Editing Closed-form expression (section)

== Dealing with non-closed-form expressions ==

=== Transformation into closed-form expressions ===

The expression:
<math display="block">f(x) = \sum_{n=0}^\infty \frac{x}{2^n}</math>
is not in closed form because the summation entails an infinite number of elementary operations. However, by summing a [[geometric series]] this expression can be expressed in the closed form:<ref>{{cite web | last=Holton | first=Glyn | title = Numerical Solution, Closed-Form Solution | url = http://www.riskglossary.com/link/closed_form_solution.htm |website=riskglossary.com | access-date = 31 December 2012 |url-status = dead | archive-url = https://web.archive.org/web/20120204082706/http://www.riskglossary.com/link/closed_form_solution.htm |archive-date = 4 February 2012 }}</ref>
<math display="block">f(x) = 2x.</math>

=== Differential Galois theory ===
{{main|Differential Galois theory}}
{{See also|Nonelementary integral}}

The integral of a closed-form expression may or may not itself be expressible as a closed-form expression. This study is referred to as [[differential Galois theory]], by analogy with algebraic Galois theory.

The basic theorem of differential Galois theory is due to [[Joseph Liouville]] in the 1830s and 1840s and hence referred to as '''[[Liouville's theorem (differential algebra)|Liouville's theorem]]'''.

A standard example of an elementary function whose antiderivative does not have a closed-form expression is: <math display="block">e^{-x^2},</math> whose one antiderivative is ([[up to]] a multiplicative constant) the [[error function]]:
<math display="block">\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^x e^{-t^2} \, dt.</math>

=== Mathematical modelling and computer simulation ===

Equations or systems too complex for closed-form or analytic solutions can often be analysed by [[mathematical model]]ling and [[computer simulation]] (for an example in physics, see<ref>{{Cite journal |last=Barsan |first=Victor |date=2018 |title=Siewert solutions of transcendental equations, generalized Lambert functions and physical applications |publisher=De Gruyter |doi=10.1515/phys-2018-0034 |doi-access=free |journal=Open Physics|volume=16 |issue=1 |pages=232–242 |bibcode=2018OPhy...16...34B |arxiv=1703.10052 }}</ref>).