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The Quarterly Journal of Mechanics and Applied Mathematics 1964 17(1):105-124; doi:10.1093/qjmam/17.1.105
© 1964 by Oxford University Press
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ON THE RELATED-EQUATION METHOD OF ASYMPTOTIC APPROXIMATION (W.K.B. OR A-A METHOD)

I. A PROPOSED NEW EXISTENCE THEOREM

E. R. PIKE

( Royal Radar Establishment, Malvern Worcs )

A review is made of developments in the theory of the asymptotic solution of ordinary, linear, second-order differential equations with respect to a large parameter. It is shown that the theory is still not in a form where it can be applied rigorously to some problems in physics, and a new existence theorem which would overcome this difficulty is conjectured. This concerns the asymptotic equivalence of solutions of equations having the same turning points (and possibly singularities) arbitrary in number, in a given region of the real axis.

In a second part of the paper typical physical applications of the theory in its modern form will be made. Earlier ‘matching’ techniques are bypassed, although for problems with two or more turning points rigorous results are not possible without the existence theorem proposed in the first part.


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