Orthogonal rational functions on a semi-infinite interval

Abstract

By applying a mapping to the Chebyshev polynomials, we define a new spectral basis: the “rational Chebyshev functions on the semi-infinite interval,” denoted by TL n ( y). Continuing earlier work by the author and by Grosch and Orszag, we show that these rational functions inherit most of the good numerical characteristics of the Chebyshev polynomials: orthogonality, completeness, exponential or “infinite order” convergence, matrix sparsity for equations with polynomial coefficients, and simplicity. Seven numerical examples illustrate their versatility. The “Charney” stability problem of meteorology, for example, is solved to show the feasibility of applying spectral methods to a semi-infinite atmosphere. For functions that are singular at both endpoints, such as K 1( y), one may combine rational Chebyshev functions with a preliminary mapping to obtain a single, exponentially convergent expansion on y ϵ [0, ∞ ]. Finally, we successfully generalize the WKB method to obtain, for the J 0 Bessel function, an amplitude-phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y, even the origin.