NDLI: Recovering Holomorphic Functions from Their Real or Imaginary Parts without the Cauchy--Riemann Equations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Shaw, William T.
Copyright Year	2004
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.
Starting Page	717
Ending Page	728
Page Count	12
File Format	PDF
ISSN	00361445
DOI	10.1137/S0036144503432151
e-ISSN	10957200
Journal	SIAM Review (SIREAD)
Issue Number	4
Volume Number	46
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-04
Access Restriction	Subscribed
Subject Keyword	Explicit machine computation and programs Cauchy--Riemann equations Integration, integrals of Cauchy type, integral representations of analytic functions Functional equations in the complex domain, iteration and composition of analytic functions complex analysis computer algebra Instructional exposition
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Computational Mathematics

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