Large mode number eigenvalues of the prolate spheroidal differential equation

摘要

We show that the eigenvalues of the prolate spheroidal differential equation of zeroth order, chi(n)(c) where c is the so-called "bandwidth" or "ellipticity" parameter, are well-approximated for large mode number n by a single function 0 in the form chi(n) similar to C(*)(2)Omega(c/c(*)) where c(*) equivalent to (pi/2)(n + 1/2). Omega is defined implicitly as the root of an algebraic equation, E(min(1, m(-1/2)),m(1/2)) = 1/rootOmega where E is the usual incomplete integral of the second kind and m = gamma(2)/Omega with gamma = c/c(*). Omega has a weak singularity at gamma = 1 proportional to (gamma - 1) log(gamma - 1) plus iterated logarithms. We give Chebyshev series for Omega accurate for gamma is an element of [0, infinity]. (C) 2003 Elsevier Inc. All rights reserved. [References: 15]