Rigorous asymptotics for the Lamé, Mathieu and spheroidal wave equations with a large parameter
Karen Ogilvie · ERA · 2016
We are interested in rigorous asymptotic results pertaining to three different differential equations which lie in the Heun class (see [1] §31). The Heun class contains those ordinary linear second-order differential equations with four regular singularities. We first investigate the Lamé equation d²w/dz² +(h − v(v + 1)k² sn²(z, k)) w = 0, z ∈ [−K,K], where 0 0 and γ² < 0. In the first case we give similar results to those previously for the prolate spheroidal wave functions Ps(z, γ²) and their corresponding eigenvalues λm/n for m, n ∈ N0 and n ≥ m as γ² →∞, and in the second we discuss the gap in theory which makes it difficult to obtain rigorous results as γ² → -∞, and how one would bridge this gap to obtain these.