Talk:Meromorphic function

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must the set of poles be finite? -- Tarquin

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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.