Irregular singular point

A singular point of a linear differential equation is irregular when it fails the regular-singular criterion. For $y^{\prime \prime }+P(x)y^{\prime }+Q(x)y=0$, a pole of order greater than one in $P$ or greater than two in $Q$ makes the point irregular.

For a first-order meromorphic linear ordinary differential equation $Y_t=A(t)Y$, with a coefficient pole of order $p>1$ in a fixed local coordinate and gauge, its Poincaré rank is $p-1$. At infinity one must transform the differential system, not just the matrix-valued function: $Y_{\lambda }=A_0{\lambda }^{m}Y+\cdots$ becomes $Y_t=-A_0{t}^{-m-2}Y+\cdots$ under $t=1/\lambda$, giving rank $m+1$. A singular gauge can lower an apparent rank; distinct leading eigenvalues prevent the elementary degeneracy in the unramified setting.