What is this thing called ``Painlevé''?

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Regular Singular Points and Movable Singularities

Recall that we are focussing on scalar ODEs governing a function y(x) of a complex variable x such as

y^{(n)}=F(y^{(n-1)}, ..., y', y, x),

where F is rational in y^{(n-1)}, ..., y', y and analytic in x except at a finite number of isolated points x_i, i=0, ..., s-1 where s is a nonnegative integer.

Definition: Suppose F is linear in y^{(n-1)}, ..., y', y:

F=G_(n-1)y^{(n-1)}+ ... + G_0 y,

where G_k are functions of x at least one of which is not analytic at x=x_0. Then x_0 is a regular singular point if

(x-x_0)^{n-k}G_k, k=0, ..., n-1

are all analytic at x=x_0.

An example is

y'=- y/x

Clearly x_0=0 is a regular singular point. The general solution is y=c/x.


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Last modified: 16 August 2002 by
N.Joshi