User talk:Robkahn131

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Cheers! prime mover (talk) 03:25, 31 March 2020 (EDT)

Sources section
Please take however long you need to explore how the "Sources" section works. --prime mover (talk) 18:19, 6 May 2020 (EDT)


 * I say to you again: please take note of the "Sources" section and see if you can work out how it is to be used. --prime mover (talk) 17:19, 7 May 2020 (EDT)

Sorry about that - I swear I'm not intentionally trying to irritate you. :) I copied a page over and forgot to delete the Sources section.


 * Some good posts, by the way, nice job. --prime mover (talk) 02:22, 8 May 2020 (EDT)


 * Thanks! I am really enjoying this creative outlet.

Proof numbering
One thing I'm puzzled about, that nobody has ever been able to answer, is this:

When people add new proofs, when there is already one or more proof in existence, they often make their new proof Proof 1, and rename / renumber all the other proofs as Proof 2, Proof 3, and so on.

I just wondered why they do this, rather than just add their new proof underneath the existing ones. Is there an inherent intrinsic natural ordering of proofs that makes it more "natural" for any given set of proofs to be ordered by? My personal view is that if there is such an ordering, then maybe we should order them in (at least approximate) chronological order of initial publication (if we have the patience to trawl through the literature to find it).

Thoughts? I ask, because this seems to be your own standard practice. --prime mover (talk) 06:34, 8 May 2020 (EDT)


 * There is a reason - Zeta of 4 had the Fourier analysis as proof 1, so I was trying to make zeta of 6 consistent with zeta of 4
 * Two additional thoughts:
 * 1) For zeta of 2, 4, 6, etc, I would think there would be consistency in the proof ordering across these topics and
 * That seems to me like it would be a rod for your own back. But if that makes sense to you, knock yourself out.


 * IMO once we have a general method of proof for any arbitrarily large $n$ such that we effectively have a recipe for zeta of any even number, that is via the Bernoulli numbers, all the other proofs are merely curiosities.
 * I completely agree. However, for reasons that I can't explain, I like to see certain curiosities.


 * Curiosities are fun and I welcome their documentation. But IMO they don't get top billing. Or at least, they don't merit renumbering everything to put them at the top. Imo.


 * Having said that, if you want to assemble all the proofs of type X for zeta 2, zeta 4, zeta 6, etc. you could always do a page for "zeta of 2n by Fourier", etc. and transclude them all. --prime mover (talk) 15:08, 8 May 2020 (EDT)


 * 2) Some proofs seem more satisfying to me. I realize that a proof is a proof, but I'd like to see the more satisfying ones up top.


 * "Satisfying"? - Different proofs put a different spotlight on the problem. But the problem of course is that what is satisfying to me may or may not be satisfying to others.


 * Indeed, in that final statement you are highly likely to be correct.


 * In this context, the "satisfying" proof is the one which covers the entire field in one go -- that is:Riemann Zeta Function at Even Integers. Then the elegant proof is the one which plugs the number into that. There's nothing more satisfying than polishing off something which is on the surface really complicated by means of a simple 3-liner: "here's one I made earlier, use that."


 * And indeed further: building a fourier series for ever more increasing n is all well and good, but it's a bit like the early days of integral calculus where whoever-it-was proved $\int x^n \rd x = \frac {x^{n + 1} } {n + 1}$ for bigger and bigger $n$ but never proved the general formula. All that hard work blown out of the water by Primitive of Power. And the technique where you equate coefficients of the Taylor series for sin x / x, again, well okay, but first you have to get those coefficients, which for larger $n$ is more and more tedious -- and even then the implementation on the proof page needs more than just handwaving, even a page like "coefficient of (whatever n) in the expansion of the Taylor series of sin x / x" or whatever -- again, not what I'd call elegant.


 * I agree again. It goes back to my own quirkiness on this one - I'm just curious to see what some of this minutae looks like.


 * Incidentally, you may wish to experiment with the technique of indenting replies to talk page posts, so as to make it clear what's a post and what's a reply. I have taken the liberty of adjusting your responses here accordingly. --prime mover (talk) 09:14, 8 May 2020 (EDT)


 * My ignorance will astound you!! :). I didn't know about the colon thing until I saw it here.


 * Separate topic - if one wanted to represent {I, -1, -I, 1} in a series, they could simply put I^k in the series to accomplish that. How does one do that for {sin(x), cos(x), -sin(x), -cos(x)}? The reason I ask - in Primitive of $x^n \cos a x$ all integrals can be removed and the result can be expressed as a power series. The pattern is obvious - the a in the denominator starts at 1 and moves up to (m+1), the x starts at m and goes down to 0. The coefficient is just a falling factorial.


 * $\frac {\sin a x} a x^6 + \frac {6 \cos a x} {a^2} x^5 - \frac {30 \sin a x} {a^3} x^4 - \frac {120 \cos a x} {a^4} x^3 + \frac {360 \sin a x} {a^5} x^2 + \frac {720 \cos a x} {a^6} x - \frac {720 \sin a x} {a^7}$


 * How about $\sequence {\dfrac {\d^n} {\d x^n} \sin x}_{n \mathop \in \N_{>0} }$?


 * How about $\displaystyle \sequence {\map {\sin} {x + \frac \pi 2 n}}_{n \mathop \in \N }$? --Julius (talk) 17:40, 8 May 2020 (EDT)


 * Heh! That's an interestingly fun little result to post up sometime: $\dfrac {\d^n} {\d x^n} \sin x = \map {\sin} {x + \dfrac {n \pi} 2}$ --prime mover (talk) 18:08, 8 May 2020 (EDT)