Abstract: | We provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions (PSWFs and OSWFs), Smn ( 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 Smn ( c , η) for large, complex c are also given. |