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 explained. 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 degree before you can get anything. My math book ("elementary differential equations". 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)

Shouldn't the portion of the article which states, "When a is a regular singular point, which by definition means that p_{n-i}(z)\, has a pole of order at most i at a..." be corrected to "When a is a regular singular point, which by definition means that p_{n-i}(z)\, has a pole of order at most (n-i) at a..." ???

Multi-dimensional equations.

There is a multi-dimensional part of the theory that worth mentionning: instead of considering one-dimensional differential equation of order n, one can consider n-dimensional first-order differential equation:

${\dot {z}}=A(t)z,\quad z\in C^{n},\,t\in C.$

Then, there are two distinct notions:

• regular singular point is an isolated singularity of A where all the solutions grow within sectors at most polynomially;
• Fuchsian singular point is an isolated singularity of A which is its pole of first order.

It is rather simple to show the implication "Fuchsian => regular" (via Gronwall's Lemma). The other implication, contrary to the one-dimensional case, is false: there are non-Fuchsian regular singularities. To construct one, it suffices to take a Fuchsian singularity, and to make a meromorphic change of variable $w=U(t)z$ with U(t) having a pole of a high order; then the new matrix B(t) will have a high-order pole, too.

This part of the theory is rather important: for instance, the Riemann-Hilbert problem considers monodromy groups for such equations. Also, a nice proof of regularity in one-dimensional case is via reduction to the multi-dimensional equation by considering the vector

$z(t)=(f(t),tf'(t),t^{2}f''(t),\dots ,t^{n-1}f^{(n-1)}(t)).$

May I suggest adding this into the article? (I'm afraid my English is not fast enough to do it myself -- I can put it into my "TODO", but no guarantee on when will it be realized...) Burivykh (talk) 00:20, 12 May 2009 (UTC)

Inhomogeneous Linear Equations

Is there a theory of regular/irregular singular points for inhomogeneous linear differential equations? I looked around the web for a few minutes and didn't find anything clear, so I figured I might as well ask some experts here. If so, it would be good to mention it on the page. — Preceding unsigned comment added by 134.134.139.74 (talk) 16:09, 27 July 2018 (UTC)