Talk:Zeros and poles

message: "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)
thread_title: Singularity

message: 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)
replies: text: 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)
thread_title:  Formal Definition 

message: 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)
replies: text: 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)

text: 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)
thread_title: Hole?

message: 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)
thread_title:  Use of 'Z' 

message: 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)
replies: text: Google books finds a few that have that usage, ie meaning a regular point.  So does Scholar. SpinningSpark 14:20, 22 October 2009 (UTC)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Poles of order 0 

message: 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)
thread_title:  Nearly-illegible math expression 

message: 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)
thread_title:  After move and merge 

message: redirects here, but the is not explained. Can someone explain it, please? — Sebastian 16:41, 17 January 2018 (UTC)
replies: text: 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)
thread_title:  Order of vanishing