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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 $$(z\sp 2-1) {{d\sp 2p} \over {dz\sp 2}} +2z {{dp} \over {dz}}- \left(\lambda+ {{\mu\sp 2} \over {z\sp 2-1}} -\gamma\sp 2(z\sp 2-1)\right)p=0, \tag *$$ where $\lambda>0$, $\mu\geq 0$ and $\gamma=iu$ $(u>0)$ 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 $-\pi<\arg z\leq\pi$. To this end he assumes that $\lambda/u\sp 2$ remains fixed and lies in the interval (0,2), more precisely: $$0\leq {\mu \over u} \leq {1\over 2} {\lambda\over {u\sp 2}}-\delta \quad \text{when} \quad 0< {\lambda \over {u\sp 2}}<1$$ while $$\sqrt{\lambda/u\sp 2-1} +\delta\leq {\mu\over u}\leq {1\over 2} {\lambda \over {u\sp 2}}-\delta \quad \text{when} \quad 1\leq {\lambda\over {u\sp 2}}<2,$$ 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.

33E15Other wave functions
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
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