Knowlpedia

Definition

Regular singular point of a second-order linear ODE

A point where the first- and zeroth-order coefficients have at most first- and second-order poles after normalization.

For y′′+p(z)y′+q(z)y=0y''+p(z)y'+q(z)y=0 on a punctured disc about zero, the origin is a regular singular point if zp(z)zp(z) and z2q(z)z^2q(z) extend to zero. Equivalently, the equation has the form

z2y′′+zP(z)y′+Q(z)y=0z^2y''+zP(z)y'+Q(z)y=0

with P,QP,Q holomorphic near zero. An ordinary point, where p,qp,q themselves extend, is included in some conventions and excluded in others.

Leading powers

Substitution of y=zry=z^r into the lowest-order terms gives the indicial equation r(r−1)+P(0)r+Q(0)=0r(r-1)+P(0)r+Q(0)=0. The Frobenius method seeks a power times a convergent series; resonances can require logarithmic terms. A regular singular equation need not have every solution smooth at zero. A particular regular branch must be selected by its boundary or normalization condition.