From a complicated function ,
we want to find a simpler s.t. .
Want to break up into a hierarchy of approximations s.t.
where are very simple and relative sizes of are known.
Definition
Suppose we have an Asymptotic Approximation which holds for all .
Then we have an asymptotic series and we write:
We make NO CLAIMS about this converging !!!!
We never fix and let .
Caution
For functions that have infinitely many zeros as the limit is taken,
this definition has problems, because we are dividing by 0.
However, we will still be approximating with them (???).
Example:
Properties
Let and
Linearity:
Multiplication: Consider
Then where
Division is possible but we don’t really use it
Integrating term by term is usually ok
Differentiating term by term is usually NOT ok
If and are different asymptotic sequences as
and suppose they both approximate ,
then the coefficients are not the same
are unique, moreover:
a_{0}=\lim_{ x \to x_{0} } \frac{f(x)}{\phi_{0}(x)}
a_{n}=\lim_{ x \to x_{0} } \left( f(x)-\sum_{k=0}^{n-1} a_{k}\phi_{k}\over\phi_{n}(x) \right)
\end{gather}