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Assessment: banner shell, Mathematics (Rater)

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== how big can the set of poles be ==

must the set of poles be finite? -- [[User:Tarquin|Tarquin]]

Nope. 1/sin(''z'') is meromorphic. [[User:AxelBoldt|AxelBoldt]]

I don't think I completely understand the definition.

Can there be an infinite number of poles?
And if so, do they have to be countable? --anon

: Yes, they can be an infinite set. Hopefully my rewrite shows that.

They really do have to be isolated? --anon
Yes, by definition. If they are not isolated, it is impossible to prove that a meromorphic function is a ratio of two holomorphic functions. [[User:Oleg Alexandrov|Oleg Alexandrov]] 17:28, 11 August 2005 (UTC)

== the number of poles of a meromorphic function must not be countable? ==

http://mathworld.wolfram.com/MeromorphicFunction.html

here they speak of

if this a different definition or is it some non trivial theorem that if your definition is true, the number of poles is countable

thanks --anon

: I don't see any contradiction between our page and mathworld. The poles of a meromorphic function must be isolated, by definition. Now, one can prove a theorem saying that a set of isolated points is finite or countable. So, is it the proof of this theorem that you are interested in? [[User:Oleg Alexandrov|Oleg Alexandrov]] 20:44, 11 August 2005 (UTC)

== Poles of sin(1/z) ==

Outside of every neighbourhood of origin the function sin(1/z) is bounded in bounded sets. Thus origin cannot be it's accumulation
point of poles. Perhaps here is a typo and it should be 1/sin(1/z)? —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:J Kataja|J Kataja]] ([[User talk:J Kataja|talk]] • [[Special:Contributions/J Kataja|contribs]]) 10:01, 9 January 2008 (UTC) 

:You're right. The function <math>\sin(1/z)</math> is actually not meromorphic in the origin, but for a different reason. -- [[User:EJ|EJ]] ([[User talk:EJ|talk]]) 12:00, 9 January 2008 (UTC)

== Elliptic Functions ==

The statement concerning elliptic functions and elliptic curves sounds wrong. What "elliptic curves" did the author have in mind? An ellipse in the plane? Then it is simply wrong: elliptic functions are the inverse functions of the functions used to calculate the area of an ellipse.

Also elliptic functions are defined on a "period parallelogram", i.e. a fundamental region of a discrete lattice in the complex plane. That does not look like an "elliptic curve" to me. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/204.119.233.250|204.119.233.250]] ([[User talk:204.119.233.250|talk]]) 21:16, 8 February 2008 (UTC) 
:See [[elliptic function]]. Is is not about ellipses. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 04:11, 9 February 2008 (UTC)

== Is meromorphic = regular? ==

This article says that "[meromorphic] functions are sometimes said to be regular functions or regular on D." However, I was unable to find a source which defines 'regular' synonymous to 'meromorphic'. Rather, most sources say that 'regular' = 'holomorphic'.[http://planetmath.org/encyclopedia/Holomorphic.html][http://eom.springer.de/a/a012240.htm][http://mathworld.wolfram.com/RegularFunction.html] Especially, the last source is mathworld which [[User:Linas]] submitted as a reference when s/he [http://en.wikipedia.org/w/index.php?title=Meromorphic_function&diff=50021368&oldid=49528032 wrote the sentence]. If nobody opposes, I'll remove the sentence within a week, which ''may'' be added to [[holomorphic function]] article with a slightly changed wording. --[[User:Acepectif|Acepectif]] ([[User talk:Acepectif|talk]]) 19:15, 16 March 2008 (UTC)

:Heh. Neither planetmath nor mathworld state what you claim they state. Springer does call a holomorphic function a "regular analytic function"; however, it is generally understood that meromorphic functions are regular except at those points at which they're not. :-) This is just common(-sense) usage; the singular points are not regular, so a function is never regular at its singular points, and so a meromorphic function is regular everywhere where its not singular. If you can figure out a way of saying this simply, that would be good.

:Please note that the phrase "regular function" enjoys broader usage than just in complex analysis; so, for example, in [[algebraic geometry]], the points of a variety that are not singular are in fact regular, and so, by analogy the functions there are also called "regular functions". (Varieties have, of course the equivalent concept of meromorphic functions as wel, etc.) [[User:Linas|linas]] ([[User talk:Linas|talk]]) 21:38, 16 March 2008 (UTC)

::First of all, it is that you should prove with sources that 'regular' = 'meromorphic' (at least often enough to be mentioned in this article), rather than that I should prove that 'regular' = 'holomorphic', to prohibit me from deleting your sentence. Despite this, if I prove the latter, it will be obvious that your sentence must change its wording, if not deleted. So let's see whether planetmath and mathworld state what I claim they state.
::For [http://planetmath.org/encyclopedia/Holomorphic.html planetmath], see the bottom of this 'holomorphic' page. "Other names: holomorphic function, regular function, complex differentiable" - these are all synonyms. You can't find the word 'regular' from the [http://planetmath.org/encyclopedia/Meromorphic.html 'meromorphic' page] of this site. Also, mathworld clearly state that "A function is termed regular iff it is analytic and single-valued throughout a region R." If it saw 'regular' is 'meromorphic', it would have said "throughout a region R except a set of isolated points."
::Even though I felt that it may be somewhat out of topic, I also wanted to mention the usage in algebraic geometry when I wrote the post above. In algebraic geometry, [[regular functions]] are everywhere-defined polynomials, i.e. they are analogues of holomorphic functions, ''not'' meromorphic ones. Rational functions or fractions of regular functions may correspond to meromorphic functions. --[[User:Acepectif|Acepectif]] ([[User talk:Acepectif|talk]]) 04:17, 17 March 2008 (UTC)

:::Sounds good to me. Do it. [[User:Linas|linas]] ([[User talk:Linas|talk]]) 16:24, 17 March 2008 (UTC)

== Strange statement ==

The article states:

* Somewhat similarly, the function
::<math> f(z) = \frac{z}{e^z - 1}</math>
: has singularities at all points in the form <math>z = 2 n \pi i</math>, for <math>n \in \mathbb{Z}</math>. However, it is not meromorphic on all of <math>\mathbb{C}</math>, since the singularity at <math>z = 0</math> is [[removable singularity|removable]]: <math>\lim_{z \rightarrow 0} f(z) = 1</math>.

Why cancellation of a zero of denominator by the zero of the numerator breaks the meromorphicity?
I would say, such <math>f</math> is meromorphic. [[User:Domitori|dima]] ([[User talk:Domitori|talk]]) 00:54, 30 September 2008 (UTC)

:Well, the definition as given in the article indeed requires all singularities to be poles, disallowing removable singularities. (OTOH, the extension of ''f'' obtained by putting ''f''(0) = 1 ''is'' meromorphic on <math>\mathbb C</math>.) For a more striking example, if you take the function which is constant 0 on <math>\mathbb C\setminus\{0\}</math>, and undefined in 0, it is not meromorphic on <math>\mathbb C</math> according to the definition. I also find this convention counterintuitive, but I am not in a position to judge which definition is more standard. — [[User:EmilJ|Emil]] [[User talk:EmilJ|J.]] 11:30, 3 October 2008 (UTC)

:: But a removable singularity ''is'' a pole, namely a pole of order 0. See e.g. the article on [[Pole_(complex_analysis)|poles]], which states: ''A pole of order 0 is a removable singularity''. And there is no requirement for the poles of a meromorphic function to have order of ''at least'' 1. Furthermore, this restriction would contradict the statement, that ''all rational functions are meromorphic'' (x/x is certainly rational and has a removable singularity at 0). I think this example is wrong and should be removed. -- (ezander) [[Special:Contributions/134.169.77.186|134.169.77.186]] ([[User talk:134.169.77.186|talk]]) 07:58, 14 September 2009 (UTC)

I agree that this example is not a good one, and I have removed it. The argument that "the definition as given in the article indeed requires all singularities to be poles, disallowing removable singularities" is not valid, as the definition specifies that we are dealing with a function "on an open subset D of the complex plane", which presumably is meant to mean that the function ''is defined'' on that open subset, in which case any removable singularities are by definition excluded. In fact I shall amend the definition to make this explicit.

Incidentally, the concept of a pole of order 0 is not a standard one: every text I have examined requires the order of a pole to be positive, and if there are any that don't then I think they must be a small minority. [[User:JamesBWatson|JamesBWatson]] ([[User talk:JamesBWatson|talk]]) 10:36, 17 September 2009 (UTC)

:Sorry, but your reformulation does not make any sense. If the function is defined in all of ''D'', then it is holomorphic on ''D'', not just meromorphic. The whole point of the concept is that a function meromorphic on ''D'' can be ''undefined'' in some points of ''D'' (which are isolated singularities). The only question is whether it is required to have poles (of positive order) in all these undefined points, or whether it can have removable singularities there. As for the poles of order 0 issue: removable singularities can also be zeros of positive order, and these are not poles of order 0, they would have to be "poles of negative order". Somehow I do not think that introducing those is a terribly good idea. — [[User:EmilJ|Emil]] [[User talk:EmilJ|J.]] 12:01, 17 September 2009 (UTC)
::Well, yes I see what you mean, and I did not express what I was trying to say accurately. Certainly it is possible to make what I said correct by taking the range of the function to be the extended complex plane, which personally I think is by far the best treatment, but I accept this is probably not the usual approach. Would it be acceptable to say "holomorphic on all D except a set of isolated points, which are poles or removable singularities"? Whatever the form of words it seems to me that removable singularities do not stop a function from being meromorphic. [[User:JamesBWatson|JamesBWatson]] ([[User talk:JamesBWatson|talk]]) 16:07, 17 September 2009 (UTC)
:::Fine with me. — [[User:EmilJ|Emil]] [[User talk:EmilJ|J.]] 17:22, 17 September 2009 (UTC)

== Series expansion of a meromorphic function ==
I dimly remember that there is a theorem which states that you can expand any meromorphic function into a summation over its poles and the residues at those poles. Surely this article here should mention such an important theorem? --[[Special:Contributions/173.180.169.187|173.180.169.187]] ([[User talk:173.180.169.187|talk]]) 16:06, 19 April 2013 (UTC)

== algebraic closure of field of meromorphuc functions? ==

If someone knows about that or perhaps could provide a reference for it at the bottom of the pae, that would be great.[[Special:Contributions/2601:7:6580:5E3:7D6D:997D:F92D:729D|2601:7:6580:5E3:7D6D:997D:F92D:729D]] ([[User talk:2601:7:6580:5E3:7D6D:997D:F92D:729D|talk]]) 11:37, 21 January 2015 (UTC)

== Unclear definition ==

I would clarify that the definition requires that the isolated singularities be poles rather than essential singularities. The fact that the phrase "the poles of the function" is in parentheses makes it ambiguous whether this is actually an additional nontrivial restriction on the definition, or simply a clarifying/explanatory comment for the reader which is being sloppy about distinguishing between poles and essential singularities.

== Analytic continuation ==

"By using analytic continuation to eliminate removable singularities": Removable singularities are by definition isolated singularities that can be eliminated by extending the function holomorphically. Mentioning the general concept of analytic continuation makes this seem more complicated than it is. Maybe one should mention [[Removable singularity#Riemann's theorem]] instead.
[[User:BSpringborn|BSpringborn]] ([[User talk:BSpringborn|talk]]) 09:19, 16 June 2017 (UTC)

== In the subsection on Riemann surfaces, it may be worth clarifying what the domains and codomains are ==

For example, I propose replacing the paragraph

> On a non-compact Riemann surface, every meromorphic function can be realized as a quotient of two (globally defined) holomorphic functions. In contrast, on a compact Riemann surface, every holomorphic function is constant, while there always exist non-constant meromorphic functions.

with

> On a non-compact [[Riemann surface]], every meromorphic function into the complex plane can be realized as a quotient of two (globally defined) holomorphic functions. In contrast, on a compact Riemann surface, every holomorphic function into the complex plane is constant, while there always exist non-constant meromorphic functions.

I had to double check the original paragraph because it didn't pass my smell test.

I haven't made this change. I'm concerned my version might be too wordy. — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Svennik|Svennik]] ([[User talk:Svennik#top|talk]] • [[Special:Contributions/Svennik|contribs]]) 21:21, 25 August 2019 (UTC) 

== "Meromorphic functions on an elliptic curve are also known as elliptic functions" ? ==

What is the reference for this, what does it mean to have "functions on an elliptic curve" and is the statement even true? [[User:Kotika98|Kotika98]] ([[User talk:Kotika98|talk]]) 12:53, 26 August 2021 (UTC)

Talk:Meromorphic function - Revision history

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