Editing Asymptotic analysis (section)

==Asymptotic expansion==
{{main|Asymptotic expansion}}
An [[asymptotic expansion]] of a function {{math|''f''(''x'')}} is in practice an expression of that function in terms of a [[series (mathematics)|series]], the [[partial sum]]s of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for {{mvar|f}}. The idea is that successive terms provide an increasingly accurate description of the order of growth of {{mvar|f}}.

In symbols, it means we have <math>f \sim g_1,</math> but also <math>f - g_1 \sim g_2</math> and <math>f - g_1 - \cdots - g_{k-1} \sim g_{k}</math> for each fixed ''k''. In view of the definition of the <math>\sim</math> symbol, the last equation means <math>f - (g_1 + \cdots + g_k) = o(g_k)</math> in the [[Big O notation#Little-o notation|little o notation]], i.e., <math>f - (g_1 + \cdots + g_k)</math> is much smaller than <math>g_k.</math>

The relation <math>f - g_1 - \cdots - g_{k-1} \sim g_{k}</math> takes its full meaning if <math>g_{k+1} = o(g_k)</math> for all ''k'', which means the <math>g_k</math> form an [[asymptotic scale]]. In that case, some authors may [[Abuse of notation|abusively]] write <math>f \sim g_1 + \cdots + g_k</math> to denote the statement <math>f - (g_1 + \cdots + g_k) = o(g_k).</math> One should however be careful that this is not a standard use of the <math>\sim</math> symbol, and that it does not correspond to the definition given in {{section link||Definition}}.

In the present situation, this relation <math>g_{k} = o(g_{k-1})</math> actually follows from combining steps ''k'' and ''k''−1; by subtracting <math>f - g_1 - \cdots - g_{k-2} = g_{k-1} + o(g_{k-1})</math> from <math>f - g_1 - \cdots - g_{k-2} - g_{k-1} = g_{k} + o(g_{k}),</math> one gets <math>g_{k} + o(g_{k})=o(g_{k-1}),</math> i.e. <math>g_{k} = o(g_{k-1}).</math>

In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. This optimal partial sum will usually have more terms as the argument approaches the limit value.

===Examples of asymptotic expansions===

* [[Gamma function]] <math display="block">\frac{e^x}{x^x \sqrt{2\pi x}} \Gamma(x+1) \sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots
 \ (x \to \infty)</math>
* [[Exponential integral]] <math display="block">xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \to \infty) </math>
* [[Error function]] <math display="block"> \sqrt{\pi}x e^{x^2}\operatorname{erfc}(x) \sim 1+\sum_{n=1}^\infty (-1)^n \frac{(2n-1)!!}{n!(2x^2)^n} \ (x \to \infty)</math> where {{math|''m''!!}} is the [[double factorial]].

===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.