UNIFORM ASYMPTOTIC EXPANSIONS FOR OBLATE SPHEROIDAL FUNCTIONS .2. NEGATIVE SEPARATION PARAMETER-LAMBDA

In [3], uniform asymptotic expansions were derived for solutions of th e oblate spheroidal wave equation dp/dz-(lambda+mu(2)/(z(2)-1)-gamma(2 )(z(2)-1))p=0, for the case where the parameter mu is real and non-neg ative, the separation parameter lambda is real and positive, and gamma is purely imaginary (gamma=iu). As u-->infinity, uniform asymptotic e xpansions were derived involving elementary, Airy and Bessel functions , these being valid in certain subdomains of the complex z plane. In t his paper the complementary case, where lambda is real and negative, i s considered. Asymptotic expansions are derived which are valid in cer tain subdomains of the half-plane \arg(z)\less than or equal to pi/2, uniformly valid for u-->infinity with lambda/u(2) fixed and negative, and 0 less than or equal to mu/u less than or equal to-1/2 lambda/u(2) -delta, where delta is an arbitrary positive constant. Explicit error bounds are available for all the approximations.