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Suppose we are given the series: . Here where .
I wish to compute . How do I find a closed form expression for the above series so that I can compute the desired derivative. Thanks-- 122.160.195.98 ( talk) 06:28, 7 March 2009 (UTC)
I have been reading the proof of the splitting lemma in the wikipedia article of that name and was wondering if anyone could help me to understand the very first part
at the very start of the proof to show that 3.(direct sum) implies 1.(left split) they take t as the natural projection of (A×C) onto A, ie. mapping (x,y) in B to x in A now why does this satisfy the condition that tq is the identity on A. similarily to show that 3. implies 2. they take u as the natural injection of C into the direct sum of A and C (A×C) ie. mapping y in C to (1,y) how does this satisfy the condition that ru is the identity on C.
It would apear to me that they mean something else by the "natural" projection and injection but i cant see what this would be??? thanks for your help and im sorry if this is badly worded —Preceding unsigned comment added by Jc235 ( talk • contribs) 16:44, 7 March 2009 (UTC)
Hi there - I'm looking at the function - the standard example for an infinitely differentiable non-analytic function - and I'm wondering exactly how you prove that the function has zeros at x=0 for all derivatives. In general, is it invalid to differentiate the function as you would normally would (assuming nice behaviour) to get, in this example, , and then simply say it may (or may not) be differentiable at the 'nasty points' such as x=0? Or are there functions which have a derivative which is defined for well behaved points, but also has a well-defined value for non-differentiable points? Apologies if that made no sense. How would you progress then to show that all are equal to 0? Do you need induction?
On continuity, I'm using the definition effectively provided by Heine as in [1] - i.e. that a function is continuous at c if : but what happens if both the function and the limit of the function are undefined at a single point and continuity holds everywhere else? Because it's not like the function has a -different- limit to its value at c explicitly, they're just both undefined - does that still make it discontinuous? Incidentally, I'm thinking of a function like , where (if I'm not being stupid) both the function and its limit are undefined at 0, but wondering more about the general case - would that make it discontinuous?
Many thanks, Spamalert101 ( talk) 21:23, 7 March 2009 (UTC)Spamalert101
Ah that's wonderful, thanks so much for all the help. As a matter of interest, I see the wikipedia article on [2]'an infinitely differentiable non-analytic function' gives that the analytic extension to has an essential singularity at the origin for - is the same true with z^2 in lieu of z? Also, another related article [3] hints at 'via a sequence of piecewise definitions, constructing from a function g(x) with g(x)= and g is infinitely differentiable - I've managed to construct such a piecewise function using and stretches, translations etc of that, but how would you go about it with ? Is there a nice way to create such a function?[User:Spamalert101|Spamalert101]] ( talk) 17:21, 10 March 2009 (UTC)Spamalert
Oh, did I make a mistake then? I sincerely doubt my own knowledge over yours! When would it become non-differentiable? I had the function so it tends to infinity on either side of ±3/2 but to 1 at ±1 and 0 at ±2 by defining the functions piecewise as above so they had a derivative which tended to 0 as x tended to ±1 or 2, for example on - would it become non-differentiable at some point then?
Spamalert101 (
talk) 06:39, 11 March 2009 (UTC)Spamalert101
Mathematics desk | ||
---|---|---|
< March 6 | << Feb | March | Apr >> | March 8 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Suppose we are given the series: . Here where .
I wish to compute . How do I find a closed form expression for the above series so that I can compute the desired derivative. Thanks-- 122.160.195.98 ( talk) 06:28, 7 March 2009 (UTC)
I have been reading the proof of the splitting lemma in the wikipedia article of that name and was wondering if anyone could help me to understand the very first part
at the very start of the proof to show that 3.(direct sum) implies 1.(left split) they take t as the natural projection of (A×C) onto A, ie. mapping (x,y) in B to x in A now why does this satisfy the condition that tq is the identity on A. similarily to show that 3. implies 2. they take u as the natural injection of C into the direct sum of A and C (A×C) ie. mapping y in C to (1,y) how does this satisfy the condition that ru is the identity on C.
It would apear to me that they mean something else by the "natural" projection and injection but i cant see what this would be??? thanks for your help and im sorry if this is badly worded —Preceding unsigned comment added by Jc235 ( talk • contribs) 16:44, 7 March 2009 (UTC)
Hi there - I'm looking at the function - the standard example for an infinitely differentiable non-analytic function - and I'm wondering exactly how you prove that the function has zeros at x=0 for all derivatives. In general, is it invalid to differentiate the function as you would normally would (assuming nice behaviour) to get, in this example, , and then simply say it may (or may not) be differentiable at the 'nasty points' such as x=0? Or are there functions which have a derivative which is defined for well behaved points, but also has a well-defined value for non-differentiable points? Apologies if that made no sense. How would you progress then to show that all are equal to 0? Do you need induction?
On continuity, I'm using the definition effectively provided by Heine as in [1] - i.e. that a function is continuous at c if : but what happens if both the function and the limit of the function are undefined at a single point and continuity holds everywhere else? Because it's not like the function has a -different- limit to its value at c explicitly, they're just both undefined - does that still make it discontinuous? Incidentally, I'm thinking of a function like , where (if I'm not being stupid) both the function and its limit are undefined at 0, but wondering more about the general case - would that make it discontinuous?
Many thanks, Spamalert101 ( talk) 21:23, 7 March 2009 (UTC)Spamalert101
Ah that's wonderful, thanks so much for all the help. As a matter of interest, I see the wikipedia article on [2]'an infinitely differentiable non-analytic function' gives that the analytic extension to has an essential singularity at the origin for - is the same true with z^2 in lieu of z? Also, another related article [3] hints at 'via a sequence of piecewise definitions, constructing from a function g(x) with g(x)= and g is infinitely differentiable - I've managed to construct such a piecewise function using and stretches, translations etc of that, but how would you go about it with ? Is there a nice way to create such a function?[User:Spamalert101|Spamalert101]] ( talk) 17:21, 10 March 2009 (UTC)Spamalert
Oh, did I make a mistake then? I sincerely doubt my own knowledge over yours! When would it become non-differentiable? I had the function so it tends to infinity on either side of ±3/2 but to 1 at ±1 and 0 at ±2 by defining the functions piecewise as above so they had a derivative which tended to 0 as x tended to ±1 or 2, for example on - would it become non-differentiable at some point then?
Spamalert101 (
talk) 06:39, 11 March 2009 (UTC)Spamalert101