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Mathematics · Glossary

What is Asymptotic expansion?

Definition 6.2 University Mathematics — Year 2 · Chapter 6 — Comparison of Functions

ff admits the asymptotic expansion

f=c1φ1+c2φ2+⋯+ckφk+o(φk)(φi+1=o(φi) in the scale)f = c_1 \varphi_1 + c_2\varphi_2 + \dots + c_k \varphi_k + o(\varphi_k) \qquad (\varphi_{i+1} = o(\varphi_i) \text{ in the scale})

when the successive remainders satisfy the displayed estimates. The coefficients are then unique: c1=lim⁡f/φ1c_1 = \lim f/\varphi_1, and inductively ci+1=lim⁡ (f−∑j≤icjφj)/φi+1c_{i+1} = \lim\,(f - \sum_{j \leq i} c_j\varphi_j)/\varphi_{i+1}.

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