Talk:Meromorphic function

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must the set of poles be finite? -- Tarquin

Nope. 1/sin(z) is meromorphic. AxelBoldt

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[edit] how big can the set of poles be

I don't think I completely understand the definition.

Can there be an infinite amount 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. Oleg Alexandrov 17:28, 11 August 2005 (UTC)

[edit] the amount 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? Oleg Alexandrov 20:44, 11 August 2005 (UTC)

[edit] 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 unsigned comment added by J Kataja (talk • contribs) 10:01, 9 January 2008 (UTC)

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

[edit] 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 unsigned comment added by 204.119.233.250 (talk) 21:16, 8 February 2008 (UTC)

See elliptic function. Is is not about ellipses. Oleg Alexandrov (talk) 04:11, 9 February 2008 (UTC)

[edit] 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'.[1][2][3] Especially, the last source is mathworld which User:Linas submitted as a reference when s/he 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. --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.) 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 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 '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. --Acepectif (talk) 04:17, 17 March 2008 (UTC)
Sounds good to me. Do it. linas (talk) 16:24, 17 March 2008 (UTC)