An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum.

Due to the lack of specificity in the above definition, different branches of mathematics often adopt more precise meanings of the term closed-form to apply to concepts therein.
For example, within differential algebra,
a function is said to be of closed-form if it is contained in some so-called Liouvillian
extension field of a field , i.e., if they are obtained from
rational functions by a finitesequence of adjunctions
of exponentials, indefinite
integrals, and algebraic functions (Churchill
and Kovacic 2006). These functions are also referred to as Liouvillian (though not
to be confused with the Liouville function),
as well as by the even more unfortunate term "elementary".

It is worth noting that the adjective "closed" is used to describe a number of mathematical notions, e.g., the notion of closed form.
Loosely speaking, a discrete function is of closed form if it shares certain essential
properties with the hypergeometric function,
a function which itself is defined to be the solution of the so-called hypergeometric
differential equation. This particular notion of closedness is completely separate
from the notion of closed-form expression as discussed above. In particular, the
hypergeometric function (and hence, any closed-form function inheriting its properties)
is considered a "special function"
and is not expressible in terms of operations which are typically viewed as "elementary."
What's more, certain agreed-upon truths like the insolvability of the quintic
fail to be true if one extends consideration to a class of functions which includes
the hypergeometric function, a result due to Klein (1877).

Baxa, C. "Diophantine Representation of the Decimal Expansions of
and ."
Math. Slovaca50, 531-539, 2000.Chow, T. Y. "What
is a Closed-Form Number?" Amer. Math. Monthly106, 440-448, 1999.Churchill,
R. C. and Kovacic, J. J. "Introduction to Differential Galois Theory."
2006. http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/intro.pdf.Klein,
C. F. "Weitere Untersuchungen Über Das Ikosaeder," Mathematische
Annalen12, 503-560, 1877.Trott, M. The
Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 105,
2004. http://www.mathematicaguidebooks.org/.Wolfram,
S. "Notes: Exact Solutions." A
New Kind of Science. Champaign, IL: Wolfram Media, p. 1133,
2002.