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I obtained a series expansions as this type

$$f(x)=g(x)^{\textstyle \sum_{i=0}^{n}\alpha_{i}x^{-i}+O\left(\tfrac{1}{x^{n+1}}\right)}$$

what is the exact name of this formula

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It's a pretty ordinary asymptotic expansion for $$\frac{\log f(x)}{\log g(x)}.$$

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Except the expansion is in terms of 1/x. Perhaps it has a special name still? Gerhard "Maybe It's A Rolyat Series" Paseman, 2012.06.15 – Gerhard Paseman Jun 15 '12 at 18:08
@Gerhard: An asymptotic expansion of $h(x)$ as $x\to\infty$ would normally involve negative powers of $x$, so I don't know why a special name should exist. – Brendan McKay Jun 16 '12 at 3:27

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