Asymptotic Solutions of Differential Equations with Transition Points or Singularities

Asymptotic solutions of d2y/dx2+[λ2p(x)+r(x,λ)]y=0 are found when λ is a large parameter and r is ``small'' in comparison with λ2p, except at a single point where either p has a simple zero, or p a pole of the first order and r a pole of the second order. The results are applied to Bessel functions, and to Hermite and Laguerre polynomials. The resulting asymptotic forms are valid uniformly in x.