User talk:Robkahn131

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Welcome to ProofWiki! Since you're new, you may want to check out the general help page. It's the best first stop to see how things are done (next to reading proofs, of course!). Please feel free to contribute to whichever area of mathematics interests you, either by adding new proofs, or fixing up existing ones. If you have any questions please feel free to contact one of the administrators, or post your question on the questions page.

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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 $\sequence {\map {\sin} {x + \dfrac \pi 2 n} }_{n \mathop \in \N }$? --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)

Details of consistency and presentation

Please note we were given a hard time and got some bad press once because one of the big-name mathematics bigshots on one of the big time mathematics websites noticed that we were not entirely consistent in the way we presented the title of the "Also see" sections. We have standardised on "Also see", as you may have noticed. However, at one point we had the occasional page here and there where it was presented as "Also See". This was all the excuse needed by that bigshot to give us a deservedly negative write-up. Hence I would urge you to make sure that if you need to add such a section, to title it "Also see" rather than "Also See". Also, while you are about it, for further consistent look and feel, please note to leave an extra space between sections. I understand how tedious it is to press the enter key an extra time, but it saves a moderator having to go in and fix it. --prime mover (talk) 02:35, 27 May 2020 (EDT)

Will do. Thanks for the feedback. --Robkahn131 (talk) 10:40, 27 May 2020 (EDT)

Proof 1 of Sum of Bernoulli Numbers by Binomial Coefficients Vanishes‎

Presumably you've got a Proof 2 coming?

And I know I keep going on about it, can you learn how the Sources section works? It's because of contributors making a mess of the source citations that we prefer people do not do major refactoring like this until we are certain they know what they're doing.

--prime mover (talk) 16:34, 28 May 2020 (EDT)

I didn't touch or do anything with the Sources. I am not clear on my misstep here. --Robkahn131 (talk) 16:38, 28 May 2020 (EDT)
You copied the Sources section from Sum of Bernoulli Numbers by Binomial Coefficients Vanishes‎ to Sum of Bernoulli Numbers by Binomial Coefficients Vanishes/‎Proof 1, which means there are now 2 pages which link to the same prev and the same next page. It is intended that the flow is a strictly linear total order of the entries in each source work. By copying a source you split the flow.
Obviously when a refactoring job like this is done (as in here, to separate out the proof into a separate page), it is necessary to have access to the actual source work(s) in the Sources section to actually make the flow good. So it is up to someone who actually has these works to fix this (as in: either the proof is included in the source work, in which case the flow will go onto the proof page, or it isn't in which case it's not).
In general we do not split the proof off from the parent page into a "Proof 1" page unless we have a "Proof 2" to add to that page. It's a pain in the arse, there's lots of ancillary work that needs to be done, and it's always Muggins who has to tidy up all that shit, because Muggins seems to be the only one who can understand how this all works (and/or has the attention to detail to make it consistent and correct).
Which is, again, why I ask: you do have a Proof 2 coming, don't you? --prime mover (talk) 16:51, 28 May 2020 (EDT)
I'm afraid I don't. I was actually attempting to be helpful - I see many other pages formatted this way. I sincerely apologize. I really like this site. Shall I put the proof back on the main page and have you delete the proof 1 page or would you prefer I leave well enough alone? Once again - sorry! --Robkahn131 (talk) 17:01, 28 May 2020 (EDT)
In general, there should in fact be very few pages which have a "Proof 1" but no "Proof 2". Which have you identified?
Don't worry, I'll put it all back the way it was.
If a result page and a proof page are separated off into parent-and-transclusion when there is only one proof, that page would be called "Proof" not "Proof 1". If there are exceptions, that usually means someone got so far with putting another proof up but never finished it, or have juggled things around through well-meaning though misguided attempts at being helpful.
One reason to deliberately separate out the proof from the result is if there is a source work which provides a result in one place in the book, then goes off to talk about something else, and then comes back to the result to prove it. To accurately model the source flow it becomes prudent to separate them out. But in that case you would just call it "Proof".
If a need arises to split up the pages like this, they will in general be identified by someone experienced who has a good reason for doing so, and (if they don't change it then and there) will apply the {{Refactor}} template to the page, or whatever other maintenance template may apply.
There are wide gaps in $\mathsf{Pr} \infty \mathsf{fWiki}$'s coverage, and a lot of scope for adding illustrative examples -- most of which that currently exist have been obtained from published works. There should be enough opportunity to considerably expand the content which should not require this radical refactoring of moving big stuff from one page to another. --prime mover (talk) 17:13, 28 May 2020 (EDT)

The eqn editor

Please try not to get into the habit of numbering every single line of every equation you write. There is only a need to give a line a number in one circumstance: and that is when you need to refer to a specific line at some stage.

Also please note that line numbers do not go in the ll column. There is a dedicated column into which line numbers are put. See {{eqn}} for how to use this.

The ll column is "traditionally" used for $\leadsto$ and $\leadstoandfrom$ as necessary (which is a good habit to get into: using $\leadsto$ before a line that follows on from a previous line).

It would also be really, really cool if you could just use the simple standard dollar delimiters on equation blocks which consist of only one line. The reason for the {{eqn}} template is to make the lines in multi-line equations line up neatly. When there is only one such line you don't need this. --prime mover (talk) 21:06, 11 March 2021 (UTC)

The n column - got it. As far as the equation template vs dollar delimiters - why does it matter? I am not being flippant - I honestly don't understand why it matters. --Robkahn131 (talk) 21:12, 11 March 2021 (UTC)
It makes for a more compact source code. For the same reason I don't get the car out to go to visit my next door neighbour. :-)
While I'm about it, try and keep the comments compact. Brevity is everything.
It would also be appreciated if you did not radically rewrite the statement of results. They are generally good as they are.
I'm in the process of recompactifying that Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function page you worked on, so please note I have it open at the moment. Any edits you make between now and when I have finished will probably be lost. --prime mover (talk) 21:21, 11 March 2021 (UTC)
Apologies, but it happened as I warned.
Another note: please do not feel you have to put empty tags in place in templates where they are not needed. (In this case, that means not leaving loads of n = lying around.) It bloats the code, causing longer loading time and larger code footprint. The exception where this is often not followed is in the case of comment tags: they are often left open in case anyone feels like filling them in. --prime mover (talk) 21:57, 11 March 2021 (UTC)
Got it - thanks for the feedback. I need to add a Fourier transform page or two to fully establish everything in Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function. --Robkahn131 (talk) 23:22, 11 March 2021 (UTC)
Be aware that such stuff may already exist, just in a slightly different form from how it is presented as you envisage it. If what needs to be done is in a genuinely new area that we have not yet properly covered, thenplease tread carefully. The page in question here was put in place by a contributor who did not share the same ideas of "build up from the bottom" and so much of his work relies upon concepts and results which do not yet exist on $\mathsf{Pr} \infty \mathsf{fWiki}$. Hence there may be rather more background material that needs to be put into place before ant of this is completely solid. --prime mover (talk) 23:37, 11 March 2021 (UTC)

Quick heads up about Fourier transforms

Are you particularly attached to the use of $\zeta$ for the dummy variable in the work on the Definition:Fourier Transform? Maybe it's a matter of personal taste, but $\zeta$ is not only a loose and ugly symbol, but also very far from being standard. Wikipedia uses $\xi$, as do we on that main page. Mathworld uses $s$, same as for Laplace transforms. Can we perhaps standardise on $s$ to make Fourier transforms consistent with Laplace transforms? If you have good reasons for being committed to a Greek symbol, then I suppose $\xi$ can be used, but I would prefer $s$ for internal consistency of integral transforms on $\mathsf{Pr} \infty \mathsf{fWiki}$, if at all possible.

Thoughts? --prime mover (talk) 23:32, 13 March 2021 (UTC)

$s$ works for me. I'll replace $\zeta$ with $s$. --Robkahn131 (talk) 00:08, 14 March 2021 (UTC)

Dirac delta function

I note that you have a different way of definining the Dirac delta function than the way the page was originally written.

What we do on $\mathsf{Pr} \infty \mathsf{fWiki}$ is not to replace a definition (which may or may not be as objectively "good" as yours) with a different definition. What we do is to add a second definition. We do this by separating off the definitions into different pages, and writing a page proving they are equivalent.

Similarly, we do not add results onto a definition page. We put them into separate pages holding results.

I understand you may have a different vision as for the direction of $\mathsf{Pr} \infty \mathsf{fWiki}$, and perhaps the structure that we currently have is intrinsically incorrect, but it is believed important (again, this also may be up for question) that the site have a consistency of structure and philosophical approach which we have tried to pin down in our style guide.

If the equivalent definitions for the delta function are genuinely cited as definitions in whatever literature you are using as the basis of this page, then each one needs to be extracted into its own subpage, and an equivalence proof written for each one.

And I take issue with the "questionable" template on the page holding the graph. You would only put "questionable" in place if it is genuinuely wrong. As far as I can tell, the definition as provided is not wrong, it's just different from how those "most sources" define it. If you can demonstrate that the way it is defined here is indeed actually incorrect, then it would be good for this to be explained. --prime mover (talk) 17:06, 14 March 2021 (UTC)

No harm or offense intended. I am a huge fan of the site. I defer to your judgement on all matters site related. My intention is not to super-impose any vision of mine, but merely to contribute whatever I can to make the site better. And sometimes my idea of "better" is NOT better!
Side note - Happy $\pi$ day!  :). --Robkahn131 (talk) 17:47, 14 March 2021 (UTC)
Oh yes of course, I always miss that. But then I see the date as $14/3$ not $3/14$. $27$th Jan is far more interesting. :-)
Work (slow) is under way to refactor the Dirac delta function page(s). --prime mover (talk) 19:46, 14 March 2021 (UTC)

Permanent links / redirects

A quick heads-up (something which I thought you'd pick up on at some stage):

When you are linking to a page with a forward slash in it, e.g. Definition:Complex Number/Polar Form, please always check to see if there is a non-slashed redirect, as in this case Definition:Polar Form of Complex Number.

We do it like this so as to make any subsequent refactoring easier. If we want to change the structure of the underlying pages (as we often seem to do) then all we need to do is change the target of the redirect within Definition:Polar Form of Complex Number, rather than have to go through every single page that link directly to Definition:Complex Number/Polar Form, which is tedious.

... so can I ask you to look at this point and see whether you can get on board with it? Cheers. --prime mover (talk) 20:51, 29 March 2021 (UTC)
Ha! Glad you're back!  :). Backslash in link equals bad - got it. I'm on board. --Robkahn131 (talk) 21:24, 29 March 2021 (UTC)
Sorry, I was settling an elderly relative into his new accommodation. Very long weekend. --prime mover (talk) 05:10, 30 March 2021 (UTC)

We also have a template {{Defof}} which was originally designed for use in the c parameter of the {{eqn}} template to streamline proofwriting and to guide towards a unified presentational style. You will see the changes I have made along these lines in Cardano's Formula/Examples/x^3 - 15x - 4.

Thanks --prime mover (talk) 08:17, 18 March 2021 (UTC)

Hat check problem

I compromised. Realised we didn't explicitly link the number of derangements with the subfactorial, so I basically completed that conceptual link, and then put back what you'd done and expanded it a bit. --prime mover (talk) 23:50, 22 March 2021 (UTC)

Nice! Looks great. --Robkahn131 (talk) 23:53, 22 March 2021 (UTC)

section gaps

... and another thing, please note that there are two linefeeds between sections. If you examine the pages that have been generated, you will see that if you add an extra linefeed, the gap is bigger. This is deliberate. There is a FAQ on the subject, I believe. --prime mover (talk) 21:13, 29 March 2021 (UTC)


Good job hacking through all that hard work proving results about all those probability distributions. It did not occur to me before how much there is to be done here. It would be something to be proud of if we can say we have the go-to repository for all this information. So, keep calm and carry on ... while I'll hover and do whatever minor tidying may be needed. --prime mover (talk) 17:13, 5 May 2021 (UTC)

Hypergeometric stuff

Good job on the Hypergeometric stuff. I've gone through and rationalised this entire section, which is now just about as tight as I can make it.

Standardising on $\map F {a, b; c; z}$ rather than the over-complicated but more technically correct $\map { {}_2 F_1} {a, b; c; z}$, which has been relegated to an "also denoted as" section.

We can always use $\map { {}_2 F_1} {a, b; c; z}$ when it is necessary to place emphasis on where it comes from. --prime mover (talk) 10:35, 18 December 2022 (UTC)

Thanks! Everything looks better. I appreciate the edits. This is interesting material IMO - more contributions on this topic are coming shortly. --Robkahn131 (talk) 16:24, 18 December 2022 (UTC)
Going on a family cruise for 2 weeks - signing off till early January :). --Robkahn131 (talk) 02:14, 23 December 2022 (UTC)
Enjoy. Happy new year. --prime mover (talk) 06:20, 23 December 2022 (UTC)

Links to pages with an equation in LateX

There's a good reason not to create a display link to a page which is totally a $\LaTeX$ expression. If you right click on it, it only allows you the option to view the source code of that link, not to open the link in another window. So if you want to open the page, you have to left click it and that navigates you away from the page you were looking at. --prime mover (talk) 23:40, 19 December 2022 (UTC)

Are you referring to the links here: Sum over k of m+r+s Choose k by n+r-s Choose n-k by r+k Choose m+n? I have an Apple mouse - there is no left or right click - it's all the same on a Mac. I assume the ask is to stick with "regular" links and not fancy links like this: $\dbinom n k = \dbinom n {n - k}$ ? --Robkahn131 (talk) 00:49, 20 December 2022 (UTC)
I see it now - I hopped on my wife's computer. I see your point. Will fix later this week. --Robkahn131 (talk) 04:27, 20 December 2022 (UTC)
Yes, that link and there's another one. Basically you can include all the $\LaTeX$ you like, as long as you include some non-$\LaTeX$ in there as well, for example Product of $\dbinom r m$ with $\dbinom m k$ works but not $\dbinom r m \dbinom m k = \dbinom r k \dbinom {r - k} {m - k}$.
It's fiddly and awkward but that's why that is. --prime mover (talk) 06:24, 20 December 2022 (UTC)

Concerning Saalschütz's Hypergeometric Theorem

In particular with regard to the historical note, you may want to review the use of the available templates to implement mentions of books and authors / mathematicians. You will see the obvious advantages (not least the fact that the actual links themselves are rendered consistently).

I renamed it to the dual attribution as that is what we try to do on $\mathsf{Pr} \infty \mathsf{fWiki}$: credit a result with as many contributors as the literature gives. --prime mover (talk) 06:29, 3 January 2023 (UTC)

More big thanks

Your work is basically awesome. Please continue to excuse the tidying up and pickiness, it's a labour of love. --prime mover (talk) 23:28, 6 January 2023 (UTC)

Thanks! I don't mind the pickiness in the least. All of us are trying to create something that will be useful to others. --Robkahn131 (talk) 03:35, 7 January 2023 (UTC)

I've got it

User:Richard47 is an AI, right?

Maybe? I'm seeing lots of different kinds of trash. There is simple trash like this: Reciprocal of 2. Then there are the bigger eyesores like this: Minimal Prime/Various Number Bases and Long Period Prime/Various Number Bases Some of the contributions have some merit. Plenty of contributors put time and energy into creating something of value. We are all creating something of value together. It seems fair and appropriate to simply delete pages that are clearly well below where they need to be. Sometimes a below par page needs to be tidied. And then sometimes we have rows and rows and rows of gibberish numbers where deletion seems like the best course. --Robkahn131 (talk) 01:25, 12 March 2023 (UTC)
I'm okay-ish with Reciprocal of 2 and Reciprocal of 5 and so on, because they merely complete the pattern of the other reciprocal pages, which IMO are worth including (and if you include one, you need to include them all or suffer from arguments over whether a particular reciprocal is notable enough to have its own page or not.
However, all those pages of numbers are being expunged. --prime mover (talk) 07:33, 14 March 2023 (UTC)

Thank You

I know you just streamlined my proof, I appreciate your work in making it more understandable. I actually came up with that proof myself and started a youtube channel just to show it off a little over a year ago. I don't know if there are other problems or proofs that are similar to this one, but I did do a little extra work on it and found a proof that works the same way for the Tribonacci numbers, and I'm sure if I wanted to I could generalize the problem, but maybe I'll do that for a research project for my PhD if nobody beats me to it. Do you know of any similar proofs to this one for solving the values of infinite sums, or do you think this is wholly unique to me? Again, I'm glad you liked the proof, and I'm glad you took the time to make it cleaner! Thank you! Asdfghjklohhnhn (talk) 02:34, 28 December 2023 (UTC)

Congratulations! Your proof is the Featured Proof on this website. How on earth did you make the connection between flipping a fair coin until you receive two heads in a row and Fibonacci numbers? Excellent observation!! I have not seen any similar proofs like this, but every time I see an infinite sum with $2^n$ in the denominator, I'm going to think about flipping a fair coin in varying scenarios to create a sample space like you did here. This proof is great! Accessible to less sophisticated folks, clever and it connects dots that you would think shouldn't be connected. Great work! --Robkahn131 (talk) 16:42, 28 December 2023 (UTC)
Thank you so much! I was actually taking a Stochastic Processes graduate class, and on the first homework assignment of the book they were asking about this specific sample space. I ended up not being able to figure out at first how to answer the problem, but made some sort of trivial mistake and couldn't figure out how to simplify this sum, so I ended up using a python library to calculate the value of the sum, amd realized it was 1, but I was super confused at first, then I came to the realization that it was the sample space and that I was doing the question wrong, but it sparked the motivation in me to share what I had found after I expand on the formality a little bit, so I wrote up a rough draft, then a final draft and recorded the video, then a little over a year later I came across this website incidentally and wanted to add this idea to the website. Also, funnily enough I saw the same problem solved by Michael Penn in the two ways I listed in Proof 3 and 4 on the same day that I posted my video about Proof 1 and 2. If I'm not mistaken he posted the video way before me though. But I had already posted the video and noticed that both ways of mine were different then both of his ways, so I was relieved and linked his video in the description of my video. Also as I was writing the proof 4 on here I noticed that Michael Penn kind of willy-nilly-ly exchanged the order of the sum without rigorous explantion on why you can, so I ended up looking up when you are able to switch the order of a sum and saw Fubini's Theorem for infinite sums and rigorously proved it was possible in proof 4. Also like I said I also proved the connection for Tribonacci numbers and flipping heads 3 times in a row, and made that my second video. After I saw that it was the featured proof I shared it on my snapchat and on the \math reddit page. So far it has gained a significant amount of traction and hopefully I broughy more foot traffic to the website, even if only temporarily. Since it has been gaining traction today, and I start my PhD in 2 weeks I was thinking about potentially doing a research paper on this question and generalizing it to N of the same outcome in a row, and an M-sided die instead of a 2-sided coin flip. This is a sort of connection that seems very easy to grasp, but unique to discover, and I feel special for discovering it. My friend was talking to me about Category Theory a few days ago, and I just realized that this proof is similar in that regard because it is using probability theory to prove a calculus problem, and can be akin to generalizing knowledge as a whole into different categories. Again, thank you, and I'm glad you and others have found it elegant! Asdfghjklohhnhn (talk) 23:48, 28 December 2023 (UTC)

Changing page name

Just checking: are you unable to use the Move option? --prime mover (talk) 22:30, 20 January 2024 (UTC)

Yes - but if I make a bad page name and I know there are no other links involved, I'd rather just move everything over to the new page name and not even bother with redirects. (i.e. Completely kill the old page name) --Robkahn131 (talk) 23:06, 20 January 2024 (UTC)
"Yes", I can use "move", but actively chose not to. --Robkahn131 (talk) 23:08, 20 January 2024 (UTC)
Well okay, I just wondered, because it makes so much more work having to do everything manually like that. No matter. --prime mover (talk) 23:17, 20 January 2024 (UTC)

Change of notation -- a shortcut

So quick off the mark you are! You must have missed my latest addition to the code: The $\LaTeX$ code for \(\harm r z\) is \harm r z .

If we write all our general harmonic numbers of order r of z (have to think of a name for this, it's not a number it's a function. General harmonic function? ) as $\harm r z$ then if we decide we change what we want it to look like, all we will need to do is change the coding of \harm. --prime mover (talk) 23:22, 27 January 2024 (UTC)