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Asymptotic expansions for the coefficient functions that arise in turning-point problems. (English) Zbl 0614.33011

Solutions of differential equations of the form w '' (ζ)=[u 2 ζ+Ψ(ζ)]w(ζ), where u is a large parameter and Ψ is holomorphic in a simply connected domain Δ which includes the point ζ=0, can be expanded in terms of Airy functions. The point ζ=0, is called the turning point. There is a well-developed theory, due to Olver, concerning uniform expansions; that is, expansions that hold uniformly with respect to ζ near the turning point. This expansion is not only valid near ζ=0 but, depending on Ψ, the ζ-domain may be rather large. Olver provided bounds on the remainders in the expansion. The present paper gives a different approach to treat the asymptotic problem, although the expansion in the same as Olver’s. The author considers ”slowly varying” functions A(u,ζ), B(u,ζ) such that

Ai(u 2/3 ζ)A(u,ζ)+u -4/3 Ai ' (u 2/3 ζ)B(u,ζ)

is an exact solution of the above differential equation, with Ai(z) a solution of Airy’s differential equation y '' (z)=zy(z). The author gives a detailed analysis on the properties of the functions A(u,ζ), B(u,ζ); these functions are called the coefficient functions, since they provide the coefficients in the asymptotic expansion. Attention is paid to the construction of error bounds, and to a comparing the new results with Olver’s approach. There is an application of the new method to Bessel functions.

Reviewer: N.M.Temme
MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
34E05Asymptotic expansions (ODE)
30E10Approximation in the complex domain
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)