Talk:Zeros and poles

Singularity

"A singularity which is not a pole is called an essential singularity. " This is not true... A singularity can be a removable singularity and even a branch point... This should be changed

In the first paragraph, wouldn't it be more accurate to say "isolated singularity" rather than just "singularity"? This is the impression I got by reading the corresponding Italian page, but I'm not confident enough to edit the page. bernie (talk) 01:44, 28 May 2015 (UTC)

Examples

Some more examples of poles or various orders should be included, as well as more information relating to what a poles order means. He Who Is 19:15, 10 June 2006 (UTC)

Agreed. The introduction is completely theoretical. Immediately providing some cursory, real-world examples, such as in RF filtering, would help keep people on board, for instance, frequencies at which the signal is completely blocked or is completely passed through.

Also, the introduction, which I think is supposed to give an overall definition, has in itself a dozen words and terms that are so specialized, those dozen words and terms link to other Wikipedia pages to find out what they mean. An introduction that is composed of a dozen words and terms that themselves require explanations is overwhelming for an introduction. We could use a little less of the exacting mathematical terminology in the introduction. Consider the audience; we came here because we're NOT experts in the field. But this introduction as only suitable for experts -- it's only understandable by people that are already familiar with the subject matter at a post-graduate level.

I might have begun with the most basic electronic circuits that have 1 (one) zero or 1 (one) pole, with the understanding that the electronic circuits have analogies to other systems. Nei1 (talk) 14:34, 31 October 2020 (UTC)

I have edited the lead for making it less WP:TECHNICAL. In particular, I have moved to the top the last sentence (the only non-technical one). I hope that the new lead is easier to understand. In any case, all technical terms are now defined in this lead, except complex differentiable. D.Lazard (talk) 18:02, 31 October 2020 (UTC)

Hi D.Lazard. Thank you for your help. I think I'm starting to understand now. It says, "... a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly..." But that definition leaves out what happens at the singularity, which is the punchline. What would you think about, "... A pole is a value of x where f(x) = ∞" Is that too simple and therefore incorrect? Or maybe it's a good start, which can be elucidated with exceptions in further paragraphs? Thanks again, Nei1 (talk) 03:35, 26 November 2020 (UTC)

The lead is not aimed for given technical definitions. It is aimed to summarize the content of the article. So, for learning what exactly is a pole, you have to go to section "Definitions". D.Lazard (talk) 09:17, 26 November 2020 (UTC)

Formal Definition

The article states: "If there exists a holomorphic function g : U → C and a nonnegative integer n such that

$f(z)={\frac {g(z)}{(z-a)^{n}}}$"

However I think g should be defined in the punctured disc around a, not necessarily in all of U. For example if f has two singularities. If this case is not being explicitly considered, then maybe it could be made more obvious? —Preceding unsigned comment added by Silasdavis (talk • contribs) 16:02, 19 November 2007 (UTC)

What does a "punctured disc" mean? If it means an open disc centred on $a$ but excluding the point $a$, then there is no problem: we can take U to be this punctured disc. JamesBWatson (talk) 09:51, 17 September 2009 (UTC)

Hole?

How does a holomorphic function have a hole. Holomorphic functions are "defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point." Clearly, you cannot differentiate at a pole. Perhaps the author meant a non-holomorphic function?

Where do you see the word "hole" in the article? If you ask how you can have a holmorphic function defined around a hole, that's very simple, the hole is not in the domain of the function. So the domain of the function is say an open disk without its center, which is an open set. Oleg Alexandrov (talk) 02:09, 13 April 2007 (UTC)

We always talk about meromorphic functions when we talk about poles we never says "Poles of a holomorphic function" we always say "Pole of a meromorphic function". We talk about holomorphic functions as functions with no pole on their domain. By omitting the pole from the domain you may say that the function become holomorphic but studying the holomorphic functions does not include studying the poles. This is in the area of study of the behavior of meromorphic function. I think the first sentence should change to "pole of a meromorphic function" i think. ~ —Preceding unsigned comment added by Bossudenotredame (talk • contribs) 04:32, 7 February 2008 (UTC)

I have talked with my professor in Riemann Surfaces and Theta function, Dr. Marco Bertola, and he agreed that it is much more appropriate. So I'm going to change it to meromorphic functions Bossudenotredame (talk) 04:49, 12 March 2008 (UTC)

Use of 'Z'

The use of the variable 'Z' is somewhat confusing after browsing articles on DSP filtering and the Z transform, for example. I would have thought 's' was more appropriate. That's all. Raidfibre (talk) 14:07, 1 June 2009 (UTC)

Poles of order 0

The article says "a pole of order 0 is a removable singularity". However, the definition implies that a pole of order 0 could be either a removable singularity or a non-singular point. This could be rectified by modifying the definition to explicitly state that a pole has to be a singularity, but I am not sure whether there is any point in such an artificial convention: it seems to me much more natural to exclude this case by requiring $n$ to be positive, rather than merely non-negative. I have no memory of ever having come across the concept of a pole of order zero before, so I have looked around to see what, if anything, is the standard usage. I found no definitions in books which allowed $n$ to be zero, and several which specified it had to be positive: I give a couple of example citations below. Online I found no definitions which allow $n$ to be zero except for ones which had wording identical to this Wikipedia article's wording or so similar that they were clearly not independent. Consequently I shall amend the definition in the article.

Here are a couple of books which specify $n$ > 0:

Schaum's Outlines: Complex Variables By Murray R Spiegel: McGraw-Hill (1964) (ISBN-10: 0070602301, ISBN-13: 978-0070602304) Page 67

Functions of a complex variable: theory and technique By George F. Carrier, Max Krook, and Carl E. Pearson: Society for Industrial Mathematics (ISBN-13: 978-0-898715-95-8, ISBN-10: 0-89871-595-4) JamesBWatson (talk) 10:24, 17 September 2009 (UTC)

Google books finds a few that have that usage, ie meaning a regular point. So does Scholar. SpinningSpark 14:20, 22 October 2009 (UTC)

Yes, you have made it clear that this usage does exist; thank you. It is worth mentioning for anyone who wishes to follow the above links that most of the hits for "pole of order zero" on both Google Books and Google Scholar do not refer to this meaning; for example some of them refer to poles in electromagnetic contexts, with a totally different meaning, and other pure mathematical meanings also occur. Also, where the meaning is the one referred to above, the full wording often suggests that the author does not regard the usage as 100% standard, as in "for convenience, we call a removable singularity a pole of order zero", and "this corresponds to ... regarding a regular point as a pole of order zero". However, the usage clearly does exist, and a very few authors include this in their definition of "pole" (e.g. Olof J. Staffans in "Well-posed linear systems"). Nevertheless it still seems to me better to leave it out of the definition in the article, both because it seems to be a minority usage and because that definition would require extra complications in some of the later parts of the article to make them consistent with the definition. JamesBWatson (talk) 14:52, 24 October 2009 (UTC)

Sorry, I had realised that some of the hits were for point charges, and should have mentioned it. I could not immediately think of a simple search string to filter them apart and assumed that readers here would be bright enough to see that for themselves. SpinningSpark 15:02, 24 October 2009 (UTC)

No criticism intended for not mentioning it, and yes, I agree that readers could realise it for themselves. However, I saw no harm in mentioning it; it is not out of the question that someone might just glance at the number of Google hits and be mislead, I have been as careless as that myself in the past. JamesBWatson (talk) 09:22, 27 October 2009 (UTC)

On reflection I have decided that, since the alternative definition (including "pole of order zero") exists, it should get at least a mention in the article, and so I have put one in. However, I still think it better to give most prominence to the more usual definition. JamesBWatson (talk) 09:28, 27 October 2009 (UTC)

Order of a pole

Shouldn't it be the largest, not the smallest, value of n? What is the order of 1/z^2 + 1/z. The definition we were given says 2. The definition listed here says 1.130.95.128.51 (talk) 05:49, 22 October 2009 (UTC)

It means the smallest n in the sense that g(z) does not contain the factor (z-a). A larger n is possible if this factor exists, but the order of the pole is not increased. In your example,

${\frac {1}{z^{2}}}+{\frac {1}{z}}={\frac {1+z}{z^{2}}}$

which does indeed have an order of 2. However,

${\frac {z(1+z)}{z^{2}}}$

has an order of one because z² does not have the lowest value of n possible.

SpinningSpark 13:59, 22 October 2009 (UTC)

Nearly-illegible math expression

In the first sentence on the page, it appears that the expression [reciprocal of z to the power of [pi]] appears, but with normal-sized text, the [pi] is essentially illegible. This is probably simple to fix, but I don't know how. (I'm using [ ] in ad hoc fashion, here.) Readers who know the subject shouldn't have trouble, but others would, I'd venture to say.

Regards, Nikevich 19:40, 25 September 2012 (UTC)

After move and merge

I have merged Zero (complex analysis) and Pole (complex analysis) into this new article. I have also rewritten the lead accordingly. The remainder of the article still deserve being updated, improved and expanded. D.Lazard (talk) 21:41, 14 January 2018 (UTC)

Order of vanishing

Order of vanishing (edit | talk | history | protect | delete | links | watch | logs | views) redirects here, but the is not explained. Can someone explain it, please? — Sebastian 16:41, 17 January 2018 (UTC)

It refers to the order (or multiplicity) of a zero. (I see that now "order of vanishing" is there, maybe added just after your comment.) — MFH:Talk 23:29, 21 May 2018 (UTC)

Lead section

The current lead section has a very confusing order and choice of content. The first couple sentences start by saying that a pole is a singularity but not an essential singularity or a branch point, and zeros aren't introduced until later. This is a vague and confusing definition, and makes the whole lead section disorienting.

Novice readers can't be expected to know what various types of singularities are (or even what "singularity" means), and even with examples included a novice is not going to immediately understand the difference from "relatively regular" behavior. Luckily an editor just corrected the long-term mistake introduced in October 2020 where 0 for the logarithm function (a logarithmic branch point) was given as a canonical example of an essential singularity, implied not to be a branch point.

It is significantly clearer to define zeros immediately at the beginning of the article, explaining that a zero is a point where a function takes the value 0. Whether or not to mention that the function must be holomorphic or complex analytic seems much less important than getting this basic point correct at the start. But if we are considering comprehensibility of jargon, the phrase "relatively regular" is not any more meaningful for less-technical readers than "meromorphic"; it just replaces a well-defined jargon word with a hand-wave that is equally unclear to everyone.

It seems fine to introduce poles either in the first paragraph or later, but a pole should immediately be defined as a point where the function's reciprocal is 0. This is a much more obvious concept for less-technical readers than the awkward hand-wave this article currently puts in its first sentence.

Whether or not we use the word 'meromorphic' in the lead section, including a paragraph defining meromorphic formally there is sort of structurally awkward, very unusual for a wikipedia article. We should perhaps replace such a definition by a shorter informal one in the lead and move any formal definition later.

The comparisons to other kinds of singularities and examples of those isn't really necessary to include in the lead at all (much less in the first sentence). The article could have a dedicated section about types of singularities ideally including figures.

Ping D.Lazard. –jacobolus (t) 16:48, 6 December 2022 (UTC)

The lead was also unnecessarily pedantic. I agree with the removal of the mention of other types of singularity. Also "zero of a function" needs to be linked, but has not to be defined here, as a reader that does not know what is a zero is likely to understand nothing in the article. I have rewritten the lead along these lines. D.Lazard (talk) 23:11, 6 December 2022 (UTC)

I find the rewrite to be no better than the previous version, from either a clarity / coherence of organization perspective, or an accessibility to non-experts perspective.

The word 'singularity' and the phrases 'holomorphic in some neighbourhood' and 'complex differentiable' are all jargon, no more comprehensible than 'meromorphic'. Saying it is the 'simplest' type of singularity is not meaningful to someone who doesn't know what a singularity is (and is unhelpful to someone who does). Explaining that the sum of poles equals the sum of zeros for a rational function seems like it should be deferred to later. The bits about neighborhoods and open sets are unnecessary here in the lead of this article (the definition at holomorphic function already clarifies that 'holomorphic' means in some domain.

I think the article should in the first sentence define zero (and bold it) as a point where a holomorphic function takes the value 0 (with or without using the word 'holomorphic'), then define pole as a zero of the reciprocal function. Adding 'technically' is not useful.

Instead of talking about holomorphic/meromorphic functions, one alternative might be to say that near a zero a function locally behaves like $a(z-z_{0})^{n}$ for some complex constant a and some positive integer n, and near a pole locally behaves like $a(z-z_{0})^{-n}$. –jacobolus (t) 03:49, 7 December 2022 (UTC)