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Conical functions with one or both parameters large. (English) Zbl 0736.33002
Conical functions are associated Legendre functions P -1/2+iτ -μ (z), Q -1/2+iτ μ (z). Uniform asymptotic expansions are derived for large values of the parameters. As τ, expansions are given which involve Bessel functions of order μ. These expansions are uniformly valid for 0μAτ (A an arbitrary positive constant), and are also uniformly valid for (z)0 in the complex case, and 0z< in the real argument case. The case μ is also considered, and expansions are given which are uniformly valid in the same z regions for 0τBμ (B an arbitrary positive constant); in the case where (z)0 and 1z<, the expansions involve Bessel functions of purely imaginary order iτ, and in the case where 0z<1 the expansions involve elementary functions. The results are derived by using the differential equation of the associated Legendre functions.
MSC:
33C05Classical hypergeometric functions, 2 F 1
33C10Bessel and Airy functions, cylinder functions, 0 F 1
34E10Perturbations, asymptotics (ODE)
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)