[00:19:30] <Millennial> "all smooth functions are analytic"
[00:19:32] <Millennial> - physicist
[00:19:40] <Polymatheia> "I wish"
[00:19:50] <Millennial> you know
[00:20:04] <Millennial> some of these people
[00:20:04] <Polymatheia> "e^(-1/x^2) for x != 0, f(0) = 0 is a counterexample"
[00:20:09] <Polymatheia> - Better physicist
[00:20:10] <Polymatheia> ;-)
[00:20:12] <Millennial> well, that's one counterexample
[00:20:16] <Millennial> it's important to know
[00:20:17] <Polymatheia> One is enough.
[00:20:22] <Polymatheia> To destroy that statement.
[00:20:25] <Millennial> exactly how you can ensure convergence
[00:20:25] <Millennial> yes
[00:20:33] <Millennial> but "destroying statements" is not what mathematicians like to do
[00:20:42] <Millennial> so, if this is not true
[00:20:45] <Millennial> how false is it, exactly?
[00:20:54] <Polymatheia> Well, I mean figuratively. I thought you guys liked counterxamples. :D
[00:20:55] <Millennial> how much regularity do i need on f to ensure that this is true
[00:20:59] <Millennial> yes, we do
[00:21:03] <Millennial> we also like explanations tho
[00:21:14] <Millennial> explanations which allow us to come up with weird counterexamples are very nice
[00:21:46] <Polymatheia> What else did you need, other than infinitely differentiable?
[00:21:48] <Polymatheia> :-D
[00:22:44] <Millennial> lol
[00:22:49] <Millennial> to be handwavy
[00:23:01] <Millennial> a smooth function is analytic if its derivatives do not get increasingly stupid
[00:23:08] <Polymatheia> LOL
[00:23:17] <Millennial> there's a very precise sense in which this is true
[00:23:34] <Millennial> you can plot the first few derivatives of exp(-1/x^2) if you'd like
[00:23:36] <Millennial> you will see what i mean
[00:23:42] <Polymatheia> I see.
[00:23:44] <Millennial> close to 0
[00:23:46] <Millennial> the derivatives blow up
[00:23:58] <Millennial> they are too big, you can't control them
[00:23:59] <Polymatheia> But _AT_ 0 they happen to exist.
[00:24:08] <Millennial> all of them exist individually, yes
[00:24:09] <Polymatheia> That is what's freaky.
[00:24:21] <Polymatheia> Nice.
[00:24:28] <Polymatheia> I may troll people in ##physics with this.
[00:24:40] <Millennial> the thing is
[00:24:45] <Millennial> the derivatives are always like
[00:24:45] <Millennial> say
[00:24:51] <Millennial> a sum of stuff of the form
[00:24:57] <Millennial> 1/x^k * exp(-1/x^2)
[00:25:03] <Polymatheia> Aha.
[00:25:16] <Millennial> and no matter how big k is
[00:25:21] <Millennial> exp(-1/x^2) decays too quickly as you go x -> 0
[00:25:29] <Millennial> so at 0, you always get 0
[00:25:35] <Millennial> however
[00:25:46] <Millennial> there is no neighborhood of 0 in which you can consistently bound the derivatives reasonably
[00:25:57] <Polymatheia> Nice.
[00:25:57] <Millennial> that is what causes exp(-1/x^2) to be non-analytic
[00:26:32] <Millennial> the precise meaning is given by the error terms of taylor's theorem
[00:26:40] <Millennial> usually, with the kth taylor polynomial
[00:27:06] <Millennial> you have an error of f^(k+1)(c)/(k+1)! (x - a)^(k+1)
[00:27:10] <Millennial> the taylor series is centered at a
[00:27:13] <Millennial> you're evaluating it at x
[00:27:21] <Polymatheia> I see.
[00:27:23] <Millennial> c is something between x and a, you can't know what it is
[00:27:31] <Millennial> but you want this to go to 0 as k -> infty, basically
[00:27:50] <Millennial> if you can, for instance, always ensure that f^(k+1)(c) remains bounded between x and a
[00:27:55] <Millennial> then this will go to 0
[00:27:59] <Millennial> because the factorial grows too qucikly
[00:28:01] <Millennial> quickly*
[00:28:15] <Millennial> examples of this include f(x) = exp(x), which has all higher derivatives equal to itself
[00:28:24] <Millennial> so you can just pick the bound on [a, x] to be exp(x)
[00:28:28] <Millennial> it's finite
[00:28:42] <Millennial> and sin(x) and cos(x), whose derivatives cycle but are always bounded between -1 and 1
[00:28:58] <Millennial> these are the most famous examples of analytic functions, i suppose
[00:31:15] <Polymatheia> I see.
[00:33:05] <Millennial> this also explains why, for instance, the taylor polynomials of sin(x) and cos(x) visually "settle" gradually, first around 0
[00:33:08] <Millennial> and as you sum more terms
[00:33:15] <Millennial> they are very accurate in a certain interval
[00:33:17] <Millennial> then they go off
[00:33:41] <Millennial> this is because if you pick (x-a) to be too large for some fixed value of k
[00:33:46] <Millennial> the remainder term can be very arge
[00:33:47] <Millennial> large*
[00:34:26] <Polymatheia> Wow.
[00:34:26] <Polymatheia> Damn.
[00:35:12] <Millennial> so, you see
[00:35:30] <Millennial> when you say "well, that claim is false, so that's it", you're not being very productive
[00:35:34] <Millennial> in general
[00:35:37] <Millennial> this is a very bad thing to do
[00:35:47] <Polymatheia> lol
[00:35:50] <Millennial> especially with such an important and interesting question
[00:36:10] <Millennial> and i believe this is more relevant to a physicist than, say, algebraic number theory
[00:36:10] <Millennial> so
[00:36:21] <Polymatheia> Depends.
[00:36:25] <Millennial> haha
[00:36:26] <Millennial> why
[00:36:40] <Polymatheia> I guess a theoretical physicist may want more details.
[00:36:58] <Millennial> may want more details in exactly how this works?
[00:37:06] <Millennial> if that happened, i would be glad
[00:37:06] <Millennial> lol
[00:37:26] <Polymatheia> lol
[00:37:39] <Polymatheia> I mean in physical theories, why integrals break down, or diverge, etc.
[00:37:56] <Millennial> yes, possible
[00:38:31] <Millennial> so you can now see
[00:38:35] <Millennial> how people came up with this stupid example
[00:38:57] <Millennial> they wanted the derivatives to get increasingly stupid, so you have to pick up a factor of at least 1/x^2 close to 0 with each iteration of the derivative
[00:39:03] <Millennial> in order to cancel the (x-a)^(k+1) term
[00:39:19] <Millennial> the simplest way to do this is to introduce exp(1/x)
[00:39:34] <Millennial> but this is not actually continuous at x = 0, so that's sort of a problem
[00:39:55] <Millennial> so we pick up a factor of 1/x^3 with each iteration, which is even better, and we use exp(-1/x^2)
[00:39:55] <Polymatheia> Haha I see.
[00:40:37] <Millennial> this is how counterexamples are born
[00:40:38] <Millennial> lol
[00:40:44] <Millennial> they aren't formulated in a vacuum, most of the time
[00:40:53] <Millennial> they come from good theories
[00:41:05] <Millennial> in a way, they are an attestment to the power of the theory for being able to predict them
[00:42:16] <Polymatheia> I see.