Asymptotic Expansion

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: \begin{gather}

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}

- An asymptotic expansion doesn't uniquely determine the function ## Complex expansions Typically, $z\in \mathbb{C}$ must be restricted to a sector of $\mathbb{C}$ around $z_{0}$. Approaching $z_{0}$ from different directions can give different approximations (even for analytic functions). This is called Stokes phenomenon We divide $\mathbb{C}$ into special sectors in which different [[Asymptotic Methods/Asymptotic Approximation\|asymptotic approximations]] hold. Their boundaries are called Stokes lines (sometimes they are called anti Stokes lines).