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Re: Extracurricular Integration Marathon Well a power series in e^(ix) IS a Fourier series. The difficulty is in getting a mode of convergence that allows us to commute the limit with the conditionally convergent integral. I did think of an idea that probably deals with it though, and will...

Re: First Year Uni Calculus Marathon And more generally, that if f: D->R, we can make sense of lim_{x->p} f(x) for any p in the closure of D, not just the interior of D. Of course these limits might not always exist.

Re: First Year Uni Calculus Marathon Nomenclature for claims is getting confusing :p. It is indeed true that the limit of f(g(t)) as t->0 is well defined.

Re: First Year Uni Calculus Marathon "And yes, I think you can just use this change of variables, in that I claim the following: Suppose f is a function defined on a set D in R^n, take n=1 if you like. Suppose E is another subset of R^n and g:E->R^n is another function. Then fog is a...

Re: First Year Uni Calculus Marathon Yes. And the reason things are okay here is that the point x=0 is not in the domain of the function f(x)=((1+x)^2-1)/x, so the countable family t near 0 for which g(t)=0 are domain holes (*) of (fog) rather than points at which we could have exceptional...

Re: First Year Uni Calculus Marathon Oh my bad, you are totally right. Take g to be the constant function g(t)=b. Then (fog) is identically f(b), which need not be L. This is also a counterexample to my other claim, but my original point about the well defined-ness of the limit of that...

Re: First Year Uni Calculus Marathon Are the counterexamples you mention counterexamples to my claim above? Or to something else? (Keeping in mind the earlier stated definition of the limit that makes sense at points that aren't necessarily interior to the domain.)

Re: First Year Uni Calculus Marathon Yes, D is the set of nonzero t such that sin(1/t) is nonzero. (The function is clearly defined here.) This is just the reals apart from a countable set of points t=1/k*pi, with k ranging over the nonzero integers. You are correct that f is not defined in...

Re: First Year Uni Calculus Marathon If this wasn't the case, we wouldn't be able to study concepts like continuity on objects like fractals, where no point is interior to the set. I admit that some courses in single variable calculus might insist that f be defined in a punctured neighbourhood...

Re: First Year Uni Calculus Marathon Sorry about the delay, I was skyping gf. I actually deleted my previous post about bait because I changed my mind. (My initial post was rather rushed and the edit was made shortly after.) I do believe the limit is the same. The issue of there being a...

Re: First Year Uni Calculus Marathon Another decent exercise with lots of different approaches is the inverse function theorem (for the purpose of this thread we will keep things single-variable, which undoubtedly opens up even more possible avenues of attack). Suppose f:R->R is C^k (has...

Re: First Year Uni Calculus Marathon Doesn't my proof of existence show that f'(a)=L, the assumed limit of f'(x) as x->a? We showed the function was differentiable at a by finding it's derivative at a, which was L.

Re: First Year Uni Calculus Marathon Bingo. Haha I figured you would realise that the previous exponent wasn't good enough when you tried to work out the details.

Re: First Year Uni Calculus Marathon You should never go just by memory! :) Do you believe that this is a counterexample? And if so, why?

Re: First Year Uni Calculus Marathon Yep. MVT implies (f(x)-f(a))/(x-a)=f'(c(x)) for some c(x) strictly between a and x. As x->a, c(x)->a, which implies that the RHS converges to a limit by the question's assumptions. So the LHS converges to a limit as x->a, which is precisely the definition...

Re: MX2 2016 Integration Marathon Such as http://www.ams.org/journals/bull/1948-54-06/S0002-9904-1948-09034-6/S0002-9904-1948-09034-6.pdf

Re: MX2 2016 Integration Marathon For sure if you just want sufficient. Prepping for a meeting right now so will work one out later, but basically you just want to use a stronger norm than the uniform norm (to avoid wiggly approximations that are uniformly close but have massive arc length)...

Re: MX2 2016 Integration Marathon This is not correct. (The flaw in your logic is that when x=1 at the endpoint of the interval, n^2.x^(2n-2) does NOT tend to zero, in fact it gets very large!) This endpoint dragging your function up is enough to make the limiting value of the integral more...

Re: Extracurricular Integration Marathon We are in the extracurricular marathon, so post a contour integration solution if you do find one. The slow decay of the integrand again makes things nontrivial (what contour do you suggest?), and the singularity is removable, not a pole so there is...

Re: Extracurricular Integration Marathon Could you clarify the source and the typical integration techniques used in this book? I will have another look at it later today if I have time.