Abstract
In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13–29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144:13–29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green’s theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.
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Research supported by FAPESP-Brazil and CNPq-Brazil.
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Aragona, J., Fernandez, R., Juriaans, S.O. et al. Differential calculus and integration of generalized functions over membranes. Monatsh Math 166, 1–18 (2012). https://doi.org/10.1007/s00605-010-0275-z
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DOI: https://doi.org/10.1007/s00605-010-0275-z
Keywords
- Colombeau algebras
- Generalized functions
- Non-Archimedean differential calculus
- Membranes
- Generalized Cauchy formula