Timeline for Finding Limits in several variables
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| May 17, 2013 at 9:45 | history | edited | J. M.'s missing motivation | CC BY-SA 3.0 | deleted 6 characters in body |
| Mar 24, 2013 at 0:05 | comment | added | Daniel Lichtblau | @whuber I suspect you are on target about that proof (my memories are a mere 20+ years dated, but I have even less of an excuse since it was my area at one time). I certainly agree that limiting behaviors in the multivariate setting are of importance, in a way that is not always evocative of the term "pathological". | |
| Mar 23, 2013 at 23:58 | comment | added | whuber | Thank you @Daniel. As I confessed earlier, I'm dredging up memories a third of a century old :-). My recollection is that the proof of the theorem relies on analysis of how limits are approached at the boundary within carefully restricted curves. (There may be multiple proofs by now; my source for this is Steven Krantz's SCV book.) Regardless of the application, the point was to indicate that these issues about limits are not mere matters of "pathology" but can be important in Analysis generally. | |
| Mar 23, 2013 at 23:53 | comment | added | Daniel Lichtblau | @whuber I believe Edge of the Wedge is for holomorphic continuation. What helps for existance of limits is restriction to pseudoconvex domains. Standard example: take z/(1-w). Then (0,1) is a point of indeterminacy. If we restrict approach paths to lie on or inside the sphere |z|^2 + |w|^2 = 1 (which is convex, therefore pseudoconvex) then it has a limit. | |
| Mar 21, 2013 at 22:30 | vote | accept | Dominic Michaelis | ||
| Mar 18, 2013 at 23:54 | comment | added | whuber | And you did a nice job with that, @Jens. But I think you might be making things look more complicated than they really are by characterizing this as an "uncountably infinite set," because in many cases it's closed and connected: it must be an interval, as the OP hints. These cases include all those where $f$ is continuous outside a closed subset of codimension $2$ or greater, such as a single point in $\mathbb{R}^2$. | |
| Mar 18, 2013 at 23:48 | comment | added | Jens | @whuber What you call a "set of possible limits" is in general going to be an uncountably infinite set. What I did in my answer is to parameterize that set by the hyperspherical angles, knowing that even that doesn't exhaust the set of possible limiting values because it assumes a radial path. You explored a different subset more appropriate for different kinds of functions. Whether those are the functions the OP is thinking about, I can't tell. | |
| Mar 18, 2013 at 17:42 | comment | added | whuber | @Jens, After re-reading the original version, I still have the impression that you and the OP may be thinking about this question differently. In the example, although a unique limit does not exist, there is nevertheless a set of possible limits that can be achieved; it is also the set of accumulation points of $f$ at $(0,0)$. The question asks--in a clear manner, in my view--how to compute that set or at least to establish that a unique limit does not exist. (My answer does not fully address this--it was offered more as an extended comment. Michael E2 probably comes closest.) | |
| Mar 18, 2013 at 17:34 | comment | added | Jens | @whuber I made that comment to get some feedback on what the intention of the question really is. As all the answers show, this is a somewhat open-ended (though interesting) discussion and it would be good to rein it in by knowing what the intended application within Mathematica is. | |
| Mar 18, 2013 at 17:10 | comment | added | whuber | @Jens "Intentionally set up" makes it sound like we're discussing a mere pathology. However, if I recall correctly (after 30 years of not thinking about it), the Edge of the Wedge Theorem of multivariate complex analysis concerns an important natural class of functions (i.e., holomorphic in some domain) which potentially have precisely these bad limiting properties at their boundaries, unless the limiting curves are carefully controlled: and therein lies the need to restrict to wedges. | |
| Mar 18, 2013 at 6:10 | answer | added | Jens | timeline score: 9 | |
| Mar 18, 2013 at 3:07 | answer | added | Michael E2 | timeline score: 12 | |
| Mar 17, 2013 at 22:42 | history | tweeted | twitter.com/#!/StackMma/status/313420225611698176 | ||
| Mar 17, 2013 at 22:08 | history | edited | m_goldberg | CC BY-SA 3.0 | added 2 characters in body |
| Mar 17, 2013 at 22:06 | answer | added | Daniel Lichtblau | timeline score: 11 | |
| Mar 17, 2013 at 21:12 | answer | added | whuber | timeline score: 44 | |
| Mar 17, 2013 at 20:56 | comment | added | Jens | With Limit, you're always restricted to a line in the larger space, and you can't make statements about the existence of the limit in the sense of the higher-dimensional space. For that, you have to show the independence of the result on the direction of the line. If you intentionally set up a function to have different limits along different lines, I don't see what else you can do with Mathematica. | |
| Mar 17, 2013 at 19:44 | history | edited | rm -rf♦ | edited tags | |
| Mar 17, 2013 at 19:36 | history | asked | Dominic Michaelis | CC BY-SA 3.0 |