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General Connection Formulae for Liouville-Green Approximations in the Complex Plane

F. W. J. Olver
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 289, No. 1364 (Jul. 27, 1978), pp. 501-548
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/75029
Page Count: 48
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General Connection Formulae for Liouville-Green Approximations in the Complex Plane
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Abstract

This paper is concerned with differential equations of the form d2w/dz2 = {u2f(u, z) + g(u, z)}w, in which u is a positive parameter and z is a complex variable ranging over a simply connected open domain D that is not necessarily one-sheeted, and may be bounded or unbounded. In the first part we assume that for each value of u the function (z-c)2-mf(u, z) is holomorphic and non-vanishing throughout D, where c is an interior point of D and m is a positive constant. It is also assumed that g(u,z) is holomorphic in D, punctured at c, and g(u,z) = O{(z-c)γ -1} as z→ c, where γ is another positive constant. Thus c is a fractional transition point of the differential equation of multiplicity (or order) m-2, and there are no other transition points in D. Uniform asymptotic approximations for the solutions, when u is large, are constructed in terms of Bessel functions of order 1/m, complete with error bounds. In the second part the Bessel function approximants are replaced by their uniform asymptotic approximations for large argument, yielding the connection formulae for the Liouville-Green (or J.W.K.B.) approximations to the solutions, again complete with error bounds. These results are then applied to solve the general problem of connecting the Liouville-Green approximations when D contains any (finite) number of transition points of arbitrary multiplicities, integral or fractional. The third, and concluding, part illustrates the theory by means of three examples. An appendix describes a numerical method for the automatic computation and plotting of the boundary curves of the Liouville-Green approximations, defined by Re ∫c zf1/2(u, t) dt = 0, where c again denotes a transition point.

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