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Article #47Exponential of an arbitrary mapping f: R^n -> R^n
From: Daniel A. Asimov
Date: Fri, 28 Aug 1992 18:12
Date: Fri, 28 Aug 1992 18:12
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1482 bytes
One way to generalize the exponential of a matrix to arbitrary continuous functions f: R^n -> R^n is as follows: exp(f): R^n -> R^n via exp(f)(x) = x + f(x) + f(f(x))/2! + f(f(f(x)))/3! +... It is easy to see that this does not necessarily converge for all x. Question: Has this version of exp(f) been defined and discussed previously in the literature? Can someone supply references and/or mention some of the elementary properties (if any) of this definition? For example, where it exists is exp(f) continuous? If f is differentiable, or even smooth, then what about exp(f) ? Are there elementary conditions under which exp(f) is invertible? For which g can one find a corresponding "log," i.e., an f such that exp(f) = g ? --Dan Asimov asimov@nas.nasa.gov Mail Stop T045-1 NASA Ames Research Center Moffett Field, CA 94035-1000 (415) 604-4799
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arc1-ca.pugleaf.net!sparky!uunet!zaphod.mps.ohio-state.edu!sdd.hp.com!ux1.cso.uiuc.edu!news.cso.uiuc.edu!usenetShow Full Headers
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Newsgroups: sci.math.research Path: arc1-ca.pugleaf.net!sparky!uunet!zaphod.mps.ohio-state.edu!sdd.hp.com!ux1.cso.uiuc.edu!news.cso.uiuc.edu!usenet From: asimov@nas.nasa.gov (Daniel A. Asimov) Subject: Exponential of an arbitrary mapping f: R^n -> R^n Message-ID: <1992Aug28.181224.14463@nas.nasa.gov> Sender: Daniel Grayson <dan@math.uiuc.edu> X-Submissions-To: sci-math-research@uiuc.edu Organization: NAS, NASA Ames Research Center, Moffett Field, CA X-Administrivia-To: sci-math-research-request@uiuc.edu Approved: Daniel Grayson <dan@math.uiuc.edu> Date: Fri, 28 Aug 1992 18:12:24 GMT Lines: 27