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Uniform asymptotic expansions for oblate spheroidal functions. II: Negative separation parameter $\lambda$. (English) Zbl 0836.33012

The author reconsiders the oblate spheroidal functions satisfying the differential equation

$\left({z}^{2}-1\right){p}^{\text{'}\text{'}}+2z{p}^{\text{'}}-\left[\lambda +\frac{{\mu }^{2}}{{z}^{2}-1}-{\gamma }^{2}\left({z}^{2}-1\right)\right]p=0$

for the case where $\lambda <0$, $\mu \ge 0$, $\gamma =iu$ with $u>0$ and $|argz|\le \pi /2$, and gives their asymptotic expansions for $u\to \infty$ while $\lambda /{u}^{2}=\text{fixed}$. The expansions are uniformly valid in certain subdomains of the $z$-plane [Notice that the case $\lambda >0$ was already studied by the author himself, ibid. 121, No. 3/4, 303-320 (1992; Zbl 0781.33010)]. Explicit error bounds are given for all the approximations. Some asymptotic behaviour of spheroidal functions in terms of elementary functions of complex arguments are also derived. These latter are used to determine relationships between the parameters $\lambda$, $\mu$, $u$ and $\nu$ (the characteristic exponent).

##### MSC:
 33E15 Other wave functions 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE) 34M99 Differential equations in the complex domain
##### Keywords:
spheroidal functions