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

The author considers the oblate spheroidal wave equation

(z 2 -1)d 2 p dz 2 +2zdp dz-λ+μ 2 z 2 -1-γ 2 (z 2 -1)p=0,(*)

where λ>0, μ0 and γ=iu (u>0) are given parameters. His aim is to derive asymptotic expansions for solutions of (*), which are uniformly valid for u in certain subdomains of -π<argzπ. To this end he assumes that λ/u 2 remains fixed and lies in the interval (0,2), more precisely:

0μ u1 2λ u 2 -δwhen0<λ u 2 <1

while

λ/u 2 -1+δμ u1 2λ u 2 -δwhen1λ u 2 <2,

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