If φn is a sequence of continuous functions on some domain, and if L is a (possibly infinite) limit point of the domain, then the sequence constitutes an asymptotic scale if for every n, . 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
The most common type of asymptotic expansion is a power series in either positive or negative terms. 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.
(x rightarrow infty)
The expression on the left is valid on the entire complex plane , while the right hand side converges only for . Multiplying by and integrating both sides yields
The integral on the left hand side can be expressed in terms of the exponential integral. The integral on the right hand side, after the substitution , may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion
Here, the right hand side is clearly not convergent for any non-zero value of t. However, by truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of for sufficiently small t. Substituting and noting that results in the asymptotic expansion given earlier in this article.
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