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In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a ''convergent'' Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a ''non-convergent'' series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as superasymptotics. The error is then typically of the form where ε is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations. See asymptotic analysis, big O notation, and little o notation for the notation used in this article. ==Formal Definition== First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion. If φ''n'' is a sequence of continuous functions on some domain, and if ''L'' is a limit point of the domain, then the sequence constitutes an asymptotic scale if for every ''n'', . (''L'' may be taken to be infinity.) In other words, a sequence of functions is an asymptotic scale if each function in the sequence grows strictly slower (in the limit ) than the preceding function. If ''f'' is a continuous function on the domain of the asymptotic scale, then ''f'' has an asymptotic expansion of order ''N'' with respect to the scale as a formal series if : or : If one or the other holds for all ''N'', then we write : In contrast to a convergent series for , wherein the series converges for any ''fixed'' in the limit , one can think of the asymptotic series as converging for ''fixed'' in the limit (with possibly infinite). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「asymptotic expansion」の詳細全文を読む スポンサード リンク
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