Matched Uniform Approximations for a Singular Boundary Point and an Interior Turning Point

The singular perturbation problem for a second order ODE with singular boundary point and interior turning point is studied. Two uniform asymptotic approximations are separately constructed around singular point and turning point. The restricted uniform approximation about the turning point corresponds to O’Malley’s uniform reduction, while that about the singular point is found by a similar method and is entirely new. Higher order matching of both dominant and recessive terms of the two approximations leads to a complete asymptotic approximation of the general solution. A subsequent paper applies these results to higher order resonance criteria.