Shaw, William T. (2004) Recovering holomorphic functions from their real or imaginary parts without the CauchyRiemann equations. SIAM Review, 46 (4). pp. 717728.

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Abstract
Students of elementary complex analysis usually begin by seeing the derivation of the CauchyRiemann 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 CauchyRiemann 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, CauchyRiemann 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:  29 May 2015 18:17 
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