Talk:Analytic continuation
From Academic Kids
Ughh. Someone should copy the bottom half of the article on Riemann surfaces into this article. The article on Riemann surfaces should be trimmed to remove the parts about analytic continuation, and just point to this article instead. Who is up for that? linas 19:43, 22 Jan 2005 (UTC)
- There also should be a mentioning of the generalization to meromorphic continuation since the link points to this article currently. - Gauge 05:16, 20 May 2005 (UTC)
extending a relation by transitivity
Can someone explain what is meant by "extending by transitivity"? Does it mean transitive closure? Do we also have to do some kind of "symmetric closure" to make this relation symmetric, as it must if we're to make an equivalence relation? -Lethe | Talk 22:33, May 31, 2005 (UTC)
Hmmm - 'not symmetric' is true, as defined. But the transitive closure is symmetric. This may really need a picture. As defined, g ≥ h can happen and the radius of convergence of h can be much smaller than that of g: so the power series for h can't 'reach' far enough. But by taking enough small steps with other functions, one can 'reach' the point about which g is defined. Say for example g is defined at 0, h at 1, and they both represent a function that has a singularity at 1.01. Then the radius of convergence of h is (at most) 0.01. You can expand h about a point like 0.99, and get a slightly larger radius of convergence, 0.015. Expand that about 0.98, say, with radius of convergence 0.02. Continuing this way, one gets the relation h R g where R is the transitive closure of ≥. Charles Matthews 06:20, 1 Jun 2005 (UTC)
- Hmm. Let me see if I have it. Transitivity fails because b can be in a's radius of convergence, and c can b in b's, but c need not be in a's. Our transitive closure will have a>c when there is an intermediate chain of germs that arrive at c. And I can sort of imagine how we can use a similar chain to see that symmetry holds too, once we have transitivity. A chain of germs getting bigger until they get a's center. Or something. I'm going to add some words to that sentence. -Lethe | Talk 11:09, Jun 1, 2005 (UTC)
