Editing Asymptotic analysis (section)

===Worked example===

Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the ordinary series
<math display="block">\frac{1}{1-w}=\sum_{n=0}^\infty w^n</math>

The expression on the left is valid on the entire complex plane <math>w \ne 1</math>, while the right hand side converges only for <math>|w|< 1</math>. Multiplying by <math>e^{-w/t}</math> and integrating both sides yields
<math display="block"> \int_0^\infty \frac{e^{-\frac{w}{t}}}{1 - w} \, dw = \sum_{n=0}^\infty t^{n+1} \int_0^\infty e^{-u} u^n \, du</math>

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 <math>u=w/t</math>, may be recognized as the [[gamma function]]. Evaluating both, one obtains the asymptotic expansion
<math display="block">e^{-\frac{1}{t}} \operatorname{Ei}\left(\frac{1}{t}\right) = \sum _{n=0}^\infty n! \; t^{n+1} </math>

Here, the right hand side is clearly not convergent for any non-zero value of ''t''. However, by keeping ''t'' small, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of <math>\operatorname{Ei}(1/t)</math>. Substituting <math>x = -1/t</math> and noting that <math>\operatorname{Ei}(x) = -E_1(-x)</math> results in the asymptotic expansion given earlier in this article.