Uniform asymptotic expansions for oblate spheroidal functions II: negative separation parameter λ

Abstract

In [3], uniform asymptotic expansions were derived for solutions of the oblate spheroidal wave equation (z² − 1)d²p/dz² + 2zdp/dz − (λ + μ²/(z² − 1)) p = 0, for the case where the parameter μ is real and non-negative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u → ∞, uniform asymptotic expansions were derived involving elementary, Airy and Bessel functions, these being valid in certain subdomains of the complex z plane. In this paper the complementary case, where λ is real and negative, is considered. Asymptotic expansions are derived which are valid in certain subdomains of the half-plane |arg (z)| ≦ π/2, uniformly valid for u → ∞ with λ /u² fixed and negative, and 0 ≦ μ/u ≦ − ½λ /u² − δ, where δ is an arbitrary positive constant. Explicit error bounds are available for all the approximations.