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Keywords:
complex Liouville-Green; WKB; asymptotic approximations
complex Liouville-Green; WKB; asymptotic approximations
Summary:
We propose a variant of the classical Liouville-Green approximation theorem for linear complex differential equations of the second order. We obtain rigorous error bounds for the asymptotics at infinity, in the spirit of F. W. J. Olver’s formulation, by using rather arbitrary $\xi $-progressive paths. This approach can provide higher flexibility in practical applications of the method.
References:
[1] Olver F. W. J.: Asymptotics and Special Functions. Academic Press, New York, 1974; reprinted by A. K. Peters, Wellesley, MA, 1997. MR 0435697 | Zbl 0308.41023
[2] Spigler R., Vianello M.: A numerical method for evaluating zeros of solutions of second-order linear differential equations. Math. Comp. 55 (1990), 591–612. MR 1035945 | Zbl 0676.65041
[3] Spigler R., Vianello M.: On the complex differential equation $Y^{\prime \prime }+G(z)Y=0$ in Banach algebras. Stud. Appl. Math. 102 (1999), 291–308. MR 1669480 | Zbl 1001.34051
[4] Thorne R. C.: Asymptotic formulae for solutions of second-order differential equations with a large parameter. J. Austral. Math. Soc. 1 (1960), 439–464. MR 0123766
[5] Vianello M.: Extensions and numerical applications of the Liouville-Green approximation. Ph. D. Thesis (in Italian), University of Padova, 1992 (advisor: R. Spigler).
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