Timeline for Convergence of series related to partial fraction expansion of cotangent function
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 12, 2023 at 5:34 | history | edited | Vincent Granville | CC BY-SA 4.0 |
[Edit removed during grace period]
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| Jan 10, 2023 at 0:33 | vote | accept | Vincent Granville | ||
| Jan 9, 2023 at 19:39 | history | edited | Vincent Granville | CC BY-SA 4.0 |
Added "purpose" section
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| Jan 9, 2023 at 19:08 | answer | added | Kostya_I | timeline score: 3 | |
| Jan 9, 2023 at 18:45 | comment | added | Vincent Granville | It sounds like your comment answers my question, unless I am mistaken. You could post it as an official answer. | |
| Jan 9, 2023 at 18:42 | comment | added | Vincent Granville | There is a generalization to this: replace $t^2-k^2$ by $\phi(t)-\phi(k)$, and $2t$ by $\phi'(t)$ (the derivative). Here $\phi$ is an arbitrary function with some limitations, say $\phi(t) = t^\alpha$ (limitations may apply). | |
| Jan 9, 2023 at 18:40 | comment | added | Conrad | RHS is right-hand side (the expression there) - actually as $\cos (\pi t)=\cos (-\pi t)$ the function $f_t$ extends by periodicity from $[-\pi, \pi]$ to a continuous function on $\mathbb R$ which of course is not $\cos t \theta$ except on $[-\pi, \pi]$ unless $t$ integral, so the convergence is uniform on $[-\pi, \pi]$ | |
| Jan 9, 2023 at 18:39 | comment | added | Vincent Granville | Thank you for your comment. I may appear as an idiot, but what is RHS? Regarding the "trivial" character of my formula, it is indeed correct if $t\in\mathbb{Z}$ (by taking the limit) and I started from there. In that sense, it is a straightforward application of L'Hopital's rule. | |
| Jan 9, 2023 at 18:31 | comment | added | Conrad | if you consider $f_t(\theta)=\cos t\theta, \theta \in [-\pi, \pi]$ and expand in Fourier series you get RHS and convergence just follows from general results on such - if $t \notin \mathbb Z$ convergence on $(-\pi, \pi)$ uniform on compacts, while at the ends we get the average value etc, while if $t$ integer by taking a limit and convergence everywhere); now if $|\theta| > \pi$ one gets that the RHS which is $2\pi$ periodic in $\theta$ converges to the periodic restriction of $f_t$ so to $f_t(\theta-2k\pi)$ where $\theta-2\pi k \in [-\pi, \pi]$ | |
| Jan 9, 2023 at 18:30 | comment | added | Vincent Granville | The goal is to provide an analytic continuation (and there are many possible ones) to a function $f(t)$ defined on integers. Here $f(t)=\cos(t\theta)$. But you could replace it by any even function: then you must replace $\cos (k\theta)$ on the right side by $f(k)$. There is a similar formula for odd functions. When combining even and odd functions, you get a more general formula. | |
| Jan 9, 2023 at 18:28 | comment | added | Iosif Pinelis | I cannot reconcile "The above equality is rather trivial" with "but the convergence of the right side towards the left side is not straightforward to me" and "If $|\theta|>\pi$, we still have convergence, but towards a different function". Also, any references to "rather trivial"? Also, in "Is that correct?", what exactly do you mean by "that"? Also, in "prove it", what exactly do you mean by "it"? | |
| Jan 9, 2023 at 18:17 | answer | added | Ofir Gorodetsky | timeline score: 6 | |
| Jan 9, 2023 at 17:59 | history | asked | Vincent Granville | CC BY-SA 4.0 |