The traditional version of the JWKB connection formula at a linear turning point is incorrect. The correct version follows from the Borel-based summability of the Airy-function asymptotic expansions. A key point is that the classically forbidden region lies on a Stokes line. The real exponentially growing solution has an explicitly complex JWKB expansion, while the explicitly real exponentially growing JWKB expansion represents a complex solution. Inconsistencies in applications of the traditional formula, such as in transmission through a barrier, are eliminated.
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Harris J. Silverstone (1985) studied this question.
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