Knowlpedia

Definition

Asymptotic expansion

Approximation to every finite order in an asymptotic scale, without a convergence requirement.

Given an (ϕj)(\phi_j), the statement

f(ε)∼∑j≥0ajϕj(ε)f(\varepsilon)\sim\sum_{j\ge0}a_j\phi_j(\varepsilon)

means that for every fixed N≥0N\ge0,

f(ε)−∑j=0Najϕj(ε)=o(∣ϕN(ε)∣).f(\varepsilon)-\sum_{j=0}^N a_j\phi_j(\varepsilon) =o(|\phi_N(\varepsilon)|).

The sum in the defining estimate is finite. The infinite expression records all these estimates; it need not converge for any fixed ε\varepsilon. For vector-valued coefficients the remainder is measured in a specified norm or family of seminorms.

Recovering coefficients

The scale makes the coefficients unique: subtracting two expansions, the first unequal coefficient would give a nonzero limit after division by its scale function. A becomes an asymptotic expansion only after an actual function and the remainder bounds have been supplied.

References