The problem of differentiating an asymptotic expansion in real powers. I: Unsatisfactory or partial results by classical approaches (Q624255)

The paper deals with the differentiation of real asymptotic expansions \(f(x)=\sum^n_{i=1} a_i x^{d_i}+O(x^\gamma)\) as \( x\to\infty\), and also of corresponding expansions with \(o\) instead of \(O\), under additional assumptions concerning \(f'\), \(f''\) or a second order differential operator of Euler type. Moreover, the possibility of higher differentiations is investigated. Examples show that some of the results are best possible.