Closed-form expression
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In mathematics, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed analytically in terms of a bounded number of certain "well-known" functions . Typically, these well-known functions are defined to be elementary functions; so infinite series, limits, and continued fractions are not permitted.
For example, the roots of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, all elementary functions. However, there are quintic equations without closed-form solutions using elementary functions.
Changing the definition of "well-known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well-known. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well-known, since numerical implementations are widely available.
Because most practical cases do not have analytical solution while have numerical one, for practical purposes analyticity is not important.
