In practical cases the assumption of small values by certain parameters may invalidate the usual method of integration by steepest descents when it would be otherwise applicable. The two ways in which this can happen are treated by using the concept of a partial asymptotic expansion, the terms of which are functions, rather than inverse powers, of the relevant variable. It is shown that the theory of Watson's lemma, and a device due to Jeffreys for estimating the ‘remainder’, can be extended to these partial expansions. The general terms are expressed in the form of Hhn or confluent hypergeometric functions, and the leading term may reduce to a complex Fresnel integral. Two examples are given from the theory of wave propagation.

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