Editing Asymptotic expansion (section)

==Properties==
===Uniqueness for a given asymptotic scale===
For a given asymptotic scale <math>\{\varphi_n(x)\}</math> the asymptotic expansion of function <math>f(x)</math> is unique.<ref name="Malham">S.J.A. Malham, "[http://www.macs.hw.ac.uk/~simonm/ae.pdf An introduction to asymptotic analysis]", [[Heriot-Watt University]].</ref> That is the coefficients <math>\{a_n\}</math> are uniquely determined in the following way:
<math display="block">\begin{align}
a_0 &= \lim_{x \to L} \frac{f(x)}{\varphi_0(x)} \\
a_1 &= \lim_{x \to L} \frac{f(x) - a_0 \varphi_0(x)} {\varphi_1(x)} \\
& \;\;\vdots \\
a_N &= \lim_{x \to L} \frac {f(x) - \sum_{n=0}^{N-1} a_n \varphi_n(x)} {\varphi_N(x)}
\end{align}</math>
where <math>L</math> is the limit point of this asymptotic expansion (may be <math>\pm \infty</math>).

===Non-uniqueness for a given function===
A given function <math>f(x)</math> may have many asymptotic expansions (each with a different asymptotic scale).<ref name="Malham"></ref>

===Subdominance===
An asymptotic expansion may be an asymptotic expansion to more than one function.<ref name="Malham"></ref>