A complete asymptotic expansion of the Jacobi functions with error bounds
Jacobi functions are in fact Gaussian hypergeometric functions of the form \({}_ 2F_ 1((\alpha +\beta +1-i\lambda),(\alpha +\beta +1+i\lambda);\alpha +1;-\sinh^ 2t)\). The asymptotic expansion of this function is given as \(\lambda\to \infty\), uniformly with respect to \(t\in [0,T]\), where T is a fixed positive parameter. Especially near the origin a Bessel function is needed as basic approximant. The complete expansion is given; for the remainder a strict bound is given together with an order estimate. This shows the asymptotic nature of the expansion. The error bound is obtained by using the Gronwall-Bellman inequality, which is applied to an integral equation for the remainder.
scientific article; zbMATH DE number 4025843