General connection formulae for Liouville-Green approximations in the complex plane

Abstract

This paper is concerned with differential equations of the form d 2 w/dz 2 = {u 2 f(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-m f(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) = 0{(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 Rc********** where c again denotes a transition point.