Talk:Picard theorem

Existence of Solutions to Linear Operators

In various wave-scattering related papers the cite "Picard's theorem" as a way of finding the range of a linear operator. E.g. here: [1] This seems to be different to the other Picard theorems already detailed. Shall I add a new page for it, or just add it on to this? 155.198.65.73 (talk) 13:40, 18 October 2010 (UTC)[reply]

Jargon

The statement of the theorem is accessible to anybody who understands the concept of an analytic function. The proof that is given in the article is not. --Aleph4 (talk) 11:54, 21 April 2009 (UTC)[reply]

I'm confused. Is it required that proofs of very hard theorems be made accessible to anybody who understands the concept of an analytic function? StatisticsMan (talk) 15:50, 25 April 2009 (UTC)[reply]

Not required, but of course desired, if possible.

If it is a very hard theorem, this should be stated clearly in the article. (We did not do the proof in my complex function theory class.) But then we cannot claim that the proof is easy. --Aleph4 (talk) 17:24, 8 May 2009 (UTC)[reply]

Here are a few specific items that should be clarified:

What is Gamma(2)?
What is $SL_{2}(\mathbb {Z} /2)$ ? Perhaps it is mentioned in SL2(C) under some different notation?
Do we know that all maps in the last paragraph in the proof (in particular the two factor maps) are holomorphic? This is probably some well-known theorem, so why not mention it? In any case we should mention unversal cover.

Aleph4 (talk) 14:13, 9 May 2009 (UTC)[reply]

The proof in Ahlfors is at the end of the book, page 306 out of 321, and he devotes an entire subsection to proving it, more than a page and a half. His proof requires the monodromy theorem, which he only proves in the last chapter of the book. I would guess there's no known proof of the theorem that's understandable by someone who just knows what an analytic function is, and the proof given here is about as simple as it gets. —Simetrical (talk • contribs) 17:19, 30 October 2009 (UTC)[reply]

One sentence is inaccurate, the other has no content.

One inaccurate sentence is as follows:

"In the case where the values of f are missing a single point, this point are called lacunary values of the function."

Well, no. In the case where the values of f are missing any points, such points are called "lacunary values" of the function.

And this sentence has no content:

"As with the little theorem, the (at most two) points that are not attained are lacunary values of the function. "

Sentences like these are misleading and create more confusion than anything else. I hope someone knowledgeable on this subject can improve the writing.50.234.60.130 (talk) 21:59, 10 December 2020 (UTC)[reply]