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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. (English) Zbl 0798.34083
The paper deals with second-order linear differential equations in some complex domain of the form d 2 W/dξ 2 =(u 2 +ψ(ξ))W where u is a large complex parameter and ψ is an analytic function. Under certain mild assumption on ψ, the authors are able to obtain convergent Liouville-Green expansions for the solutions in terms of inverse factorials using Laplace transforms and the integral equation method or, alternatively, the inverse Laplace transform. These expansions converge for (u)>0, uniformly for ξ in a certain subdomain of the domain of asymptotic validity. Finally, the general result is used to derive convergent Liouville-Green expansions for the modified Bessel functions I ν (νz) and K ν (νz) of large order ν.

MSC:
34L10Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE)
33C10Bessel and Airy functions, cylinder functions, 0 F 1