On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

We provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions (PSWFs and OSWFs), S_mn(c, η), and their eigenvalues, λ_mn, for arbitrary complex size parameter c in the asymptotic regime of large |c|, m and n fixed. The ability to calculate these SWFs for large and complex size parameters is important for many applications in mathematics, engineering, and physics. For arbitrary arg(c), the PSWFs and their eigenvalues are accurately expressed by established prolate-type or oblate-type asymptotic expansions. However, determining the proper expansion type is dependent upon finding spheroidal branch points, c^mn_○;r, in the complex c-plane where the PSWF alternates expansion type due to analytic continuation. We implement a numerical search method for tabulating these branch points as a function of spheroidal parameters m, n, and arg(c). The resulting table allows rapid determination of the appropriate asymptotic expansion type of the SWFs. Normalizations, which are dependent on c, are derived for both the prolate- and oblate-type asymptotic expansions and for both (n−m) even and odd. The ordering for these expansions is different from the original ordering of the SWFs and is dictated by the location of c^mn_○;r. We document this ordering for the specific case of arg(c) =π/4, which occurs for the diffusion equation in spheroidal coordinates. Some representative values of λ_mn and S_mn(c, η) for large, complex c are also given.