Sufficiency Condition for the Validity of the WKB Approximation

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Abstract
In spite of the many applications—in theoretical physics and elsewhere—of this well-known method of approximating to solutions of the differential equation W″ + fW = 0, when f is slowly varying through a region where f ≠ 0, no simple sufficient condition for its validity appears yet to have been given. In the present paper, such a condition is derived. Also, the connection formulas establishing the relation between the constants in the WKB approximation to a given solution in the various regions of the complex plane delineated by the ``Stokes lines,'' which converge on a simple zero of f, are rederived, as existing arguments are open to criticism on at least two major grounds. Finally, a sufficient condition is given for the existence of a common region of validity for the approximation based on the series solutions of the differential equation around a zero of f and the WKB approximation valid sufficiently far from this zero.

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