Editing Asymptotic expansion (section)

==Formal definition==

First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion.

If <math>\ \varphi_n\ </math> is a sequence of [[continuous function]]s on some domain, and if <math>\ L\ </math> is a [[limit point]] of the domain, then the sequence constitutes an '''asymptotic scale''' if for every {{mvar|n}},

:<math>\varphi_{n+1}(x) = o(\varphi_n(x)) \quad (x \to L)\ .</math> 

(<math>\ L\ </math> 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 <math>\ x \to L\ </math>) than the preceding function.

If <math>\ f\ </math> is a continuous function on the domain of the asymptotic scale, then {{mvar|f}} has an asymptotic expansion of order <math>\ N\ </math> with respect to the scale as a formal series 

:<math> \sum_{n=0}^N a_n \varphi_{n}(x) </math> 

if

:<math> f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = O(\varphi_{N}(x)) \quad (x \to L) </math>

or the weaker condition

:<math> f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = o(\varphi_{N-1}(x)) \quad (x \to L)\ </math>

is satisfied. Here, <math>o</math> is the [[little o]] notation. If one or the other holds for all <math>\ N\ </math>, then we write{{cn|date=November 2017}}

:<math> f(x) \sim \sum_{n=0}^\infty a_n \varphi_n(x) \quad (x \to L)\ .</math>

In contrast to a convergent series for <math>\ f\ </math>, wherein the series converges for any ''fixed'' <math>\ x\ </math> in the limit <math>N \to \infty</math>, one can think of the asymptotic series as converging for ''fixed'' <math>\ N\ </math> in the limit <math>\ x \to L\ </math> (with <math>\ L\ </math> possibly infinite).