Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations
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...
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| Formato: | Journal article |
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2004
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| _version_ | 1826283939597123584 |
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| author | Shaw, W |
| author_facet | Shaw, W |
| author_sort | Shaw, W |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T01:06:22Z |
| format | Journal article |
| id | oxford-uuid:8b79f92b-73db-4a55-82f1-2c6eed22a1f6 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:06:22Z |
| publishDate | 2004 |
| record_format | dspace |
| spelling | oxford-uuid:8b79f92b-73db-4a55-82f1-2c6eed22a1f62022-03-26T22:38:17ZRecovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:8b79f92b-73db-4a55-82f1-2c6eed22a1f6Mathematical Institute - ePrints2004Shaw, WStudents 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. |
| spellingShingle | Shaw, W Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations |
| title | Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations |
| title_full | Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations |
| title_fullStr | Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations |
| title_full_unstemmed | Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations |
| title_short | Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations |
| title_sort | recovering holomorphic functions from their real or imaginary parts without the cauchy riemann equations |
| work_keys_str_mv | AT shaww recoveringholomorphicfunctionsfromtheirrealorimaginarypartswithoutthecauchyriemannequations |