Knowlpedia

Definition

Asymptotic scale

An ordered sequence whose successive functions are negligible relative to their predecessors.

An asymptotic scale at ε=0+\varepsilon=0^+ is a sequence of eventually nonzero scalar functions (ϕj)j≥0(\phi_j)_{j\ge0} such that

ϕj+1=o(∣ϕj∣)(ε↓0)\phi_{j+1}=o(|\phi_j|)\qquad(\varepsilon\downarrow0)

for every fixed jj. Thus the order of the sequence records successively smaller contributions in the sense of .

Examples

The powers 1,ε,ε2,…1,\varepsilon,\varepsilon^2,\ldots form a scale. So do εaj\varepsilon^{a_j} for any strictly increasing sequence of real exponents. Scales may also contain logarithms and need not consist of integral powers.

References