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The Quarterly Journal of Mechanics and Applied Mathematics 1979 32(2):187-204; doi:10.1093/qjmam/32.2.187
© 1979 by Oxford University Press
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UNIFORM APPROXIMATIONS TO SOLUTIONS OF A LINEAR SECOND-ORDER DIFFERENTIAL EQUATION OUTSIDE A VANISHINGLY SMALL REGION CONTAINING TWO ASYMPTOTICALLY COINCIDENT TRANSITION POINTS

P. BALDWIN

( Department of Engineering Mathematics, University of Newcastle upon Tyne )

Approximate solutions of the differential equation


where v is a complex constant and {alpha}({lambda}) = o({lambda}–2/3) as {lambda} -> {infty}, are obtained for large |{lambda}|. The solutions are uniformly valid for any arg z and arg {lambda}, outside a vanishingly small neighbourhood of the origin, which contains the singularity {alpha}({lambda}). Error estimates for both the solution and its derivative are given.

The comparison-equation method is used, the comparison equation being found by setting {alpha} = 0 in the given equation. A new feature in the application of this method is a representation of the modified Bessel functions, closely allied to their asymptotic expansions, which allows bounds to be placed on the functions for any order and argument.


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