Formal expansion - Knowlpedia

A formal expansion $\sum_{j\ge0}\varepsilon^j a_j$ specifies a sequence of coefficients and an indeterminate $\varepsilon$ . Addition is coefficientwise; when coefficients can be multiplied, multiplication uses

(ab)_n=\sum_{j=0}^n a_jb_{n-j}.

Each output coefficient is a finite sum. No numerical value of $\varepsilon$ , convergence, or underlying function is asserted.

Equations order by order

Substitution into a polynomial equation and comparison of coefficients can recursively determine the $a_j$ . A finite differential polynomial admits the same procedure with coefficient functions and their derivatives. Producing a function with this formal expansion requires an additional realization or remainder argument; an asymptotic expansion states that analytic conclusion.