Rigorous asymptotics for the Lamé, Mathieu and spheroidal wave equations with a large parameter
Ogilvie, Karen Anna
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We are interested in rigorous asymptotic results pertaining to three different differential equations which lie in the Heun class (see  §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 < k < 1, sn(z, k) is a Jacobi elliptic function, and K = ∫ 1 0 dz/√(1 − z²)(1 − k²z²) is the complete elliptic integral of the first kind. We obtain rigorous uniform asymptotic approximations complete with error bounds for the Lamé functions Ecm/v (z, k²) and Esm/v+¹ (z, k²) for z ∈ [0,K] and m ∈ N0, and rigorous approximations for their respective eigenvalues am/v and bm/v+¹, as v →∞. Then we obtain asymptotic expansions for the Lamé functions complete with error bounds, which hold only in a shrinking neighbourhood of the origin as v →∞. We also find corresponding expansions for the eigenvalues complete with order estimates for the errors. Then finally we give rigorous result for the exponentially small difference between the eigenvalues bm/v+¹ and am/v as v →∞. Second we investigate Mathieu’s equation d²w/dz² + (λ − 2h² cos 2z) w = 0, z ∈ [0, π], and obtain analogous results for the Mathieu functions cem(h, z) and sem+1(h, z) and their corresponding eigenvalues am and bm+1 for m ∈ N0 as h→∞, which are derived from those of Lamé ’s equation by considering a limiting case. Lastly we investigate the spheroidal wave equation d /dz ((1 − z²) dw/dz) + ( λ+ γ²(1 − z²) − μ²/1 − z²) w = 0, z ∈ [−1, 1], and consider separately the cases where γ² > 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.