What is this thing called ``Painlevé''?
ordinary points 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 \( y^{(n)}=F(y^{(n-1)}, ..., y', y, x), \quad (0.3)\) 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 \(x=x_0\). Consider the example (compare this with Equation (0.2) in Section 0.1)\( y'=- y/x \qquad\qquad (0.4)\) Clearly \(x_0=0\) is a regular singular point. The general solution is \(y=c/x\).
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