Uniform asymptotic solutions of boundary and eigenvalue problems for a second-order ordinary differential equation with singular and turning points (Q1108433)

A uniform asymptotic solution to the boundary value problem \[ u''(x,\lambda)+\lambda^ 2\hat R(x,\lambda)u=\lambda^ 2G(x,\lambda)u(z_ i,\lambda)+\eta u(z_ i,\lambda)=\sigma^{(i)},\quad i=1,2, \] and \(x\in I=[z_ 1,z_ 2]\), is developed by Langer's method. \(\hat R(x,\lambda)\) can have poles at the end points of I. The case in which \(\hat R(x,\lambda)=E- U(x,\lambda)\), where E is an unknown eigenvalue and U has poles in I, is also discussed. The asymptotic nature of the formal solution to the boundary value problem is investigated, and specific examples are solved.