Abstract
This paper concerns asymptotic approximations and expansions in cases where the usual Poincaré power series in 1 / x 1/x do not suffice because there may be more than one comparability class of functions that are very large or very small. The attempt to find asymptotic approximations in terms of real powers of given representatives of the comparability classes fails in general, but the situation can be saved by the adjunction of suitable real power products of the original functions, at the possible cost of an increase in the number of comparability classes.
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