
Deposited On:05 Nov 2004
Last Modified:20 Jul 2009 14:18

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