@Chris'ssis fyi, that integral i posted before (the one i said was in terms of elliptic integrals) is one of the parts of a bigger calculation, which is some rather difficult contour integral
what i'm trying to do now is figure out how to compute/sensibly approximate the last, really difficult part
right now i'm trying to figure out the right way to say what the integration cycle is
it's from 0 to 2pi in the real part, but there's an imaginary offset
if the square root out front were in the numerator, i'd say the imaginary part was negatie and chosen such that the square root vanishes when the real part is $\pi$. but in this case, i think i should say it's some sufficiently small $\epsilon>0$ above that
it needs to lie below (in imaginary part) all the poles of the function inside the logarithm, but still be above the branch point in the negative half-strip.
not exactly the nicest contour integral i've had to deal with :)
(i call that a contour integral since it corresponds to a closed cycle when I either take $z=e^{i k}$ or take the domain to be a cylinder rather than a strip)
Hi guys. I've got a small reference request for some well-known result. I'm looking for a paper by Bonner from 1910 establishing the invariance of dimension under homeomorphisms.
@robjohn In my opinion the best way to tackle to whole story is to begin by letting $-\log(\sin(x))=y$. After that some integration by parts combined with other substitutions (especially Weierstrass substitution) should work pretty fine. It's just much to write, that's all. It's not just "maybe it works" but it works that way.
2 of the components are of the type $$\int_1^{\infty} \frac{q^s}{(1+q)} \ dq=\frac{1}{2} \left(\psi ^{(0)}\left(\frac{1}{2}-\frac{s}{2}\right)-\psi ^{(0)}\left(-\frac{s}{2}\right)\right) $$
yeah, but nowadays such a result can be given as an exercise in a first topilogy class - I doubt the paper gives any profound insight into compactness. More likely it's interesting because it demonstrates the development of the idea - a historian's pursuit :)
@Chris'ssis one of the approaches I tried involved computing $$\int_0^{\pi/2}\frac{x}{\sin(x)}\,\mathrm{d}x$$ :-) That's when I thought that Catalan's constant might be in the answer
@robjohn Well, one cannot avoid here the use of a row of polylogarithms ... (unfortunately) At a certain point, one integral has to be computed in terms of polylogarithms.
@robjohn If talking of closed forms, it's clear that the hypergeomtric form can be brought down to more pieces where the most sophisticated one is the trilogarithm with the complex argument.
@robjohn Yes, but it that series can be split such that you get some known closed forms + something that is less sophisticated than the hypergeometric one you have?
@Chris'ssis Polylogarithms are the same. At certain arguments, they have nice values, but both Polylogarithms and Hypergeometrics are rewritten series.
@Chris'ssis I like answers devoid of Polylogarithms and Hypergeometrics, but if that is all you can do, then that is fine. Someone may find a nice value for the particular argument you need.
i'm trying to remember what form the FTC takes on in the context of differential forms. my instinct would be "all differential forms are locally exact" but i suspect i'm remembering wrong
@Ted Allow me to give you a brief run-down of the weather headed my way: 1-2 inches of rain today through tomorrow morning, temps dropping all day on Wednesday, 1-3 inches of snow/ice accumulating Wed PM/Thurs AM
@TedShifrin: i guess what i probably have in mind is just the statement that, if I integrate a function, then i can differentiate to get that function back. which i suppose should generalize to something like: you can take an $n$-form, integrate with respect to some variable to get an $(n-1)$-form, and then observe that $d$ maps this back to the $n$-form.
@TedShifrin Thank you. Of course, I will need to stay physically healthy to do that, and that should not be a problem. I just need to solve my mental problems and then I can start living again.
@TedShifrin Thank you. You know, all these years, maybe I made many many mistakes in trying to solve my mental problems, but I just want to say that I think I have tried my best, and I will continue trying.
@DanielFischer Hi just a quick question. Do you know why is it that for parametric curves, say $r(t)$, it follows that the derivative $r'(t)$ is orthogonal to the curve but the second derivative $r''(t)$ is not necessarily orthogonal to the derivative $r'(t)$?
hmm. for the FTC, though, we essentially have $F=\int_0^x \omega \implies dF=\omega$ where $\omega=f(x)\,dx$. though in that case one is just integrating along the real line.
LetF be an antiderivative of f,as in the statement of the theorem. Now define a new function gas follows: g(x)= x a f(t)dt 3 ByFTC PartI,g is continuous on [a,b]and differentiable on(a,b)and g(x)=f(x) for every x in(a,b). now define another new function as follows: h(x)=g(x)−F(x) Then h is continuous on[a,b]and differentiable on (a,b) as a difference of two functions with those two.Moreover,ifx∈(a,b),h(x)=g(x)−F(x),but g(x)=f(x)byFTCPartI,andF(x)=f(x)bydefinitionofantiderivative.
Or you could develop a bordism theory, which is highly nontrivial.
With some more algebraic topology you personally can calculate $\Omega^2(pt)$. $\Omega^3$ will still be out of your reach without developing tools built for such a purpiose.
The way one normally builds it up for CW complexes is by using Meyer-Vietoris, and now that one no longer has simple values on the point, I highly doubt values on CW complexes will be uniquely determined by values on a point.
@TedShifrin: i guess the relevant example for 2-forms is again a simple line current, but now considering the magnetic flux rather than the circulation
bah, i'm feeling silly for not being able to see it
I see... Can we conclude that the number of digits is $\lfloor \log_{10} N \rfloor +1$ from the formula $N=a_0 10^0 +a_1 10^1 + \dots + a_n 10^n, a_n \leq 9$ ??
