Asymptotic Expansion

1 Asymptotic Expansion

An asymptotic expansion (Poincaré expansion) is a formal series of functions such that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular (often infinite) point.

The most common type of asymptotic expansion is a Power Series in either positive or negative powers. Methods of generating expansions include the Euler-Maclaurin Summation Formula, and integral transforms such as the Laplace Transform or the Mellin Transform. Additionally, repeated integration by parts will often lead to an asymptotic expansion.

1.1 Asymptotic Expansion

Let \(\varphi_{n}\) is a sequence of continuous functions on some domain and let \(L\) be a limit point of the domain. Then the sequence constitutes an asymptotic scale if for every \(n\) \[ \varphi_{n+1}(x)=o(\varphi_{n}(x));\quad x \to L, \] where \(L\) can be taken to be \(\infty\). In other words, each function in the sequence grows strictly slower (as \(x \to L\)) than the preceding function.