Talk:Regular singular point

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I don't know why it is thought that the Newton polygon connection is 'not worth mentioning'. This aspect occurs in contemporary research (e.g. [1]). Charles Matthews 10:06, 2 April 2006 (UTC)

Perhaps 'not worth mentioning' is a bit too subjective. Nevertheless, the paragraph was, in my opinion, not clearly enough explaned. What are the axes? Did i refer to the index of the coefficient? etc.... Also, there are many point worth mentioning of which this is just one. I don't think it is relevant for a short discussion on regularity. I hope my deletion doesn't annoy you too much. GeometryJim 10:45, 2 April 2006 (UTC)

Well, it annoyed me. If there are many other points worth mentioning, why not mention some? One starts an article, and hopes to see it go forward, not just to be picked apart. Charles Matthews 19:28, 2 April 2006 (UTC)

Fair enough. I'll put it back GeometryJim 13:48, 3 April 2006 (UTC)

Isn't there a more simple way to talk about regular singular points when it comes to ODE's? This article reads like you need a math degeree before you can get anything. My math book ("elementary differential eqations". Boyce. DiPrima, 7ed 2001.) define says that for a second order linear ode p(x)y' ' + q(z)y' + r(x)y = 0, x0 is regular singular if lim(x - x0)q/p->0 is finite and lim(x - x0)^2*r/p is finite. this is the related to euler equations and frobenius series. anton

It's true this definition of "regular singular point" is a bit sophisticated:

Then amongst singular points, an important distinction is made between a regular singular point, where there are meromorphic function solutions in Laurent series, and an irregular singular point, where the full solution set requires functions with an essential singularity.

I also think it's wrong!

Saying a differential equation has a regular singular point if it has "meromorphic function solutions in Laurent series" is a (not terrifically clear) way of saying that its solutions have at worst a pole of finite order at the point in question. But later, the article comes out and admits that:

To be strictly accurate, solutions will be a Laurent series at a multiplied by a power

(z − a)^r

where r need not be an integer

In other words, "to be strictly accurate", the solution can have not just a pole but a branch point! And indeed, it's branch points that really come up in practice, e.g. in the Riemann-Hilbert problem. So, making the reader fight through the whole article to see this is a bit unfair.

The definition given by Boyce and dePrima is fine for 2nd-order ODE, no good for higher-order ones. But, I bet most people reading this will be interested in the 2nd-order case. So, it might be good to start with that case and then move on to the general case.

I'd fix the article myself if I were 100% sure I knew what I was doing.

--John Baez 21:38, 25 August 2007 (UTC)