Differential calculus and integration of generalized functions over membranes |
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Authors: | Jorge Aragona Roseli Fernandez Stanley O Juriaans Michael Oberguggenberger |
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Institution: | 1. Instituto de Matem??tica e Estat??stica, Universidade de S?o Paulo, CP 66281, S?o Paulo, CEP 05314-970, Brazil 2. Arbeitsbereich f??r Technische Mathematik, Universit?t Innsbruck, 6020, Innsbruck, Austria
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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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