Editing Closed-form expression (section)

==Comparison of different classes of expressions ==

The closed-form expressions do not include [[infinite series]] or [[continued fraction]]s; neither includes [[integral]]s or [[limit of a sequence|limits]]. Indeed, by the [[Stone–Weierstrass theorem]], any [[continuous function]] on the [[unit interval]] can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.

Similarly, an [[equation]] or [[system of equations]] is said to have a '''closed-form solution''' [[if and only if]] at least one [[equation solving|solution]] can be expressed as a closed-form expression; and it is said to have an '''analytic solution''' if and only if at least one solution can be expressed as an analytic expression. There is a subtle distinction between a "closed-form ''function''" and a "[[#Closed-form number|closed-form ''number'']]" in the discussion of a "closed-form solution", discussed in {{Harv|Chow|1999}} and [[#Closed-form number|below]]. A closed-form or analytic solution is sometimes referred to as an '''explicit solution'''.

{{Mathematical expressions}}