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Uniform asymptotic expansions for oblate spheroidal functions. I: Positive separation parameter $\lambda$. (English) Zbl 0781.33010

The author considers the oblate spheroidal wave equation

$\left({z}^{2}-1\right)\frac{{d}^{2}p}{d{z}^{2}}+2z\frac{dp}{dz}-\left(\lambda +\frac{{\mu }^{2}}{{z}^{2}-1}-{\gamma }^{2}\left({z}^{2}-1\right)\right)p=0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $\lambda >0$, $\mu \ge 0$ and $\gamma =iu$ $\left(u>0\right)$ are given parameters. His aim is to derive asymptotic expansions for solutions of (*), which are uniformly valid for $u\to \infty$ in certain subdomains of $-\pi . To this end he assumes that $\lambda /{u}^{2}$ remains fixed and lies in the interval (0,2), more precisely:

$0\le \frac{\mu }{u}\le \frac{1}{2}\frac{\lambda }{{u}^{2}}-\delta \phantom{\rule{1.em}{0ex}}\text{when}\phantom{\rule{1.em}{0ex}}0<\frac{\lambda }{{u}^{2}}<1$

while

$\sqrt{\lambda /{u}^{2}-1}+\delta \le \frac{\mu }{u}\le \frac{1}{2}\frac{\lambda }{{u}^{2}}-\delta \phantom{\rule{1.em}{0ex}}\text{when}\phantom{\rule{1.em}{0ex}}1\le \frac{\lambda }{{u}^{2}}<2,$

where $\delta >0$ is an arbitrary small constant. By applying three different Liouville transformations he obtains three types of expansions, which involve elementary, Airy and Bessel functions, respectively.

##### MSC:
 3.3e+16 Other wave functions 3.4e+21 Asymptotic singular perturbations, turning point theory, WKB methods (ODE)