Editing Closed-form expression (section)

{{Redirect|Closed formula|"closed formula" in the sense of a logic formula with no free variables|Sentence (mathematical logic)}}
{{Use American English|date = January 2019}}
{{Short description|Mathematical formula involving a given set of operations}}
{{more citations needed|date=June 2014}}

In [[mathematics]], an [[mathematical expression|expression]] or [[equation]] is in '''closed form''' if it is formed with [[Constant (mathematics)|constants]], [[Variable (mathematics)|variables]], and a [[set (mathematics)|set]] of [[function (mathematics)|functions]] considered as ''basic'' and connected by arithmetic operations ({{itco|{{math|+, −, ×, /}}}}, and [[exponentiation#Integer|integer powers]]) and [[function composition]]. Commonly, the basic functions that are allowed in closed forms are [[Nth root|''n''th root]], [[exponential function]], [[logarithm]], and [[trigonometric functions]].{{efn|[[Hyperbolic functions]], [[inverse trigonometric functions]] and [[inverse hyperbolic functions]] are also allowed, since they can be expressed in terms of the preceding ones.}} However, the set of basic functions depends on the context. For example, if one adds [[polynomial root]]s to the basic functions, the functions that have a closed form are called [[elementary function]]s.

The ''closed-form problem'' arises when new ways are introduced for specifying [[mathematical objects]], such as [[limit (mathematics)|limit]]s, [[series (mathematics)|series]], and [[integral]]s: given an object specified with such tools, a natural problem is to find, if possible, a ''closed-form expression'' of this object; that is, an expression of this object in terms of previous ways of specifying it.