Tito Omburo

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Could you tell me why partial derivative continuity is completely wrong?

Hi Tito.

I have seen your restoration of my latest modification on the page Cauchy-Riemann equations. Could you tell me this change (adding partial derivative continuity for Cauchy-Riemann equations to make a sufficient condition for complex differentiability) is completely wrong?

I think this change is logical and supported by the following references. Please provide me your explanation. If not, then I will restore the modification.

(Ref.[1]) Chapter 10 "Complex Calculus", Mathematical Physics, A Modern Introduction to Its Foundations, 2nd, Sadri Hassani, 2013.

(Ref.[2]) Chapter 11.2 "Cauchy-Riemann Conditions", Mathematical Methods for Physicists - A Comprehensive Guide, 7th, Arfken, Weber, and Harris, 2013. Goodphy (talk) 23:01, 11 September 2023 (UTC)

The sources are cited in the article itself, which even includes a proof. Complex differentiability at a point is equivalent to the following two conditions: (1) the real and imaginary parts of the function are differentiable (as real functions of two variables) at the point, (2) they satisfy the Cauchy-Riemann equations at the point. There is no requirement of continuity. None. (Continuity of the partial derivatives would guarantee that the function is real differentiable at the point, but that is not a necessary condition, and so it's wrong to include it in a discussion of "necessary and sufficient conditions" for complex differentiability.)

An example of a complex differentiable function with discontinuous partials is $f(z)=z^{2}e^{i/|z|}$ at $z=0$. Tito Omburo (talk) 23:45, 11 September 2023 (UTC)

Hi Tito Omburo.

I agree that the C-R (Cauchy-Riemann) equations are a necessary condition for complex differentiability.

However, I do not agree that the C-R equations is solely a sufficient condition for complex differentiability. It is possible that there is a function which is complex-differentiable while their partial derivatives with respect to real variables x and y are not continuous, but it does not mean that the C-R equation alone is the sufficient condition. If there is a partial derivative jump at z = x + iy, then the mathematical expression after the sentence "This is equivalent to the existence of the following linear approximation" of the chapter "Complex differentiability" in this Wikipedia page is not guaranteed because of the jump. This is also supported by referenced I have mentioned before in this talk.

Thus, I think the statement like "the C-R equations is equivalent to the complex differentiability" is a danger statement. The two references that I have mentioned do not say that way. Goodphy (talk) 10:18, 12 September 2023 (UTC)

I gave two necessary conditions, but you seem to be ignoring the first one. Real differentiability is defined as the existence of a linear approximation (see differentiable function#Differentiability in higher dimensions). Tito Omburo (talk) 10:28, 12 September 2023 (UTC)

Hi.

I have slept on your feedback, and the following conclusion is made via some investigation:

Real differentiability (differentiable with respect to real variables) is a stronger condition than the existence of partial derivatives for a 2 or higher dimensional function domain. And if the real and imaginary parts of a complex function are real-differentiable, then their sum is of course differentiable. There is not many things earned from this.

So, I suggest modifying this Wikipedia page (Cauchy-Riemann equations) such that the page will use partial derivatives of u (real part of a complex function f) and v (imaginary part of f) instead of real differentiability. This prevents readers from studying higher dimensional differentiability definitions to understand the what C-R equations for (saving time). I believe this modification will also match with university textbook descriptions including the mentioned two references. Goodphy (talk) 11:01, 13 September 2023 (UTC)

I oppose this proposed change. You cannot modify the necessary and sufficient conditions for complex differentiability, well covered by the literature and existing article, to suit your personal distaste for standard definitions from calculus. (As I have already shown, your proposal would actually give incorrect necessary conditions.) Note that, moreover, the discussion of Goursat's theorem also doesn't assume continuity of the partial derivatives, so this is a pretty important point for the article as a whole! In fact, one does not even need to assume differentiability, by the Looman-Menchoff theorem, provided you are in an open set. Since the bulk of applications concern this case, your beef with the correct necessary and sufficient conditions seems particularly silly. Tito Omburo (talk) 11:30, 13 September 2023 (UTC)

Hi. I appreciate your effort and expertise to improve this page! Can I ask a question? For a complex function f = u +iv, Dose the current section "Complex differentiability" explicitly show the real differentiability of u and v as a necessary condition for the complex differentiability of f?. Goodphy (talk) 12:08, 14 September 2023 (UTC)

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Weyl connection

I reverted your edits to Weyl connection because the subject does not seem to be notable. Just two citations fails WP:GNG. Please read WP:SPINOUT regarding how to develop the content. Also, many editors are not numerate so it would help if you collaborated with an experienced editor at Wikipedia:WikiProject Mathematics. If you hadn't already, you might consider joining that project. Chris Troutman (talk) 16:17, 17 October 2023 (UTC)

I have added more references. Is there some specific number of references you would be happy with for a stub? (I note that the article that you restored the redirect to actually had fewer references than the one you happily deleted and threatened its author with blocking.) Tito Omburo (talk) 16:24, 17 October 2023 (UTC)

There is not a firm number for WP:GNG, but I don't think the citations (from reliable sources) are quite enough. Please read WP:OTHERCRAPEXISTS regarding the rest of your complaint. I'm just trying to get a barnstar from NPP. Your edit triggered my patrol. Other content on Wikipedia does not concern me. Chris Troutman (talk) 19:00, 17 October 2023 (UTC)

There are now almost two dozen sources, including standard textbooks, journal articles, and even the Springer Encyclopedia. I humbly sumbit that this is about as notable as it gets. No idea why you've decided to pick this weird beef with me. I hope you get your "barnstar" and feel really good about all the bullying you did to get it. Tito Omburo (talk) 20:02, 17 October 2023 (UTC)

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Paley-Wiener

I put a single dash not because I thought Paley and Wiener had a compound name as a single person - but because Google searching for Paley-Wiener did not return your page, and inside of Wikipedia I also couldn't find it. Few people will be searching for commonly spelled Paley-Wiener with a special character long dash... Fritsebits (talk) 13:30, 24 May 2024 (UTC)

per MOS:DASH:

Use an en dash for the names of two or more entities in an attributive compound.

the Seifert–van Kampen theorem; the Alpher–Bethe–Gamow theory

Tito Omburo (talk) 13:33, 24 May 2024 (UTC)

Erdős–Delange theorem moved to draftspace

Thanks for your contributions to Erdős–Delange theorem. Unfortunately, I do not think it is ready for publishing at this time because it has no sources. I have converted your article to a draft which you can improve, undisturbed for a while.

Please see more information at Help:Unreviewed new page. When the article is ready for publication, please click on the "Submit your draft for review!" button at the top of the page OR move the page back. C F A 💬 22:10, 22 June 2024 (UTC)

Regarding your removal of eulers number edit

E (mathematical constant)

you say the self derivative and integral property is discussed in the article, but that is not a reason to remove it being in the introduction. It is arguably the defining characteristic of e in mathematics. Why does it being the limit of (1+1/n)^n , or the limit of the infinite series of reciprocal factorials deserve being in the intro but not this, if all of this is discussed elsewhere further down ? The main defining property making e useful in math needs to be in the intro for someone looking it up for the first time Mfuentes6293 (talk) 01:24, 7 September 2024 (UTC)

I happen to think both of the things you mention should be moved out of the lead of the article and into the first section, where the property of the natural exponential function being its own derivative is discussed. Tito Omburo (talk) 10:40, 7 September 2024 (UTC)

I think if anything, the definitions as infinite series are not worthy of being in the intro but the self derivative property is important enough to include. It is like discussing pi without mentioning circles in the intro Mfuentes6293 (talk) 23:49, 7 September 2024 (UTC)

I've added that it's the base of the natural exponential, and removed from the lead the limit and series. The definition of the natural exponential via the derivative property is discussed in the first section. Tito Omburo (talk) 12:25, 8 September 2024 (UTC)

Swinnerton-Dyer polynomial

Great job on this page. KudosToploftical (talk) 14:39, 26 October 2024 (UTC)

Your reversion of my edit

I just got notified that you reverted my edit of Swinnerton-Dyer polynomial. You said that it was broken in Chrome, but on my computer under Chrome version 131, it looked perfectly fine. Could you explain what the problem was in more detail, or perhaps post a screenshot? — LucasBrown 01:09, 16 November 2024 (UTC)

The symbol $\mathbb {Z} _{2}^{|P|}$ does not render properly in Chrome 131 Mobile. The Z is huge, and the exponents are squeezed together so that it looks like $\mathbb {Z} _{2^{|P|}}$. MathML does not render reliably on all browsers and browser configurations. For example, depending on the browser, the two different stylings of g: g and $g$ may appear different from each other. Tito Omburo (talk) 01:46, 16 November 2024 (UTC)

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